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This patch fixes a typo in the src/hyper/Makefile.
Apparently only part of the changes made the last commit.

Tim

=======================================================================
diff --git a/changelog b/changelog
index 51a4abc..daecc5d 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080102 tpd src/hyper/Makefile.pamphlet fix typo for axbook
 20071230 tpd src/hyper/Makefile prevent spurious remake of axbook (7052)
 20071230 tpd src/input/summation.input update tests with new mathml output
 20071229 acr src/algebra/mathml.spad fix sum ambiguity bug in mathml (7057)
diff --git a/src/hyper/Makefile.pamphlet b/src/hyper/Makefile.pamphlet
index 95d03f3..646757a 100644
--- a/src/hyper/Makefile.pamphlet
+++ b/src/hyper/Makefile.pamphlet
@@ -1133,7 +1133,7 @@ ${HYPER}/axbook: ${IN}/axbook.tgz
 <<environment>>
 all: ${INCLUDES} ${BITMAPS} ${OBJS} ${SCRIPTS} ${BINFILES} \
      ${HYPER}/pages/ht.db ${DOCFILES} ${HYPER}/rootpage.xhtml \
-     ${HYPER}/axbook/book-index.xhtml ${MNT}/${SYS}/doc/bookvol11.dvi \
+     ${HYPER}/axbook ${MNT}/${SYS}/doc/bookvol11.dvi \
      ${HYPER}/bigbayou.png ${HYPER}/doctitle.png
 	@ echo 155 finished ${IN}
 



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Handle besselK differently. Apply Waldek's patches. Fix 7090/355.

(1) -> D(besselK(a,x),x)

        - besselK(a + 1,x) - besselK(a - 1,x)
   (1)  -------------------------------------
                          2
                                                     Type: Expression Integer
(2) -> D(besselK(a,x),a)

   (2)  besselK  (a,x)
               ,1
                                                     Type: Expression Integer
(3) -> integrate(D(besselK(a,x),a),a)

   (3)  besselK(a,x)
                                          Type: Union(Expression Integer,...)
(4) -> limit(D(besselK(a,x),a),a=1/2)

   (4)  "failed"
                                                    Type: Union("failed",...)
=======================================================================
diff --git a/changelog b/changelog
index daecc5d..8acaf43 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080103 wxh src/algebra/sf.spad handle besselK (7090/355)
+20080103 wxh src/algebra/op.spad handle besselK (7090/355)
+20080103 wxh src/algebra/combfunc.spad handle besselK (7090/355)	
 20080102 tpd src/hyper/Makefile.pamphlet fix typo for axbook
 20071230 tpd src/hyper/Makefile prevent spurious remake of axbook (7052)
 20071230 tpd src/input/summation.input update tests with new mathml output
diff --git a/src/algebra/combfunc.spad.pamphlet b/src/algebra/combfunc.spad.pamphlet
index c7d54b7..9da05e9 100644
--- a/src/algebra/combfunc.spad.pamphlet
+++ b/src/algebra/combfunc.spad.pamphlet
@@ -83,6 +83,53 @@ CombinatorialFunction(R, F): Exports == Implementation where
     binomial   : (F, F) -> F
       ++ binomial(n, r) returns the number of subsets of r objects
       ++ taken among n objects, i.e. n!/(r! * (n-r)!);
+@
+
+We currently simplify binomial coefficients only for non-negative integral
+second argument, using the formula
+$$ \binom{n}{k}=\frac{1}{k!}\prod_{i=0..k-1} (n-i),$$
+except if the second argument is symbolic: in this case [[binomial(n,n)]] is
+simplified to one.
+ 
+Note that there are at least two different ways to define binomial coefficients
+for negative integral second argument. One way, particular suitable for
+combinatorics, is to set the binomial coefficient equal to zero for negative
+second argument. This is, partially, also the approach taken in
+[[combinat.spad]], where we find
+
+\begin{verbatim}
+   binomial(n, m) ==
+      n < 0 or m < 0 or m > n => 0
+      m = 0 => 1
+\end{verbatim}
+
+Of course, here [[n]] and [[m]] are integers. This definition agrees with the
+recurrence
+
+$$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}.$$
+
+Alternatively, one can use the formula
+$$ \binom{n}{k}=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}, $$
+and leave the case where $k\in\mathbb Z$, $n\in\mathbb Z$ and $k \leq n < 0$
+undefined, since the limit does not exist in this case:
+
+Since we then have that $n-k+1\geq 1$, $\Gamma(n-k+1)$ is finite. So it is
+sufficient to consider $\frac{\Gamma(n+1)}{\Gamma(k+1)}$. On the one hand, we
+have
+$$\lim_{n_0\to n} \lim_{k_0\to k}\frac{\Gamma(n_0+1)}{\Gamma(k_0+1)} = 0,$$
+since for any non-integral $n_0$, $\Gamma(n_0+1)$ is finite. On the other
+hand,
+$$\lim_{k_0\to k} \lim_{n_0\to n}\frac{\Gamma(n_0+1)}{\Gamma(k_0+1)}$$
+does not exist, since for non-integral $k_0$, $\Gamma(k_0+1)$ is finite while
+$\Gamma(n_0+1)$ is unbounded.
+
+However, since for $k\in\mathbb Z$, $n\in\mathbb Z$ and $0 < k < n$ both
+definitions agree, one could also combine them. This is what, for example,
+Mathematica does. It seems that MuPAD sets [[binomial(n,n)=1]] for all
+arguments [[n]], and returns [[binomial(-2, n)]] unevaluated. Provisos may help
+here.
+
+<<package COMBF CombinatorialFunction>>=
     permutation: (F, F) -> F
       ++ permutation(n, r) returns the number of permutations of
       ++ n objects taken r at a time, i.e. n!/(n-r)!;
@@ -517,17 +564,33 @@ dummy variable is introduced to make the indexing variable \lq local\rq.
 
       if R has RetractableTo(Z) and F has Algebra(Fraction(Z)) then
          iibinom l ==
+           (s:=retractIfCan(second l)@Union(R,"failed")) case R and
+              (t:=retractIfCan(s)@Union(Z,"failed")) case Z and t>0 =>
+                ans:=1::F
+                for i in 0..t-1 repeat
+                    ans:=ans*(first l - i::R::F)
+                (1/factorial t) * ans
            (s:=retractIfCan(first l-second l)@Union(R,"failed")) case R and
-             (t:=retractIfCan(s)@Union(Z,"failed")) case Z and s>0=>
-              ans:=1::F
-              for i in 1..t repeat
-                  ans:=ans*(second l+i::R::F)
-              (1/factorial t) * ans
+             (t:=retractIfCan(s)@Union(Z,"failed")) case Z and t>0 =>
+                ans:=1::F
+                for i in 1..t repeat
+                    ans:=ans*(second l+i::R::F)
+                (1/factorial t) * ans
            (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or
              (r2 := retractIfCan(second l)@Union(R,"failed")) case "failed"
                => ibinom l
            binomial(r1::R, r2::R)::F
 
+@
+
+[[iibinom]] checks those cases in which the binomial coefficient may be
+evaluated explicitly. Note that up to [[patch--51]], the case where the second
+argument is a positive integer was not checked.(Issue~\#336) Currently, the
+naive iterative algorithm is used to calculate the coefficient, there is room
+for improvement here.
+
+<<package COMBF CombinatorialFunction>>=
+
       else
          iibinom l ==
            (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or
@@ -622,6 +685,7 @@ FunctionalSpecialFunction(R, F): Exports == Implementation where
   OP  ==> BasicOperator
   K   ==> Kernel F
   SE  ==> Symbol
+  SPECIALDIFF  ==> "%specialDiff"
 
   Exports ==> with
     belong? : OP -> Boolean
@@ -635,46 +699,73 @@ FunctionalSpecialFunction(R, F): Exports == Implementation where
     Gamma   : F -> F
       ++ Gamma(f) returns the formal Gamma function applied to f
     Gamma   : (F,F) -> F
-      ++ Gamma(a,x) returns the incomplete Gamma function applied to a and x.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ Gamma(a,x) returns the incomplete Gamma function applied to a and x
     Beta:      (F,F) -> F
       ++ Beta(x,y) returns the beta function applied to x and y
     digamma:   F->F
       ++ digamma(x) returns the digamma function applied to x 
     polygamma: (F,F) ->F
-      ++ polygamma(x,y) returns the polygamma function applied to x and y.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ polygamma(x,y) returns the polygamma function applied to x and y
     besselJ:   (F,F) -> F
-      ++ besselJ(x,y) returns the besselj function applied to x and y.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ besselJ(x,y) returns the besselj function applied to x and y
     besselY:   (F,F) -> F
-      ++ besselY(x,y) returns the bessely function applied to x and y.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ besselY(x,y) returns the bessely function applied to x and y
     besselI:   (F,F) -> F
-      ++ besselI(x,y) returns the besseli function applied to x and y.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ besselI(x,y) returns the besseli function applied to x and y
     besselK:   (F,F) -> F
-      ++ besselK(x,y) returns the besselk function applied to x and y.
-      ++ Concerning differentiation, it is regarded as a function in the
-      ++ second argument only.
+      ++ besselK(x,y) returns the besselk function applied to x and y
     airyAi:    F -> F
       ++ airyAi(x) returns the airyai function applied to x 
     airyBi:    F -> F
       ++ airyBi(x) returns the airybi function applied to x
 
+@
+
+In case we want to have more special function operators here, do not forget to
+add them to the list [[specop]] in [[CommonOperators]].  Otherwise they will
+not have the \lq special\rq\ attribute and will not be recognised here.  One
+effect could be that
+\begin{verbatim}
+myNewSpecOp(1::Expression Integer)::Expression DoubleFloat
+\end{verbatim}
+might not re-evaluate the operator.
+
+<<package FSPECF FunctionalSpecialFunction>>=
     iiGamma : F -> F
       ++ iiGamma(x) should be local but conditional;
     iiabs     : F -> F
       ++ iiabs(x) should be local but conditional;
+    iiBeta     : List F -> F
+      ++ iiGamma(x) should be local but conditional;
+    iidigamma  : F -> F
+      ++ iidigamma(x) should be local but conditional;
+    iipolygamma: List F -> F
+      ++ iipolygamma(x) should be local but conditional;
+    iiBesselJ  : List F -> F
+      ++ iiBesselJ(x) should be local but conditional;
+    iiBesselY  : List F -> F
+      ++ iiBesselY(x) should be local but conditional;
+    iiBesselI  : List F -> F
+      ++ iiBesselI(x) should be local but conditional;
+    iiBesselK  : List F -> F
+      ++ iiBesselK(x) should be local but conditional;
+    iiAiryAi   : F -> F
+      ++ iiAiryAi(x) should be local but conditional;
+    iiAiryBi   : F -> F
+      ++ iiAiryBi(x) should be local but conditional;
 
   Implementation ==> add
-    iabs     : F -> F
-    iGamma:     F -> F
+    iabs      : F -> F
+    iGamma    : F -> F
+    iBeta     : (F, F) -> F
+    idigamma  : F -> F
+    iiipolygamma: (F, F) -> F
+    iiiBesselJ  : (F, F) -> F
+    iiiBesselY  : (F, F) -> F
+    iiiBesselI  : (F, F) -> F
+    iiiBesselK  : (F, F) -> F
+    iAiryAi   : F -> F
+    iAiryBi   : F -> F
 
     opabs       := operator("abs"::Symbol)$CommonOperators
     opGamma     := operator("Gamma"::Symbol)$CommonOperators
@@ -732,6 +823,17 @@ FunctionalSpecialFunction(R, F): Exports == Implementation where
       x < 0 => kernel(opabs, -x)
       kernel(opabs, x)
 
+    iBeta(x, y) == kernel(opBeta, [x, y])
+    idigamma x == kernel(opdigamma, x)
+    iiipolygamma(n, x) == kernel(oppolygamma, [n, x])
+    iiiBesselJ(x, y) == kernel(opBesselJ, [x, y])
+    iiiBesselY(x, y) == kernel(opBesselY, [x, y])
+    iiiBesselI(x, y) == kernel(opBesselI, [x, y])
+    iiiBesselK(x, y) == kernel(opBesselK, [x, y])
+    iAiryAi x == kernel(opAiryAi, x)
+    iAiryBi x == kernel(opAiryBi, x)
+
+
     -- Could put more conditional special rules for other functions here
 
     if R has abs : R -> R then
@@ -750,6 +852,54 @@ FunctionalSpecialFunction(R, F): Exports == Implementation where
         (r:=retractIfCan(x)@Union(R,"failed")) case "failed" => iGamma x
         Gamma(r::R)::F
 
+      iiBeta l ==
+        (r:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (s:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iBeta(first l, second l)
+        Beta(r::R, s::R)::F
+
+      iidigamma x ==
+        (r:=retractIfCan(x)@Union(R,"failed")) case "failed" => idigamma x
+        digamma(r::R)::F
+
+      iipolygamma l ==
+        (s:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (r:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iiipolygamma(first l, second l)
+        polygamma(s::R, r::R)::F
+
+      iiBesselJ l ==
+        (r:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (s:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iiiBesselJ(first l, second l)
+        besselJ(r::R, s::R)::F
+
+      iiBesselY l ==
+        (r:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (s:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iiiBesselY(first l, second l)
+        besselY(r::R, s::R)::F
+
+      iiBesselI l ==
+        (r:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (s:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iiiBesselI(first l, second l)
+        besselI(r::R, s::R)::F
+
+      iiBesselK l ==
+        (r:=retractIfCan(first l)@Union(R,"failed")) case "failed" or _
+        (s:=retractIfCan(second l)@Union(R,"failed")) case "failed" _
+            => iiiBesselK(first l, second l)
+        besselK(r::R, s::R)::F
+
+      iiAiryAi x ==
+        (r:=retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAi x
+        airyAi(r::R)::F
+
+      iiAiryBi x ==
+        (r:=retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBi x
+        airyBi(r::R)::F
+
     else
       if R has RetractableTo Integer then
         iiGamma x ==
@@ -759,39 +909,113 @@ FunctionalSpecialFunction(R, F): Exports == Implementation where
       else
         iiGamma x == iGamma x
 
+      iiBeta l == iBeta(first l, second l)
+      iidigamma x == idigamma x 
+      iipolygamma l == iiipolygamma(first l, second l)
+      iiBesselJ l == iiiBesselJ(first l, second l) 
+      iiBesselY l == iiiBesselY(first l, second l)
+      iiBesselI l == iiiBesselI(first l, second l)
+      iiBesselK l == iiiBesselK(first l, second l)
+      iiAiryAi x == iAiryAi x
+      iiAiryBi x == iAiryBi x
+
     -- Default behaviour is to build a kernel
     evaluate(opGamma, iiGamma)$BasicOperatorFunctions1(F)
     evaluate(opabs, iiabs)$BasicOperatorFunctions1(F)
+--    evaluate(opGamma2    ,iiGamma2   )$BasicOperatorFunctions1(F)
+    evaluate(opBeta      ,iiBeta     )$BasicOperatorFunctions1(F)
+    evaluate(opdigamma   ,iidigamma  )$BasicOperatorFunctions1(F)
+    evaluate(oppolygamma ,iipolygamma)$BasicOperatorFunctions1(F)
+    evaluate(opBesselJ   ,iiBesselJ  )$BasicOperatorFunctions1(F)
+    evaluate(opBesselY   ,iiBesselY  )$BasicOperatorFunctions1(F)
+    evaluate(opBesselI   ,iiBesselI  )$BasicOperatorFunctions1(F)
+    evaluate(opBesselK   ,iiBesselK  )$BasicOperatorFunctions1(F)
+    evaluate(opAiryAi    ,iiAiryAi   )$BasicOperatorFunctions1(F)
+    evaluate(opAiryBi    ,iiAiryBi   )$BasicOperatorFunctions1(F)
+@
+
+\subsection{differentiation of special functions}
+
+In the following we define the symbolic derivatives of the special functions we
+provide.  The formulas we use for the Bessel functions can be found in Milton
+Abramowitz and Irene A. Stegun, eds.  (1965). Handbook of Mathematical
+Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN
+0-486-61272-4, Equations~9.1.27 and 9.6.26.  Up to [[patch--50]] the formula
+for $K$ missed the minus sign.  (Issue~\#355)
+
+We do not attempt to provide formulas for the derivative with respect to the
+first argument currently.  Instead, we leave such derivatives unevaluated.
 
+<<package FSPECF FunctionalSpecialFunction>>=
     import Fraction Integer
     ahalf:  F    := recip(2::F)::F
     athird: F    := recip(2::F)::F
     twothirds: F := 2*recip(3::F)::F
+@
+
+We need to get hold of the differentiation operator as modified by
+[[FunctionSpace]]. Otherwise, for example, display will be ugly.  We accomplish
+that by differentiating an operator, which will certainly result in a single
+kernel only.
+
+<<package FSPECF FunctionalSpecialFunction>>=
+    dummyArg: SE := new()$SE
+    opdiff := operator first kernels D((operator(new()$SE)$BasicOperator)
+                                            (dummyArg::F), dummyArg)
+@
+
+The differentiation operator [[opdiff]] takes three arguments corresponding to
+$$
+F_{,i}(a_1,a_2,\dots,a_n):
+$$
+\begin{enumerate}
+\item $F(a_1,...,dm,...a_n)$, where the $i$\textsuperscript{th} argument is a
+  dummy variable,
+\item $dm$, the dummy variable, and
+\item $a_i$, the point at which the differential is evaluated.
+\end{enumerate}
+
+In the following, it seems to be safe to use the same dummy variable
+troughout.  At least, this is done also in [[FunctionSpace]], and did not cause
+problems.
 
-    lzero(l: List F): F == 0
+The operation [[symbolicGrad]] returns the first component of the gradient of
+[[op l]].
+
+<<package FSPECF FunctionalSpecialFunction>>=
+    dm := new()$SE :: F
 
-    iBesselJGrad(l: List F): F ==
+    iBesselJ(l: List F, t: SE): F ==
         n := first l; x := second l
-        ahalf * (besselJ (n-1,x) - besselJ (n+1,x))
-    iBesselYGrad(l: List F): F ==
+        differentiate(n, t)*kernel(opdiff, [opBesselJ [dm, x], dm, n])
+          + differentiate(x, t) * ahalf * (besselJ (n-1,x) - besselJ (n+1,x))
+
+    iBesselY(l: List F, t: SE): F ==
         n := first l; x := second l
-        ahalf * (besselY (n-1,x) - besselY (n+1,x))
-    iBesselIGrad(l: List F): F ==
+        differentiate(n, t)*kernel(opdiff, [opBesselY [dm, x], dm, n])
+          + differentiate(x, t) * ahalf * (besselY (n-1,x) - besselY (n+1,x))
+
+    iBesselI(l: List F, t: SE): F ==
         n := first l; x := second l
-        ahalf * (besselI (n-1,x) + besselI (n+1,x))
-    iBesselKGrad(l: List F): F ==
+        differentiate(n, t)*kernel(opdiff, [opBesselI [dm, x], dm, n])
+          + differentiate(x, t)* ahalf * (besselI (n-1,x) + besselI (n+1,x))
+
+    iBesselK(l: List F, t: SE): F ==
         n := first l; x := second l
-        - ahalf * (besselK (n-1,x) + besselK (n+1,x))
+        differentiate(n, t)*kernel(opdiff, [opBesselK [dm, x], dm, n])
+          - differentiate(x, t)* ahalf * (besselK (n-1,x) + besselK (n+1,x))
+
 @
-The formulas above for the Bessel functions can be found in Milton Abramowitz
-and Irene A. Stegun, eds.  (1965). Handbook of Mathematical Functions with
-Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4,
-Equations~9.1.27 and 9.6.26.  Up to [[patch--50]] the formula for $K$ missed
-the minus sign.  (Issue~\#355)
+
+For the moment we throw an error if we try to differentiate [[polygamma]] with
+respect to the first argument.
+
 <<package FSPECF FunctionalSpecialFunction>>=
-    ipolygammaGrad(l: List F): F ==
-        n := first l; x := second l
-        polygamma(n+1, x)
+    ipolygamma(l: List F, x: SE): F ==
+        member?(x, variables first l) =>
+            error "cannot differentiate polygamma with respect to the first argument"
+        n := first l; y := second l
+        differentiate(y, x)*polygamma(n+1, y)
     iBetaGrad1(l: List F): F ==
         x := first l; y := second l
         Beta(x,y)*(digamma x - digamma(x+y))
@@ -800,20 +1024,36 @@ the minus sign.  (Issue~\#355)
         Beta(x,y)*(digamma y - digamma(x+y))
 
     if F has ElementaryFunctionCategory then
-      iGamma2Grad(l: List F):F ==
+      iGamma2(l: List F, t: SE): F ==
         a := first l; x := second l
-        - x ** (a - 1) * exp(-x)
-      derivative(opGamma2, [lzero, iGamma2Grad])
+        differentiate(a, t)*kernel(opdiff, [opGamma2 [dm, x], dm, a])
+          - differentiate(x, t)* x ** (a - 1) * exp(-x)
+      setProperty(opGamma2, SPECIALDIFF, iGamma2@((List F, SE)->F) 
+                                                 pretend None)
+@
+
+Finally, we tell Axiom to use these functions for differentiation.  Note that
+up to [[patch--50]], the properties for the Bessel functions were set using
+[[derivative(oppolygamma, [lzero, ipolygammaGrad])]], where [[lzero]] returned
+zero always.  Trying to replace [[lzero]] by a function that returns the first
+component of the gradient failed, it resulted in an infinite loop for
+[[integrate(D(besselJ(a,x),a),a)]].
 
+<<package FSPECF FunctionalSpecialFunction>>=
     derivative(opabs,       abs(#1) * inv(#1))
     derivative(opGamma,     digamma #1 * Gamma #1)
     derivative(opBeta,      [iBetaGrad1, iBetaGrad2])
     derivative(opdigamma,   polygamma(1, #1))
-    derivative(oppolygamma, [lzero, ipolygammaGrad])
-    derivative(opBesselJ,   [lzero, iBesselJGrad])
-    derivative(opBesselY,   [lzero, iBesselYGrad])
-    derivative(opBesselI,   [lzero, iBesselIGrad])
-    derivative(opBesselK,   [lzero, iBesselKGrad])
+    setProperty(oppolygamma, SPECIALDIFF, ipolygamma@((List F, SE)->F)
+                                                     pretend None)
+    setProperty(opBesselJ, SPECIALDIFF, iBesselJ@((List F, SE)->F) 
+                                                 pretend None)
+    setProperty(opBesselY, SPECIALDIFF, iBesselY@((List F, SE)->F) 
+                                                 pretend None)
+    setProperty(opBesselI, SPECIALDIFF, iBesselI@((List F, SE)->F) 
+                                                 pretend None)
+    setProperty(opBesselK, SPECIALDIFF, iBesselK@((List F, SE)->F) 
+                                                 pretend None)
 
 @
 \section{package SUMFS FunctionSpaceSum}
diff --git a/src/algebra/op.spad.pamphlet b/src/algebra/op.spad.pamphlet
index e443d3c..51d4d6c 100644
--- a/src/algebra/op.spad.pamphlet
+++ b/src/algebra/op.spad.pamphlet
@@ -460,9 +460,11 @@ BasicOperator(): Exports == Implementation where
 -- property EQUAL? contains a function f: (BOP, BOP) -> Boolean
 -- such that f(o1, o2) is true iff o1 = o2
     op1 = op2 ==
+      (EQ$Lisp)(op1, op2) => true
       name(op1) ^= name(op2) => false
       op1.narg ^= op2.narg => false
-      brace(keys properties op1)^=$Set(String) brace(keys properties op2) => false
+      brace(keys properties op1)^=$Set(String) _
+                     brace(keys properties op2) => false
       (func := property(op1, EQUAL?)) case None =>
                    ((func::None) pretend (($, $) -> Boolean)) (op1, op2)
       true
@@ -721,7 +723,8 @@ CommonOperators(): Exports == Implementation where
     combop  := [opfact, opperm, opbinom, oppow,
                                          opsum, opdsum, opprod, opdprod]
     specop  := [opGamma, opGamma2, opBeta, opdigamma, oppolygamma, opabs,
-                opBesselJ, opBesselY, opBesselI, opBesselK]
+                opBesselJ, opBesselY, opBesselI, opBesselK, opAiryAi,
+                 opAiryBi]
     anyop   := [oppren, opdiff, opbox, opquote]
     allop   := concat(concat(concat(concat(concat(
                             algop,elemop),primop),combop),specop),anyop)
@@ -746,7 +749,23 @@ CommonOperators(): Exports == Implementation where
     dpi l    == "%pi"::Symbol::O
     dfact x  == postfix("!"::Symbol::O, (ATOM(x)$Lisp => x; paren x))
     dquote l == prefix(quote(first(l)::O), rest l)
+@
+It is certainly an abuse of OutputForm to produce a Gamma as done below.
+Originally, it was even worse (Issue~\#6):
+\begin{verbatim}
     dgamma l == prefix(hconcat("|"::Symbol::O, overbar(" "::Symbol::O)), l)
+\end{verbatim}
+which was TeXed to
+$${|{\overline{\ }}}
+\left(
+{x}
+\right).
+$$
+
+The right thing would be to introduce Greek letters in OutputForm, but
+that should be coordinated with the new mathml package
+<<package COMMONOP CommonOperators>>=
+    dgamma l == prefix(super("|"::Symbol::O, "-"::Symbol::O), l)
     setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)
 
     dexp x ==
diff --git a/src/algebra/sf.spad.pamphlet b/src/algebra/sf.spad.pamphlet
index 8ed563d..571ccd0 100644
--- a/src/algebra/sf.spad.pamphlet
+++ b/src/algebra/sf.spad.pamphlet
@@ -993,7 +993,8 @@ o $AXIOM/doc/src/algebra/sf.spad.dvi
 -- I've put some timing comparisons in the notes for the Float
 -- domain about the difference in speed between the two domains.
 DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath,
-   TranscendentalFunctionCategory, ConvertibleTo InputForm) with
+   TranscendentalFunctionCategory, SpecialFunctionCategory, _
+   ConvertibleTo InputForm) with
       _/   : (%, Integer) -> %
         ++ x / i computes the division from x by an integer i.
       _*_* : (%,%) -> %
@@ -1082,16 +1083,18 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath,
      base() = 2 => precision()
      base() = 16 => 4*precision()
      wholePart(precision()*log2(base()::%))::PositiveInteger
-   max()            == MOST_-POSITIVE_-LONG_-FLOAT$Lisp
-   min()            == MOST_-NEGATIVE_-LONG_-FLOAT$Lisp
+   max()            == MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp
+   min()            == MOST_-NEGATIVE_-DOUBLE_-FLOAT$Lisp
    order(a) == precision() + exponent a - 1
-   0                == FLOAT(0$Lisp,MOST_-POSITIVE_-LONG_-FLOAT$Lisp)$Lisp
-   1                == FLOAT(1$Lisp,MOST_-POSITIVE_-LONG_-FLOAT$Lisp)$Lisp
+   0                == FLOAT(0$Lisp,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp
+   1                == FLOAT(1$Lisp,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp
    -- rational approximation to e accurate to 23 digits
-   exp1()           == FLOAT(534625820200,MOST_-POSITIVE_-LONG_-FLOAT$Lisp)$Lisp / FLOAT(196677847971,MOST_-POSITIVE_-LONG_-FLOAT$Lisp)$Lisp
-   pi()             == PI$Lisp
+   exp1()  == FLOAT(534625820200,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp / _
+              FLOAT(196677847971,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp
+   pi()    == FLOAT(PI$Lisp,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp
    coerce(x:%):OutputForm == 
-     outputForm(FORMAT(NIL$Lisp,format,x)$Lisp pretend DoubleFloat)
+     x >= 0 => message(FORMAT(NIL$Lisp,format,x)$Lisp pretend String)
+     - (message(FORMAT(NIL$Lisp,format,-x)$Lisp pretend String))
    convert(x:%):InputForm == convert(x pretend DoubleFloat)$InputForm
    x < y            == (x<y)$Lisp
    - x              == (-x)$Lisp
@@ -1107,7 +1110,7 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath,
    log10 x          == checkComplex log(x)$Lisp
    x:% ** i:Integer == EXPT(x,i)$Lisp
    x:% ** y:%       == checkComplex EXPT(x,y)$Lisp
-   coerce(i:Integer):% == FLOAT(i,MOST_-POSITIVE_-LONG_-FLOAT$Lisp)$Lisp
+   coerce(i:Integer):% == FLOAT(i,MOST_-POSITIVE_-DOUBLE_-FLOAT$Lisp)$Lisp
    exp x            == EXP(x)$Lisp
    log x            == checkComplex LN(x)$Lisp
    log2 x           == checkComplex LOG2(x)$Lisp
@@ -1145,8 +1148,22 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath,
    SFSFUN           ==> DoubleFloatSpecialFunctions()
    sfx              ==> x pretend DoubleFloat
    sfy              ==> y pretend DoubleFloat
-   Gamma x          == Gamma(sfx)$SFSFUN pretend %
+   airyAi x         == airyAi(sfx)$SFSFUN pretend %
+   airyBi x         == airyBi(sfx)$SFSFUN pretend %
+   besselI(x,y)     == besselI(sfx,sfy)$SFSFUN pretend %
+   besselJ(x,y)     == besselJ(sfx,sfy)$SFSFUN pretend %
+   besselK(x,y)     == besselK(sfx,sfy)$SFSFUN pretend %
+   besselY(x,y)     == besselY(sfx,sfy)$SFSFUN pretend %
    Beta(x,y)        == Beta(sfx,sfy)$SFSFUN pretend %
+   digamma x        == digamma(sfx)$SFSFUN pretend %
+   Gamma x          == Gamma(sfx)$SFSFUN pretend %
+-- not implemented in SFSFUN
+--   Gamma(x,y)       == Gamma(sfx,sfy)$SFSFUN pretend %
+   polygamma(x,y)   ==
+       if (n := retractIfCan(x:%):Union(Integer, "failed")) case Integer _
+          and n >= 0
+       then polygamma(n::Integer::NonNegativeInteger,sfy)$SFSFUN pretend %
+       else error "polygamma: first argument should be a nonnegative integer"
 
    wholePart x            == FIX(x)$Lisp
    float(ma,ex,b)   == ma*(b::%)**ex



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daly@axiom-developer.org writes:

> Handle besselK differently. 

> +20080103 wxh src/algebra/sf.spad handle besselK (7090/355)
> +20080103 wxh src/algebra/op.spad handle besselK (7090/355)
> +20080103 wxh src/algebra/combfunc.spad handle besselK (7090/355)	

These patches are mine, not waldek's.

Martin




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The patch fixes the besselk patch. I wrongly attributed the patch
to Waldek when it should properly have been attributed to Martin.

Tim

========================================================================
diff --git a/changelog b/changelog
index f6db6a4..7c5d8fe 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080104 tpd changelog correct the author of besselk patches
 20080104 mxr src/input/repa6.input fix function names (7093)
 20080104 mxr src/input/knot2.input fix function names (7093)
 20080104 mxr src/input/grpthry.input fix function names (7093)
@@ -6,9 +7,9 @@
 20080104 mxr src/input/easter.input fix function names (7093)
 20080104 mxr src/input/collect.input fix function names (7093)
 20080104 mxr src/input/calculus2.input fix function names (7093)
-20080103 wxh src/algebra/sf.spad handle besselK (7090/355)
-20080103 wxh src/algebra/op.spad handle besselK (7090/355)
-20080103 wxh src/algebra/combfunc.spad handle besselK (7090/355)	
+20080103 mxr src/algebra/sf.spad handle besselK (7090/355)
+20080103 mxr src/algebra/op.spad handle besselK (7090/355)
+20080103 mxr src/algebra/combfunc.spad handle besselK (7090/355)	
 20080102 tpd src/hyper/Makefile.pamphlet fix typo for axbook
 20071230 tpd src/hyper/Makefile prevent spurious remake of axbook (7052)
 20071230 tpd src/input/summation.input update tests with new mathml output



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Martin fixed a number of misnamed functions in the input files.
These patches are redone and applied to the regression test files.

=======================================================================
diff --git a/changelog b/changelog
index 8acaf43..f6db6a4 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,11 @@
+20080104 mxr src/input/repa6.input fix function names (7093)
+20080104 mxr src/input/knot2.input fix function names (7093)
+20080104 mxr src/input/grpthry.input fix function names (7093)
+20080104 mxr src/input/exsum.input fix function names (7093)
+20080104 mxr src/input/exlap.input fix function names (7093)
+20080104 mxr src/input/easter.input fix function names (7093)
+20080104 mxr src/input/collect.input fix function names (7093)
+20080104 mxr src/input/calculus2.input fix function names (7093)
 20080103 wxh src/algebra/sf.spad handle besselK (7090/355)
 20080103 wxh src/algebra/op.spad handle besselK (7090/355)
 20080103 wxh src/algebra/combfunc.spad handle besselK (7090/355)	
diff --git a/src/input/calculus2.input.pamphlet b/src/input/calculus2.input.pamphlet
index 3db6893..031f0ec 100644
--- a/src/input/calculus2.input.pamphlet
+++ b/src/input/calculus2.input.pamphlet
@@ -370,7 +370,7 @@ eq := differentiate(y(x), x, 3) - sin(differentiate(y(x), x, 2)) * exp(y(x)) = c
 --S 29 of 112
 seriesSolve(eq, y, x = 0, [1, 0, 0])
 --R 
---R   Compiling function %B with type List UnivariateTaylorSeries(
+--I   Compiling function %B with type List UnivariateTaylorSeries(
 --R      Expression Integer,x,0) -> UnivariateTaylorSeries(Expression 
 --R      Integer,x,0) 
 --R
@@ -388,7 +388,7 @@ seriesSolve(eq, y, x = 0, [1, 0, 0])
 --S 30 of 112
 x := operator 'x
 --R 
---R   Compiled code for %B has been cleared.
+--I   Compiled code for %B has been cleared.
 --R
 --R   (4)  x
 --R                                                          Type: BasicOperator
@@ -417,10 +417,10 @@ eq2 := differentiate(y(t), t) = x(t) * y(t)
 --S 33 of 112
 seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])
 --R 
---R   Compiling function %D with type List UnivariateTaylorSeries(
+--I   Compiling function %D with type List UnivariateTaylorSeries(
 --R      Expression Integer,t,0) -> UnivariateTaylorSeries(Expression 
 --R      Integer,t,0) 
---R   Compiling function %E with type List UnivariateTaylorSeries(
+--I   Compiling function %E with type List UnivariateTaylorSeries(
 --R      Expression Integer,t,0) -> UnivariateTaylorSeries(Expression 
 --R      Integer,t,0) 
 --R
@@ -493,37 +493,27 @@ laplace((cos(a*t) - cos(b*t))/t, t, s)
 --E 39
 
 --S 40 of 112
-laplace(exp(a*t+b)*ei(c*t), t, s)
---R 
---R   There are no library operations named ei 
---R      Use HyperDoc Browse or issue
---R                                 )what op ei
---R      to learn if there is any operation containing " ei " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ei 
---R      with argument type(s) 
---R                             Polynomial Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+laplace(exp(a*t+b)*Ei(c*t), t, s)
+--R
+--R          b    s + c - a
+--R        %e log(---------)
+--R                   c
+--R   (7)  -----------------
+--R              s - a
+--R                                                     Type: Expression Integer
 --E 40
 
 --S 41 of 112
-laplace(a*ci(b*t) + c*si(d*t), t, s)
---R 
---R   There are no library operations named ci 
---R      Use HyperDoc Browse or issue
---R                                 )what op ci
---R      to learn if there is any operation containing " ci " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ci 
---R      with argument type(s) 
---R                             Polynomial Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+laplace(a*Ci(b*t) + c*Si(d*t), t, s)
+--R
+--R               2    2
+--R              s  + b             d
+--R        a log(-------) + 2c atan(-)
+--R                  2              s
+--R                 b
+--R   (8)  ---------------------------
+--R                     2s
+--R                                                     Type: Expression Integer
 --E 41
 
 --S 42 of 112
@@ -533,7 +523,7 @@ laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)
 --R                                    2
 --R          4     2 2    4           t           3
 --R        (s  + 2a s  + a )laplace(%e  ,t,s) + 2a
---R   (7)  ----------------------------------------
+--R   (9)  ----------------------------------------
 --R                      4     2 2    4
 --R                     s  + 2a s  + a
 --R                                                     Type: Expression Integer
@@ -637,9 +627,9 @@ integrate(g, x)
 --R 
 --R
 --R           x      +--------+
---R         ++  log(\|b + %G a  + 1)
---R   (4)   |   -------------------- d%G
---R        ++            %G
+--I         ++  log(\|b + %G a  + 1)
+--I   (4)   |   -------------------- d%G
+--I        ++            %G
 --R                                          Type: Union(Expression Integer,...)
 --E 51
 
diff --git a/src/input/collect.input.pamphlet b/src/input/collect.input.pamphlet
index 5e92eda..9e07896 100644
--- a/src/input/collect.input.pamphlet
+++ b/src/input/collect.input.pamphlet
@@ -91,10 +91,71 @@ e := reverse [i**3 for i in 10..0 by -2 | even? i]
 --R                                                           Type: List Integer
 --E 9
 
+--S 10 of 55
+[x**3 - y for x in b | even? x for y in e]
+--R
+--R   (10)  [0,- 56,- 448]
+--R                                                           Type: List Integer
+--E 10
+
+--S 11 of 55
+f := [i**3 for i in 0..]
+--R
+--R   (11)  [0,1,8,27,64,125,216,343,512,729,...]
+--R                                              Type: Stream NonNegativeInteger
+--E 11
+
+--S 12 of 55
+[i**3 for i in 0..10]
+--R
+--R   (12)  [0,1,8,27,64,125,216,343,512,729,1000]
+--R                                                Type: List NonNegativeInteger
+--E 12
+
+--S 13 of 55
+[i**3 for i in 0.. while i < 11]
+--R
+--R   (13)  [0,1,8,27,64,125,216,343,512,729,...]
+--R                                              Type: Stream NonNegativeInteger
+--E 13
+
+--S 14 of 55
+[i**3 for i in 0.. for x in 0..10]
+--R
+--R   (14)  [0,1,8,27,64,125,216,343,512,729,...]
+--R                                              Type: Stream NonNegativeInteger
+--E 14
+
+--S 15 of 55
+[ [i**j for j in 0..3] for i in 0..]
+--R
+--R   (15)
+--R   [[1,0,0,0], [1,1,1,1], [1,2,4,8], [1,3,9,27], [1,4,16,64], [1,5,25,125],
+--R    [1,6,36,216], [1,7,49,343], [1,8,64,512], [1,9,81,729], ...]
+--R                                         Type: Stream List NonNegativeInteger
+--E 15
+
+--S 16 of 55
+[ [i**j for j in 0..] for i in 0..3]
+--R
+--R   (16)
+--R   [[1,0,0,0,0,0,0,0,0,0,...], [1,1,1,1,1,1,1,1,1,1,...],
+--R    [1,2,4,8,16,32,64,128,256,512,...],
+--R    [1,3,9,27,81,243,729,2187,6561,19683,...]]
+--R                                           Type: List Stream Fraction Integer
+--E 16
+
+--S 17 of 55
+brace [i**3 for i in 10..0 by -2]
+--R
+--R   (17)  {0,8,64,216,512,1000}
+--R                                                 Type: Set NonNegativeInteger
+--E 17
+
 -- Input generated from ContinuedFractionXmpPage
 )clear all
 
---S 10 of 55
+--S 18 of 55
 c := continuedFraction(314159/100000)
 --R 
 --R
@@ -102,17 +163,17 @@ c := continuedFraction(314159/100000)
 --R   (1)  3 + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
 --R            | 7     | 15     | 1     | 25     | 1     | 7     | 4
 --R                                              Type: ContinuedFraction Integer
---E 10
+--E 18
 
---S 11 of 55
+--S 19 of 55
 partialQuotients c
 --R 
 --R
 --R   (2)  [3,7,15,1,25,1,7,4]
 --R                                                         Type: Stream Integer
---E 11
+--E 19
 
---S 12 of 55
+--S 20 of 55
 convergents c
 --R 
 --R
@@ -120,9 +181,9 @@ convergents c
 --R   (3)  [3,--,---,---,----,----,-----,------]
 --R            7 106 113 2931 3044 24239 100000
 --R                                                Type: Stream Fraction Integer
---E 12
+--E 20
 
---S 13 of 55
+--S 21 of 55
 approximants c
 --R 
 --R
@@ -131,17 +192,17 @@ approximants c
 --R   (4)  [3,--,---,---,----,----,-----,------]
 --R            7 106 113 2931 3044 24239 100000
 --R                                                Type: Stream Fraction Integer
---E 13
+--E 21
 
---S 14 of 55
+--S 22 of 55
 pq := partialQuotients(1/c)
 --R 
 --R
 --R   (5)  [0,3,7,15,1,25,1,7,4]
 --R                                                         Type: Stream Integer
---E 14
+--E 22
 
---S 15 of 55
+--S 23 of 55
 continuedFraction(first pq,repeating [1],rest pq)
 --R 
 --R
@@ -149,9 +210,9 @@ continuedFraction(first pq,repeating [1],rest pq)
 --R   (6)  +---+ + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
 --R        | 3     | 7     | 15     | 1     | 25     | 1     | 7     | 4
 --R                                              Type: ContinuedFraction Integer
---E 15
+--E 23
 
---S 16 of 55
+--S 24 of 55
 z:=continuedFraction(3,repeating [1],repeating [3,6])
 --R 
 --R
@@ -164,17 +225,17 @@ z:=continuedFraction(3,repeating [1],repeating [3,6])
 --R     +---+ + ...
 --R     | 6
 --R                                              Type: ContinuedFraction Integer
---E 16
+--E 24
 
---S 17 of 55
+--S 25 of 55
 dens:Stream Integer := cons(1,generate((x+->x+4),6))
 --R 
 --R
 --R   (8)  [1,6,10,14,18,22,26,30,34,38,...]
 --R                                                         Type: Stream Integer
---E 17
+--E 25
 
---S 18 of 55
+--S 26 of 55
 cf := continuedFraction(0,repeating [1],dens)
 --R 
 --R
@@ -187,9 +248,9 @@ cf := continuedFraction(0,repeating [1],dens)
 --R     +----+ + +----+ + ...
 --R     | 34     | 38
 --R                                              Type: ContinuedFraction Integer
---E 18
+--E 26
 
---S 19 of 55
+--S 27 of 55
 ccf := convergents cf
 --R 
 --R
@@ -197,9 +258,9 @@ ccf := convergents cf
 --R   (10)  [0,1,-,--,----,-----,------,--------,---------,-----------,...]
 --R              7 71 1001 18089 398959 10391023 312129649 10622799089
 --R                                                Type: Stream Fraction Integer
---E 19
+--E 27
 
---S 20 of 55
+--S 28 of 55
 eConvergents := [2*e + 1 for e in ccf]
 --R 
 --R
@@ -207,9 +268,9 @@ eConvergents := [2*e + 1 for e in ccf]
 --R   (11)  [1,3,--,---,----,-----,-------,--------,---------,-----------,...]
 --R               7  71 1001 18089  398959 10391023 312129649 10622799089
 --R                                                Type: Stream Fraction Integer
---E 20
+--E 28
 
---S 21 of 55
+--S 29 of 55
 eConvergents :: Stream Float
 --R 
 --R
@@ -219,17 +280,17 @@ eConvergents :: Stream Float
 --R    2.7182818284 590458514, 2.7182818284 590452348, 2.7182818284 590452354,
 --R    ...]
 --R                                                           Type: Stream Float
---E 21
+--E 29
 
---S 22 of 55
+--S 30 of 55
 exp 1.0
 --R 
 --R
 --R   (13)  2.7182818284 590452354
 --R                                                                  Type: Float
---E 22
+--E 30
 
---S 23 of 55
+--S 31 of 55
 cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])
 --R 
 --R
@@ -242,9 +303,9 @@ cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])
 --R     +-----+ + +-----+ + ...
 --R     |  2      |  2
 --R                                              Type: ContinuedFraction Integer
---E 23
+--E 31
 
---S 24 of 55
+--S 32 of 55
 ccf := convergents cf
 --R 
 --R
@@ -252,9 +313,9 @@ ccf := convergents cf
 --R   (15)  [1,-,--,---,---,----,-----,-----,------,--------,...]
 --R            2 13  76 263 2578 36979 33976 622637 11064338
 --R                                                Type: Stream Fraction Integer
---E 24
+--E 32
 
---S 25 of 55
+--S 33 of 55
 piConvergents := [4/p for p in ccf]
 --R 
 --R
@@ -262,9 +323,9 @@ piConvergents := [4/p for p in ccf]
 --R   (16)  [4,-,--,---,----,-----,------,------,-------,--------,...]
 --R            3 15 105  315  3465  45045  45045  765765 14549535
 --R                                                Type: Stream Fraction Integer
---E 25
+--E 33
 
---S 26 of 55
+--S 34 of 55
 piConvergents :: Stream Float
 --R 
 --R
@@ -274,9 +335,9 @@ piConvergents :: Stream Float
 --R    3.2837384837 384837385, 3.0170718170 718170718, 3.2523659347 188758953,
 --R    3.0418396189 294022111, ...]
 --R                                                           Type: Stream Float
---E 26
+--E 34
 
---S 27 of 55
+--S 35 of 55
 continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
 --R 
 --R
@@ -284,15 +345,15 @@ continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
 --R   (18)  - 90 + 59%i + +---------+ + +-----------+
 --R                       | 1 - 2%i     | - 1 + 2%i
 --R                                      Type: ContinuedFraction Complex Integer
---E 27
+--E 35
 
---S 28 of 55
+--S 36 of 55
 r : Fraction UnivariatePolynomial(x,Fraction Integer)
 --R 
 --R                                                                   Type: Void
---E 28
+--E 36
 
---S 29 of 55
+--S 37 of 55
 r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
 --R 
 --R
@@ -302,9 +363,9 @@ r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
 --R          2
 --R         x  - 7x + 12
 --R                      Type: Fraction UnivariatePolynomial(x,Fraction Integer)
---E 29
+--E 37
 
---S 30 of 55
+--S 38 of 55
 continuedFraction r
 --R 
 --R
@@ -314,9 +375,9 @@ continuedFraction r
 --R             | - x - -     | -- x - --
 --R             | 4     8     |  3      3
 --R             Type: ContinuedFraction UnivariatePolynomial(x,Fraction Integer)
---E 30
+--E 38
 
---S 31 of 55
+--S 39 of 55
 [i*i for i in convergents(z) :: Stream Float]
 --R 
 --R
@@ -326,77 +387,6 @@ continuedFraction r
 --R    11.0000000017 53603304, 10.9999999999 12099531, 11.0000000000 04406066,
 --R    ...]
 --R                                                           Type: Stream Float
---E 31
-
---S 32 of 55
-[x**3 - y for x in b | even? x for y in e]
---R 
---R 
---RDaly Bug
---R   AXIOM cannot iterate with x over your form now. Perhaps you should 
---R      try using a conversion to make sure your form is a list or 
---R      stream, for example.
---E 32
-
---S 33 of 55
-f := [i**3 for i in 0..]
---R 
---R
---R   (23)  [0,1,8,27,64,125,216,343,512,729,...]
---R                                              Type: Stream NonNegativeInteger
---E 33
-
---S 34 of 55
-[i**3 for i in 0..10]
---R 
---R
---R   (24)  [0,1,8,27,64,125,216,343,512,729,1000]
---R                                                Type: List NonNegativeInteger
---E 34
-
---S 35 of 55
-[i**3 for i in 0.. while i < 11]
---R 
---R
---R   (25)  [0,1,8,27,64,125,216,343,512,729,...]
---R                                              Type: Stream NonNegativeInteger
---E 35
-
---S 36 of 55
-[i**3 for i in 0.. for x in 0..10]
---R 
---R
---R   (26)  [0,1,8,27,64,125,216,343,512,729,...]
---R                                              Type: Stream NonNegativeInteger
---E 36
-
---S 37 of 55
-[[i**j for j in 0..3] for i in 0..]
---R 
---R
---R   (27)
---R   [[1,0,0,0], [1,1,1,1], [1,2,4,8], [1,3,9,27], [1,4,16,64], [1,5,25,125],
---R    [1,6,36,216], [1,7,49,343], [1,8,64,512], [1,9,81,729], ...]
---R                                         Type: Stream List NonNegativeInteger
---E 37
-
---S 38 of 55
-[[i**j for j in 0..] for i in 0..3]
---R 
---R
---R   (28)
---R   [[1,0,0,0,0,0,0,0,0,0,...], [1,1,1,1,1,1,1,1,1,1,...],
---R    [1,2,4,8,16,32,64,128,256,512,...],
---R    [1,3,9,27,81,243,729,2187,6561,19683,...]]
---R                                           Type: List Stream Fraction Integer
---E 38
-
---S 39 of 55
-brace [i**3 for i in 10..0 by -2]
---R 
---R
---R   (29)  {0,8,64,216,512,1000}
---R                                                 Type: Set NonNegativeInteger
 --E 39
 
 -- Input for page ForCollectionDetailPage
diff --git a/src/input/easter.input.pamphlet b/src/input/easter.input.pamphlet
index 5db2309..64c9c44 100644
--- a/src/input/easter.input.pamphlet
+++ b/src/input/easter.input.pamphlet
@@ -28,7 +28,7 @@
 \section{Numbers}
 Let's begin by playing with numbers: infinite precision integers
 <<*>>=
---S 1 of 201
+--S 1 of 200
 factorial(50)
 --R 
 --R
@@ -36,7 +36,7 @@ factorial(50)
 --R                                                        Type: PositiveInteger
 --E 1
 
---S 2 of 201
+--S 2 of 200
 factor(%)
 --R 
 --R
@@ -48,7 +48,7 @@ factor(%)
 @
 Infinite precision rational numbers
 <<*>>=
---S 3 of 201
+--S 3 of 200
 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10
 --R 
 --R
@@ -61,7 +61,7 @@ Infinite precision rational numbers
 @
 Arbitrary precision floating point numbers
 <<*>>=
---S 4 of 201
+--S 4 of 200
 digits(50);
 --R 
 --R
@@ -71,7 +71,7 @@ digits(50);
 @
 This number is nearly an integer
 <<*>>=
---S 5 of 201
+--S 5 of 200
 exp(sqrt(163.)*%pi)
 --R 
 --R
@@ -79,7 +79,7 @@ exp(sqrt(163.)*%pi)
 --R                                                                  Type: Float
 --E 5
 
---S 6 of 201
+--S 6 of 200
 digits(20);
 --R 
 --R
@@ -89,7 +89,7 @@ digits(20);
 @
 Special functions
 <<*>>=
---S 7 of 201
+--S 7 of 200
 besselJ(2, 1 + %i)
 --R 
 --R
@@ -100,7 +100,7 @@ besselJ(2, 1 + %i)
 @
 Complete decimal expansion of a rational number
 <<*>>=
---S 8 of 201
+--S 8 of 200
 decimal(1/7)
 --R 
 --R
@@ -112,7 +112,7 @@ decimal(1/7)
 @
 Continued fractions
 <<*>>=
---S 9 of 201
+--S 9 of 200
 continuedFraction(3.1415926535)
 --R 
 --R
@@ -125,7 +125,7 @@ continuedFraction(3.1415926535)
 @
 Simplify an expression with nested square roots
 <<*>>=
---S 10 of 201
+--S 10 of 200
 sqrt(2*sqrt(3) + 4)
 --R 
 --R
@@ -135,7 +135,7 @@ sqrt(2*sqrt(3) + 4)
 --R                                                        Type: AlgebraicNumber
 --E 10
 
---S 11 of 201
+--S 11 of 200
 simplify(%)
 --R 
 --R
@@ -148,7 +148,7 @@ simplify(%)
 @
 Try a more complicated example (from the Putnam exam)
 <<*>>=
---S 12 of 201
+--S 12 of 200
 sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))))
 --R 
 --R
@@ -161,7 +161,7 @@ sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))))
 --R                                                        Type: AlgebraicNumber
 --E 12
 
---S 13 of 201
+--S 13 of 200
 simplify(%)
 --R 
 --R
@@ -177,7 +177,7 @@ simplify(%)
 @
 Cardinal numbers
 <<*>>=
---S 14 of 201
+--S 14 of 200
 2*Aleph(0) - 3
 --R 
 --R
@@ -190,7 +190,7 @@ Cardinal numbers
 Numbers are nice, but symbols allow for variability---try some high school
 algebra: rational simplification
 <<*>>=
---S 15 of 201
+--S 15 of 200
 (x**2 - 4)/(x**2 + 4*x + 4)
 --R 
 --R
@@ -203,7 +203,7 @@ algebra: rational simplification
 @
 This example requires more sophistication
 <<*>>=
---S 16 of 201
+--S 16 of 200
 (%e**x - 1)/(%e**(x/2) + 1)
 --R 
 --R
@@ -217,7 +217,7 @@ This example requires more sophistication
 --R                                                     Type: Expression Integer
 --E 16
 
---S 17 of 201
+--S 17 of 200
 normalize(%)
 --R 
 --R
@@ -231,7 +231,7 @@ normalize(%)
 @
 Expand and factor polynomials
 <<*>>=
---S 18 of 201
+--S 18 of 200
 (x + 1)**20
 --R 
 --R
@@ -247,7 +247,7 @@ Expand and factor polynomials
 --R                                                     Type: Polynomial Integer
 --E 18
 
---S 19 of 201
+--S 19 of 200
 D(%, x)
 --R 
 --R
@@ -263,7 +263,7 @@ D(%, x)
 --R                                                     Type: Polynomial Integer
 --E 19
 
---S 20 of 201
+--S 20 of 200
 factor(%)
 --R 
 --R
@@ -272,7 +272,7 @@ factor(%)
 --R                                            Type: Factored Polynomial Integer
 --E 20
 
---S 21 of 201
+--S 21 of 200
 x**100 - 1
 --R 
 --R
@@ -281,7 +281,7 @@ x**100 - 1
 --R                                                     Type: Polynomial Integer
 --E 21
 
---S 22 of 201
+--S 22 of 200
 factor(%)
 --R 
 --R
@@ -300,7 +300,7 @@ factor(%)
 @
 Factor polynomials over finite fields and field extensions
 <<*>>=
---S 23 of 201
+--S 23 of 200
 p:= x**4 - 3*x**2 + 1
 --R 
 --R
@@ -309,7 +309,7 @@ p:= x**4 - 3*x**2 + 1
 --R                                                     Type: Polynomial Integer
 --E 23
 
---S 24 of 201
+--S 24 of 200
 factor(p)
 --R 
 --R
@@ -318,14 +318,14 @@ factor(p)
 --R                                            Type: Factored Polynomial Integer
 --E 24
 
---S 25 of 201
+--S 25 of 200
 phi:= rootOf(phi**2 - phi - 1);
 --R 
 --R
 --R                                                        Type: AlgebraicNumber
 --E 25
 
---S 26 of 201
+--S 26 of 200
 factor(p, [phi])
 --R 
 --R
@@ -333,7 +333,7 @@ factor(p, [phi])
 --R                                    Type: Factored Polynomial AlgebraicNumber
 --E 26
 
---S 27 of 201
+--S 27 of 200
 factor(p :: Polynomial(PrimeField(5)))
 --R 
 --R
@@ -342,7 +342,7 @@ factor(p :: Polynomial(PrimeField(5)))
 --R                                       Type: Factored Polynomial PrimeField 5
 --E 27
 
---S 28 of 201
+--S 28 of 200
 expand(%)
 --R 
 --R
@@ -354,7 +354,7 @@ expand(%)
 @
 Partial fraction decomposition
 <<*>>=
---S 29 of 201
+--S 29 of 200
 (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
 --R 
 --R
@@ -366,7 +366,7 @@ Partial fraction decomposition
 --R                                            Type: Fraction Polynomial Integer
 --E 29
 
---S 30 of 201
+--S 30 of 200
 padicFraction(
    partialFraction(numerator(%) :: UnivariatePolynomial(x, Fraction Integer),
                    factor(denominator(%) :: Polynomial Integer) ::
@@ -384,7 +384,7 @@ padicFraction(
 \section{Trigonometry}
 Trigonometric manipulations---these are typically difficult for students
 <<*>>=
---S 31 of 201
+--S 31 of 200
 r:= cos(3*x)/cos(x)
 --R 
 --R
@@ -394,7 +394,7 @@ r:= cos(3*x)/cos(x)
 --R                                                     Type: Expression Integer
 --E 31
 
---S 32 of 201
+--S 32 of 200
 real(complexNormalize(%))
 --R 
 --R
@@ -403,7 +403,7 @@ real(complexNormalize(%))
 --R                                                     Type: Expression Integer
 --E 32
 
---S 33 of 201
+--S 33 of 200
 real(normalize(simplify(complexNormalize(r))))
 --R 
 --R
@@ -414,7 +414,7 @@ real(normalize(simplify(complexNormalize(r))))
 @
 Use rewrite rules
 <<*>>=
---S 34 of 201
+--S 34 of 200
 sincosAngles:= rule _
   (cos((n | integer?(n)) * x) == _
       cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x); _
@@ -428,7 +428,7 @@ sincosAngles:= rule _
 --R                            Type: Ruleset(Integer,Integer,Expression Integer)
 --E 34
 
---S 35 of 201
+--S 35 of 200
 sincosAngles r
 --R 
 --R
@@ -437,7 +437,7 @@ sincosAngles r
 --R                                                     Type: Expression Integer
 --E 35
 
---S 36 of 201
+--S 36 of 200
 r:= 'r;
 --R 
 --R
@@ -448,7 +448,7 @@ r:= 'r;
 \section{Determining Zero Equivalence}
 The following expressions are all equal to zero
 <<*>>=
---S 37 of 201
+--S 37 of 200
 sqrt(997) - (997**3)**(1/6)
 --R 
 --R
@@ -456,7 +456,7 @@ sqrt(997) - (997**3)**(1/6)
 --R                                                        Type: AlgebraicNumber
 --E 37
 
---S 38 of 201
+--S 38 of 200
 sqrt(999983) - (999983**3)**(1/6)
 --R 
 --R
@@ -464,7 +464,7 @@ sqrt(999983) - (999983**3)**(1/6)
 --R                                                        Type: AlgebraicNumber
 --E 38
 
---S 39 of 201
+--S 39 of 200
 (2**(1/3) + 4**(1/3))**3 - 6*(2**(1/3) + 4**(1/3)) - 6
 --R 
 --R
@@ -473,7 +473,7 @@ sqrt(999983) - (999983**3)**(1/6)
 --R                                                        Type: AlgebraicNumber
 --E 39
 
---S 40 of 201
+--S 40 of 200
 simplify(%)
 --R 
 --R
@@ -485,7 +485,7 @@ simplify(%)
 @
 This expression is zero for $x, y > 0$ and $n$ not equal to zero
 <<*>>=
---S 41 of 201
+--S 41 of 200
 x**(1/n)*y**(1/n) - (x*y)**(1/n)
 --R 
 --R
@@ -496,7 +496,7 @@ x**(1/n)*y**(1/n) - (x*y)**(1/n)
 --R                                                     Type: Expression Integer
 --E 41
 
---S 42 of 201
+--S 42 of 200
 normalize(%)
 --R 
 --R
@@ -508,7 +508,7 @@ normalize(%)
 See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'',
 CACM, Volume 14, Number 8, August 1971
 <<*>>=
---S 43 of 201
+--S 43 of 200
 expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x))
 --R 
 --R
@@ -518,7 +518,7 @@ expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x))
 --R                                                     Type: Expression Integer
 --E 43
 
---S 44 of 201
+--S 44 of 200
 complexNormalize(%)
 --R 
 --R
@@ -573,7 +573,7 @@ complexNormalize(%)
 @
 Use a roundabout method---show that expr is a constant equal to zero
 <<*>>=
---S 45 of 201
+--S 45 of 200
 D(expr, x)
 --R 
 --R
@@ -590,7 +590,7 @@ D(expr, x)
 --R                                                     Type: Expression Integer
 --E 45
 
---S 46 of 201
+--S 46 of 200
 simplify(real(complexNormalize(expand(simplify(%)))))
 --R 
 --R
@@ -611,7 +611,7 @@ simplify(real(complexNormalize(expand(simplify(%)))))
 --R                                                     Type: Expression Integer
 --E 46
 
---S 47 of 201
+--S 47 of 200
 normalize(eval(expr, x = 0))
 --R 
 --R
@@ -619,7 +619,7 @@ normalize(eval(expr, x = 0))
 --R                                                     Type: Expression Integer
 --E 47
 
---S 48 of 201
+--S 48 of 200
 log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1))
 --R 
 --R
@@ -632,7 +632,7 @@ log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1))
 --R                                                     Type: Expression Integer
 --E 48
 
---S 49 of 201
+--S 49 of 200
 simplify(%)
 --R 
 --R
@@ -645,7 +645,7 @@ simplify(%)
 --R                                                     Type: Expression Integer
 --E 49
 
---S 50 of 201
+--S 50 of 200
 (4*r + 4*sqrt(r) + 1)**(sqrt(r)/(2*sqrt(r) + 1)) _
    * (2*sqrt(r) + 1)**(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1
 --R 
@@ -659,7 +659,7 @@ simplify(%)
 --R                                                     Type: Expression Integer
 --E 50
 
---S 51 of 201
+--S 51 of 200
 normalize(%)
 --R 
 --R
@@ -671,13 +671,13 @@ normalize(%)
 \section{The Complex Domain}
  Complex functions---separate into their real and imaginary parts
 <<*>>=
---S 52 of 201
+--S 52 of 200
 rectform(z) == real(z) + %i*imag(z)
 --R 
 --R                                                                   Type: Void
 --E 52
 
---S 53 of 201
+--S 53 of 200
 rectform(log(3 + 4*%i))
 --R 
 --R   Compiling function rectform with type Expression Complex Integer -> 
@@ -691,7 +691,7 @@ rectform(log(3 + 4*%i))
 --R                                             Type: Expression Complex Integer
 --E 53
 
---S 54 of 201
+--S 54 of 200
 simplify(rectform(tan(x + %i*y)))
 --R 
 --R
@@ -710,7 +710,7 @@ September 1991.  This first expression can simplify to
 $\sqrt{(x y)}/\sqrt{(x)}$,
 but no further in general (consider what happens when x, y = -1).
 <<*>>=
---S 55 of 201
+--S 55 of 200
 sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z))
 --R 
 --R
@@ -726,7 +726,7 @@ sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z))
 @
 If $z = -1$, $\sqrt(1/z)$ is not equal to $1/\sqrt(z)$
 <<*>>=
---S 56 of 201
+--S 56 of 200
 sqrt(1/z) - 1/sqrt(z)
 --R 
 --R
@@ -743,7 +743,7 @@ sqrt(1/z) - 1/sqrt(z)
 @
 If $z = 3 \pi i$, $\log(\exp(z))$ is not equal to $z$
 <<*>>=
---S 57 of 201
+--S 57 of 200
 log(%e**z)
 --R 
 --R
@@ -751,7 +751,7 @@ log(%e**z)
 --R                                                     Type: Expression Integer
 --E 57
 
---S 58 of 201
+--S 58 of 200
 normalize(%)
 --R 
 --R
@@ -762,7 +762,7 @@ normalize(%)
 @
 The principal value of this expression is $(10 - 4 \pi) i$
 <<*>>=
---S 59 of 201
+--S 59 of 200
 log(%e**(10*%i))
 --R 
 --R
@@ -771,7 +771,7 @@ log(%e**(10*%i))
 --R                                             Type: Expression Complex Integer
 --E 59
 
---S 60 of 201
+--S 60 of 200
 normalize(%)
 --R 
 --R
@@ -783,7 +783,7 @@ normalize(%)
 @
 If $z = \pi$, $\arctan(\tan(z))$ is not equal to $z$
 <<*>>=
---S 61 of 201
+--S 61 of 200
 atan(tan(z))
 --R 
 --R
@@ -794,7 +794,7 @@ atan(tan(z))
 @
 If $z = 2 \pi i$, $\sqrt(\exp(z))$ is not equal to $\exp(z/2)$
 <<*>>=
---S 62 of 201
+--S 62 of 200
 sqrt(%e**z) - %e**(z/2)
 --R 
 --R
@@ -809,7 +809,7 @@ sqrt(%e**z) - %e**(z/2)
 \section{Equations}
 Manipulate an equation using a natural syntax
 <<*>>=
---S 63 of 201
+--S 63 of 200
 (x = 0)/2 + 1
 --R 
 --R
@@ -822,7 +822,7 @@ Manipulate an equation using a natural syntax
 @
 Solve various nonlinear equations---this cubic polynomial has all real roots
 <<*>>=
---S 64 of 201
+--S 64 of 200
 radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x)
 --R 
 --R
@@ -869,7 +869,7 @@ radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x)
 --R                                       Type: List Equation Expression Integer
 --E 64
 
---S 65 of 201
+--S 65 of 200
 map(e +-> lhs(e) = rectform(rhs(e)), %)
 --R 
 --R   Compiling function rectform with type Expression Integer -> 
@@ -924,7 +924,7 @@ map(e +-> lhs(e) = rectform(rhs(e)), %)
 @
 Some simple seeming problems can have messy answers
 <<*>>=
---S 66 of 201
+--S 66 of 200
 eqn:= x**4 + x**3 + x**2 + x + 1 = 0
 --R 
 --R
@@ -933,7 +933,7 @@ eqn:= x**4 + x**3 + x**2 + x + 1 = 0
 --R                                            Type: Equation Polynomial Integer
 --E 66
 
---S 67 of 201
+--S 67 of 200
 radicalSolve(eqn, x)
 --R 
 --R
@@ -1284,7 +1284,7 @@ radicalSolve(eqn, x)
 @
 Check one of the answers
 <<*>>=
---S 68 of 201
+--S 68 of 200
 eval(eqn, %.1)
 --R 
 --R
@@ -1373,7 +1373,7 @@ eval(eqn, %.1)
 --R                                            Type: Equation Expression Integer
 --E 68
 
---S 69 of 201
+--S 69 of 200
 %e**(2*x) + 2*%e**x + 1 = z
 --R 
 --R
@@ -1382,7 +1382,7 @@ eval(eqn, %.1)
 --R                                            Type: Equation Expression Integer
 --E 69
 
---S 70 of 201
+--S 70 of 200
 solve(%, x)
 --R 
 --R
@@ -1394,7 +1394,7 @@ solve(%, x)
 @
 This equation is already factored and so {\sl should} be easy to solve
 <<*>>=
---S 71 of 201
+--S 71 of 200
 (x + 1) * (sin(x)**2 + 1)**2 * cos(3*x)**3 = 0
 --R 
 --R
@@ -1403,7 +1403,7 @@ This equation is already factored and so {\sl should} be easy to solve
 --R                                            Type: Equation Expression Integer
 --E 71
 
---S 72 of 201
+--S 72 of 200
 solve(%, x)
 --R 
 --R
@@ -1417,7 +1417,7 @@ solve(%, x)
 The following equations have an infinite number of solutions (let $n$ be an
 arbitrary integer): $z = 0 [+ n 2 \pi i]$
 <<*>>=
---S 73 of 201
+--S 73 of 200
 solve(%e**z = 1, z)
 --R 
 --R
@@ -1428,7 +1428,7 @@ solve(%e**z = 1, z)
 @
 $x = \pi/4 [+ n \pi]$
 <<*>>=
---S 74 of 201
+--S 74 of 200
 solve(sin(x) = cos(x), x)
 --R 
 --R
@@ -1438,7 +1438,7 @@ solve(sin(x) = cos(x), x)
 --R                                       Type: List Equation Expression Integer
 --E 74
 
---S 75 of 201
+--S 75 of 200
 solve(tan(x) = 1, x)
 --R 
 --R
@@ -1451,7 +1451,7 @@ solve(tan(x) = 1, x)
 @
 $x = 0$, $0 [+ n \pi, + n 2 \pi]$
 <<*>>=
---S 76 of 201
+--S 76 of 200
 solve(sin(x) = tan(x), x)
 --R 
 --R
@@ -1462,7 +1462,7 @@ solve(sin(x) = tan(x), x)
 @
 This equation has no solutions
 <<*>>=
---S 77 of 201
+--S 77 of 200
 solve(sqrt(x**2 + 1) = x - 2, x)
 --R 
 --R
@@ -1473,7 +1473,7 @@ solve(sqrt(x**2 + 1) = x - 2, x)
 @
 Solve a system of linear equations
 <<*>>=
---S 78 of 201
+--S 78 of 200
 eq1:=   x +   y +   z =  6
 --R 
 --R
@@ -1481,7 +1481,7 @@ eq1:=   x +   y +   z =  6
 --R                                            Type: Equation Polynomial Integer
 --E 78
 
---S 79 of 201
+--S 79 of 200
 eq2:= 2*x +   y + 2*z = 10
 --R 
 --R
@@ -1489,7 +1489,7 @@ eq2:= 2*x +   y + 2*z = 10
 --R                                            Type: Equation Polynomial Integer
 --E 79
 
---S 80 of 201
+--S 80 of 200
 eq3:=   x + 3*y +   z = 10
 --R 
 --R
@@ -1500,17 +1500,17 @@ eq3:=   x + 3*y +   z = 10
 @
 Note that the solution is parametric
 <<*>>=
---S 81 of 201
+--S 81 of 200
 solve([eq1, eq2, eq3], [x, y, z])
 --R 
 --R
---R   (81)  [[x= - %BU + 4,y= 2,z= %BU]]
+--I   (81)  [[x= - %BU + 4,y= 2,z= %BU]]
 --R                         Type: List List Equation Fraction Polynomial Integer
 --E 81
 @
 Solve a system of nonlinear equations
 <<*>>=
---S 82 of 201
+--S 82 of 200
 eq1:= x**2*y + 3*y*z - 4 = 0
 --R 
 --R
@@ -1519,7 +1519,7 @@ eq1:= x**2*y + 3*y*z - 4 = 0
 --R                                            Type: Equation Polynomial Integer
 --E 82
 
---S 83 of 201
+--S 83 of 200
 eq2:= -3*x**2*z + 2*y**2 + 1 = 0
 --R 
 --R
@@ -1528,7 +1528,7 @@ eq2:= -3*x**2*z + 2*y**2 + 1 = 0
 --R                                            Type: Equation Polynomial Integer
 --E 83
 
---S 84 of 201
+--S 84 of 200
 eq3:= 2*y*z**2 - z**2 - 1 = 0
 --R 
 --R
@@ -1540,7 +1540,7 @@ eq3:= 2*y*z**2 - z**2 - 1 = 0
 @
 Solving this by hand would be a nightmare
 <<*>>=
---S 85 of 201
+--S 85 of 200
 solve([eq1, eq2, eq3], [x, y, z])
 --R 
 --R
@@ -1562,7 +1562,7 @@ solve([eq1, eq2, eq3], [x, y, z])
 @
 \section{Matrix Algebra}
 <<*>>=
---S 86 of 201
+--S 86 of 200
 m:= matrix([[a, b], [1, a*b]])
 --R 
 --R
@@ -1575,7 +1575,7 @@ m:= matrix([[a, b], [1, a*b]])
 @
 Invert the matrix
 <<*>>=
---S 87 of 201
+--S 87 of 200
 minv:= inverse(m)
 --R 
 --R
@@ -1591,7 +1591,7 @@ minv:= inverse(m)
 --R                          Type: Union(Matrix Fraction Polynomial Integer,...)
 --E 87
 
---S 88 of 201
+--S 88 of 200
 m * minv
 --R 
 --R
@@ -1604,7 +1604,7 @@ m * minv
 @
 Define a Vandermonde matrix (useful for doing polynomial interpolations)
 <<*>>=
---S 89 of 201
+--S 89 of 200
 matrix([[1,    1,    1,    1   ], _
         [w,    x,    y,    z   ], _
         [w**2, x**2, y**2, z**2], _
@@ -1623,7 +1623,7 @@ matrix([[1,    1,    1,    1   ], _
 --R                                              Type: Matrix Polynomial Integer
 --E 89
 
---S 90 of 201
+--S 90 of 200
 determinant(%)
 --R 
 --R
@@ -1645,7 +1645,7 @@ determinant(%)
 @
 The following formula implies a general result
 <<*>>=
---S 91 of 201
+--S 91 of 200
 factor(%)
 --R 
 --R
@@ -1656,7 +1656,7 @@ factor(%)
 @
 Compute the eigenvalues of a matrix from its characteristic polynomial
 <<*>>=
---S 92 of 201
+--S 92 of 200
 m:= matrix([[ 5, -3, -7], _
             [-2,  1,  2], _
             [ 2, -3, -4]])
@@ -1670,7 +1670,7 @@ m:= matrix([[ 5, -3, -7], _
 --R                                                         Type: Matrix Integer
 --E 92
 
---S 93 of 201
+--S 93 of 200
 characteristicPolynomial(m, lambda)
 --R 
 --R
@@ -1679,7 +1679,7 @@ characteristicPolynomial(m, lambda)
 --R                                                     Type: Polynomial Integer
 --E 93
 
---S 94 of 201
+--S 94 of 200
 solve(% = 0, lambda)
 --R 
 --R
@@ -1687,7 +1687,7 @@ solve(% = 0, lambda)
 --R                              Type: List Equation Fraction Polynomial Integer
 --E 94
 
---S 95 of 201
+--S 95 of 200
 m:= 'm;
 --R 
 --R
@@ -1698,7 +1698,7 @@ m:= 'm;
 \section{Sums and Products}
 \subsection{Sums: finite and infinite}
 <<*>>=
---S 96 of 201
+--S 96 of 200
 summation(k**3, k = 1..n)
 --R 
 --R
@@ -1710,7 +1710,7 @@ summation(k**3, k = 1..n)
 --R                                                     Type: Expression Integer
 --E 96
 
---S 97 of 201
+--S 97 of 200
 sum(k**3, k = 1..n)
 --R 
 --R
@@ -1721,7 +1721,7 @@ sum(k**3, k = 1..n)
 --R                                            Type: Fraction Polynomial Integer
 --E 97
 
---S 98 of 201
+--S 98 of 200
 limit(sum(1/k**2 + 1/k**3, k = 1..n), n = %plusInfinity)
 --R 
 --R
@@ -1731,7 +1731,7 @@ limit(sum(1/k**2 + 1/k**3, k = 1..n), n = %plusInfinity)
 @
 \subsection{Products}
 <<*>>=
---S 99 of 201
+--S 99 of 200
 product(k, k = 1..n)
 --R 
 --R
@@ -1747,7 +1747,7 @@ product(k, k = 1..n)
 \section{Calculus}
 \subsection{Limits --- start with a famous example}
 <<*>>=
---S 100 of 201
+--S 100 of 200
 limit((1 + 1/n)**n, n = %plusInfinity)
 --R 
 --R
@@ -1755,7 +1755,7 @@ limit((1 + 1/n)**n, n = %plusInfinity)
 --R                        Type: Union(OrderedCompletion Expression Integer,...)
 --E 100
 
---S 101 of 201
+--S 101 of 200
 limit((1 - cos(x))/x**2, x = 0)
 --R 
 --R
@@ -1769,21 +1769,21 @@ limit((1 - cos(x))/x**2, x = 0)
 Apply the chain rule---this is important for PDEs and many other
 applications
 <<*>>=
---S 102 of 201
+--S 102 of 200
 y:= operator('y);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 102
 
---S 103 of 201
+--S 103 of 200
 x:= operator('x);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 103
 
---S 104 of 201
+--S 104 of 200
 D(y(x(t)), t, 2)
 --R 
 --R
@@ -1798,7 +1798,7 @@ D(y(x(t)), t, 2)
 @
 \subsection{Indefinite Integrals}
 <<*>>=
---S 105 of 201
+--S 105 of 200
 1/(x**3 + 2)
 --R 
 --R
@@ -1812,7 +1812,7 @@ D(y(x(t)), t, 2)
 @
 This would be very difficult to do by hand
 <<*>>=
---S 106 of 201
+--S 106 of 200
 integrate(%, x)
 --R 
 --R
@@ -1830,7 +1830,7 @@ integrate(%, x)
 --R                                          Type: Union(Expression Integer,...)
 --E 106
 
---S 107 of 201
+--S 107 of 200
 D(%, x)
 --R 
 --R
@@ -1844,7 +1844,7 @@ D(%, x)
 @
 This example involves several symbolic parameters
 <<*>>=
---S 108 of 201
+--S 108 of 200
 integrate(1/(a + b*cos(x)), x)
 --R 
 --R
@@ -1870,7 +1870,7 @@ integrate(1/(a + b*cos(x)), x)
 --R                                     Type: Union(List Expression Integer,...)
 --E 108
 
---S 109 of 201
+--S 109 of 200
 map(simplify, map(f +-> D(f, x), %))
 --R 
 --R
@@ -1883,7 +1883,7 @@ map(simplify, map(f +-> D(f, x), %))
 @
 Calculus on a non-smooth (but well defined) function
 <<*>>=
---S 110 of 201
+--S 110 of 200
 D(abs(x), x)
 --R 
 --R
@@ -1893,13 +1893,13 @@ D(abs(x), x)
 --R                                                     Type: Expression Integer
 --E 110
 
---S 111 of 201
+--S 111 of 200
 integrate(abs(x), x)
 --R 
 --R
 --R             x
 --R           ++
---R   (111)   |   abs(%J)d%J
+--I   (111)   |   abs(%J)d%J
 --R          ++
 --R                                          Type: Union(Expression Integer,...)
 --E 111
@@ -1907,13 +1907,13 @@ integrate(abs(x), x)
 @
 Calculus on a piecewise defined function
 <<*>>=
---S 112 of 201
+--S 112 of 200
 a(x) == if x < 0 then -x else x
 --R 
 --R                                                                   Type: Void
 --E 112
 
---S 113 of 201
+--S 113 of 200
 D(a(x), x)
 --R 
 --R   Compiling function a with type Variable x -> Polynomial Integer 
@@ -1922,7 +1922,7 @@ D(a(x), x)
 --R                                                     Type: Polynomial Integer
 --E 113
 
---S 114 of 201
+--S 114 of 200
 integrate(a(x), x)
 --R 
 --R
@@ -1938,7 +1938,7 @@ integrate(a(x), x)
 The following two integrals should be equivalent.  The correct solution is
 $[(1 + x)^(3/2) + (1 - x)^(3/2)] / 3$
 <<*>>=
---S 115 of 201
+--S 115 of 200
 integrate(x/(sqrt(1 + x) + sqrt(1 - x)), x)
 --R 
 --R
@@ -1949,7 +1949,7 @@ integrate(x/(sqrt(1 + x) + sqrt(1 - x)), x)
 --R                                          Type: Union(Expression Integer,...)
 --E 115
 
---S 116 of 201
+--S 116 of 200
 integrate((sqrt(1 + x) - sqrt(1 - x))/2, x)
 --R 
 --R
@@ -1964,7 +1964,7 @@ integrate((sqrt(1 + x) - sqrt(1 - x))/2, x)
 \subsection{Definite Integrals}
 The following two functions have a pole at zero
 <<*>>=
---S 117 of 201
+--S 117 of 200
 integrate(1/x, x = -1..1)
 --R 
 --R 
@@ -1976,7 +1976,7 @@ integrate(1/x, x = -1..1)
 --R
 --E 117
 
---S 118 of 201
+--S 118 of 200
 integrate(1/x**2, x = -1..1)
 --R 
 --R 
@@ -1993,7 +1993,7 @@ Different branches of the square root need to be chosen in the intervals
 [0, 1] and [1, 2].  The correct results are $4/3$, $[4 - \sqrt{(8)}]/3$,
 $[8 - \sqrt{(8)}]/3$, respectively.
 <<*>>=
---S 119 of 201
+--S 119 of 200
 integrate(sqrt(x + 1/x - 2), x = 0..1)
 --R 
 --R
@@ -2001,7 +2001,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..1)
 --R                                         Type: Union(pole: potentialPole,...)
 --E 119
 
---S 120 of 201
+--S 120 of 200
 integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole")
 --R 
 --R
@@ -2011,7 +2011,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole")
 --R                    Type: Union(f1: OrderedCompletion Expression Integer,...)
 --E 120
 
---S 121 of 201
+--S 121 of 200
 integrate(sqrt(x + 1/x - 2), x = 1..2)
 --R 
 --R
@@ -2019,7 +2019,7 @@ integrate(sqrt(x + 1/x - 2), x = 1..2)
 --R                                         Type: Union(pole: potentialPole,...)
 --E 121
 
---S 122 of 201
+--S 122 of 200
 integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole")
 --R 
 --R
@@ -2030,7 +2030,7 @@ integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole")
 --R                    Type: Union(f1: OrderedCompletion Expression Integer,...)
 --E 122
 
---S 123 of 201
+--S 123 of 200
 integrate(sqrt(x + 1/x - 2), x = 0..2)
 --R 
 --R
@@ -2038,7 +2038,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..2)
 --R                                         Type: Union(pole: potentialPole,...)
 --E 123
 
---S 124 of 201
+--S 124 of 200
 integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole")
 --R 
 --R
@@ -2052,7 +2052,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole")
 @
 \subsection{Contour integrals}
 <<*>>=
---S 125 of 201
+--S 125 of 200
 integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity)
 --R 
 --R
@@ -2060,7 +2060,7 @@ integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity)
 --R                                         Type: Union(pole: potentialPole,...)
 --E 125
 
---S 126 of 201
+--S 126 of 200
 integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity, "noPole")
 --R 
 --R
@@ -2071,7 +2071,7 @@ integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity, "noPole")
 @
 \subsection{Integrand with a branch point}
 <<*>>=
---S 127 of 201
+--S 127 of 200
 integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity)
 --R 
 --R
@@ -2079,7 +2079,7 @@ integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity)
 --R                                         Type: Union(pole: potentialPole,...)
 --E 127
 
---S 128 of 201
+--S 128 of 200
 integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity, "noPole")
 --R 
 --R
@@ -2090,7 +2090,7 @@ integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity, "noPole")
 @
 Multiple integrals: volume of a tetrahedron
 <<*>>=
---S 129 of 201
+--S 129 of 200
 integrate(integrate(integrate(1, z = 0..c*(1 - x/a - y/b)), _
                     y = 0..b*(1 - x/a)), _
           x = 0..a)
@@ -2106,7 +2106,7 @@ integrate(integrate(integrate(1, z = 0..c*(1 - x/a - y/b)), _
 \subsection{Series}
 Taylor series---this first example comes from special relativity
 <<*>>=
---S 130 of 201
+--S 130 of 200
 1/sqrt(1 - (v/c)**2)
 --R 
 --R
@@ -2121,7 +2121,7 @@ Taylor series---this first example comes from special relativity
 --R                                                     Type: Expression Integer
 --E 130
 
---S 131 of 201
+--S 131 of 200
 series(%, v = 0)
 --R 
 --R
@@ -2132,7 +2132,7 @@ series(%, v = 0)
 --R                        Type: UnivariatePuiseuxSeries(Expression Integer,v,0)
 --E 131
 
---S 132 of 201
+--S 132 of 200
 1/%**2
 --R 
 --R
@@ -2143,7 +2143,7 @@ series(%, v = 0)
 --R                        Type: UnivariatePuiseuxSeries(Expression Integer,v,0)
 --E 132
 
---S 133 of 201
+--S 133 of 200
 tsin:= series(sin(x), x = 0)
 --R 
 --R
@@ -2153,7 +2153,7 @@ tsin:= series(sin(x), x = 0)
 --R                        Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
 --E 133 
 
---S 134 of 201
+--S 134 of 200
 tcos:= series(cos(x), x = 0)
 --R 
 --R
@@ -2166,7 +2166,7 @@ tcos:= series(cos(x), x = 0)
 @
 Note that additional terms will be computed as needed
 <<*>>=
---S 135 of 201
+--S 135 of 200
 tsin/tcos
 --R 
 --R
@@ -2176,7 +2176,7 @@ tsin/tcos
 --R                        Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
 --E 135
 
---S 136 of 201
+--S 136 of 200
 series(tan(x), x = 0)
 --R 
 --R
@@ -2192,7 +2192,7 @@ Look at the Taylor series around $x = 1$
 
 )set streams calculate 1
 
---S 137  of 201
+--S 137 of 200
 log(x)**a*exp(-b*x)
 --R 
 --R
@@ -2201,7 +2201,7 @@ log(x)**a*exp(-b*x)
 --R                                                     Type: Expression Integer
 --E 137
 
---S 138 of 201
+--S 138 of 200
 series(%, x = 1)
 --R 
 --R 
@@ -2218,7 +2218,7 @@ series(%, x = 1)
 @ 
 Compare the Taylor series of two different formulations of a function
 <<*>>=
---S 139 of 201
+--S 139 of 200
 taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
 --R 
 --R 
@@ -2230,7 +2230,7 @@ taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
 --R
 --E 139
 
---S 140 of 201
+--S 140 of 200
 % - taylor(log(sinh(z) * cosh(z + w)), z = 0)
 --R 
 --R 
@@ -2246,7 +2246,7 @@ taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
 \subsection{Power series}
 Compute the general formula
 <<*>>=
---S 141 of 201
+--S 141 of 200
 log(sin(x)/x)
 --R 
 --R
@@ -2256,7 +2256,7 @@ log(sin(x)/x)
 --R                                                     Type: Expression Integer
 --E 141
 
---S 142 of 201
+--S 142 of 200
 series(%, x = 0)
 --R 
 --R
@@ -2266,7 +2266,7 @@ series(%, x = 0)
 --R                        Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
 --E 142
 
---S 143 of 201
+--S 143 of 200
 exp(-x)*sin(x)
 --R 
 --R
@@ -2275,7 +2275,7 @@ exp(-x)*sin(x)
 --R                                                     Type: Expression Integer
 --E 143
 
---S 144 of 201
+--S 144 of 200
 series(%, x = 0)
 --R 
 --R
@@ -2289,14 +2289,14 @@ series(%, x = 0)
 Derive an explicit Taylor series solution of y as a function of x from the
 following implicit relation
 <<*>>=
---S 145 of 201
+--S 145 of 200
 y:= operator('y);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 145
 
---S 146 of 201
+--S 146 of 200
 x = sin(y(x)) + cos(y(x))
 --R 
 --R
@@ -2304,7 +2304,7 @@ x = sin(y(x)) + cos(y(x))
 --R                                            Type: Equation Expression Integer
 --E 146
 
---S 147 of 201
+--S 147 of 200
 seriesSolve(%, y, x = 1, 0)
 --R 
 --R 
@@ -2321,7 +2321,7 @@ seriesSolve(%, y, x = 1, 0)
 @ 
 \subsection{Pade (rational function) approximation}
 <<*>>=
---S 148 of 201
+--S 148 of 200
 pade(1, 1, taylor(exp(-x), x = 0))
 --R 
 --R
@@ -2335,7 +2335,7 @@ pade(1, 1, taylor(exp(-x), x = 0))
 \section{Transforms}
 \subsection{Laplace and inverse Laplace transforms}
 <<*>>=
---S 149 of 201
+--S 149 of 200
 laplace(cos((w - 1)*t), t, s)
 --R 
 --R
@@ -2346,7 +2346,7 @@ laplace(cos((w - 1)*t), t, s)
 --R                                                     Type: Expression Integer
 --E 149
 
---S 150 of 201
+--S 150 of 200
 inverseLaplace(%, s, t)
 --R 
 --R
@@ -2360,14 +2360,14 @@ inverseLaplace(%, s, t)
 \section{Difference and Differential Equations}
 \subsection{Second order linear recurrence equation}
 <<*>>=
---S 151 of 201
+--S 151 of 200
 r:= operator('r);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 151
 
---S 152 of 201
+--S 152 of 200
 r(n + 2) - 2 * r(n + 1) + r(n) = 2
 --R 
 --R
@@ -2375,7 +2375,7 @@ r(n + 2) - 2 * r(n + 1) + r(n) = 2
 --R                                            Type: Equation Expression Integer
 --E 152
 
---S 153 of 201
+--S 153 of 200
 [%, r(0) = 1, r(1) = m]
 --R 
 --R
@@ -2389,14 +2389,14 @@ r(n + 2) - 2 * r(n + 1) + r(n) = 2
 \subsection{Second order ODE with initial conditions}
 solve first using Laplace transforms
 <<*>>=
---S 154 of 201
+--S 154 of 200
 f:= operator('f);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 154
 
---S 155 of 201
+--S 155 of 200
 ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t)
 --R 
 --R
@@ -2406,7 +2406,7 @@ ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t)
 --R                                            Type: Equation Expression Integer
 --E 155
 
---S 156 of 201
+--S 156 of 200
 map(e +-> laplace(e, t, s), %)
 --R 
 --R
@@ -2420,7 +2420,7 @@ map(e +-> laplace(e, t, s), %)
 @
 Now, solve the ODE directly
 <<*>>=
---S 157 of 201
+--S 157 of 200
 solve(ode, f, t = 0, [0, 0])
 --R 
 --R
@@ -2433,14 +2433,14 @@ solve(ode, f, t = 0, [0, 0])
 @
 \subsection{First order linear ODE}
 <<*>>=
---S 158 of 201
+--S 158 of 200
 y:= operator('y);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 158
 
---S 159 of 201
+--S 159 of 200
 x**2 * D(y(x), x) + 3*x*y(x) = sin(x)/x
 --R 
 --R
@@ -2450,7 +2450,7 @@ x**2 * D(y(x), x) + 3*x*y(x) = sin(x)/x
 --R                                            Type: Equation Expression Integer
 --E 159
 
---S 160 of 201
+--S 160 of 200
 solve(%, y, x)
 --R 
 --R
@@ -2464,7 +2464,7 @@ solve(%, y, x)
 @
 \subsection{Nonlinear ODE}
 <<*>>=
---S 161 of 201
+--S 161 of 200
 D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0
 --R 
 --R
@@ -2474,7 +2474,7 @@ D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0
 --R                                            Type: Equation Expression Integer
 --E 161
 
---S 162 of 201
+--S 162 of 200
 solve(%, y, x)
 --R 
 --R 
@@ -2489,7 +2489,7 @@ solve(%, y, x)
 @
 A simple parametric ODE
 <<*>>=
---S 163 of 201
+--S 163 of 200
 D(y(x, a), x) = a*y(x, a)
 --R 
 --R
@@ -2498,7 +2498,7 @@ D(y(x, a), x) = a*y(x, a)
 --R                                            Type: Equation Expression Integer
 --E 163
 
---S 164 of 201
+--S 164 of 200
 solve(%, y, x);
 --R 
 --R 
@@ -2515,7 +2515,7 @@ solve(%, y, x);
 This problem has nontrivial solutions
 $y(x) = A \sin([\pi/2 + n \pi] x)$ for $n$ an arbitrary integer.
 <<*>>=
---S 165 of 201
+--S 165 of 200
 solve(D(y(x), x, 2) + k**2*y(x) = 0, y, x)
 --R 
 --R
@@ -2528,14 +2528,14 @@ solve(D(y(x), x, 2) + k**2*y(x) = 0, y, x)
 @
 \subsection{System of two linear, constant coefficient ODEs}
 <<*>>=
---S 166 of 201
+--S 166 of 200
 x:= operator('x);
 --R 
 --R
 --R                                                          Type: BasicOperator
 --E 166
 
---S 167 of 201
+--S 167 of 200
 system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) + y(t)]
 --R 
 --R
@@ -2548,7 +2548,7 @@ system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) + y(t)]
 @
 Check the answer. Triangular system of two ODEs
 <<*>>=
---S 168 of 201
+--S 168 of 200
 system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _
           D(y(t), t) = x(t) - y(t)]
 --R 
@@ -2562,8 +2562,8 @@ system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _
 @
 Try solving this system one equation at a time
 <<*>>=
---S 169 of 201
-solve(system.1, x, t)
+--S 169 of 200
+s:=solve(system.1, x, t)
 --R 
 --R
 --R                                   t            t
@@ -2571,185 +2571,125 @@ solve(system.1, x, t)
 --RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
 --E 169
 
---S 170 of 201
-isTimes(subst(%.basis.1, cos(t) = sqrt(1 - sin(t)**2)))
+--S 170 of 200
+eq1 := x(t) = C1 * s.basis.1
 --R 
 --R
---R   (162)  "failed"
---R                                                    Type: Union("failed",...)
+--R                     t               t
+--R   (162)  x(t)= C1 %e sin(t) + 2C1 %e
+--R                                            Type: Equation Expression Integer
 --E 170
 
---S 171 of 201
-reduce(*, cons(subst(
-   factors(factor(subst(%.1**2, sin(t) = u) :: Polynomial Integer)).1.factor,
-                     u = sin(t)),
-               rest(%)))
---R 
---R   There are 30 exposed and 3 unexposed library operations named elt 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                               )display op elt
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
+--S 171 of 200
+s1:=solve(map(e +-> subst(e, eq1), system.2), y, t)
 --R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named elt 
---R      with argument type(s) 
---R                                   failed
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (163)
+--R                      - t   t 2                              - t   t 2
+--R                2C1 %e   (%e ) sin(t) + (- C1 cos(t) + 5C1)%e   (%e )
+--R   [particular= ------------------------------------------------------,
+--R                                           5
+--R              - t
+--R    basis= [%e   ]]
+--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
 --E 171
 
---S 172 of 201
-x(t) = C1 * %
---R 
---R   There are 34 exposed and 23 unexposed library operations named * 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                                )display op *
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
+--S 172 of 200
+eq2 := y(t) = simplify(s1.particular) + C2 * s1.basis.1
 --R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named * 
---R      with argument type(s) 
---R                                 Variable C1
---R                                   failed
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R                      t                              t         - t
+--R                2C1 %e sin(t) + (- C1 cos(t) + 5C1)%e  + 5C2 %e
+--R   (164)  y(t)= --------------------------------------------------
+--R                                         5
+--R                                            Type: Equation Expression Integer
 --E 172
 
---S 173 of 201
-solve(map(e +-> subst(e, %), system.2), y, t)
---R 
---R   There are 3 exposed and 0 unexposed library operations named subst 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                              )display op subst
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R   Cannot find a definition or applicable library operation named subst
---R      with argument type(s) 
---R                             Expression Integer
---R                   Union(List Expression Integer,"failed")
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
---R   AXIOM will attempt to step through and interpret the code.
+--S 173 of 200
+map(e +-> rightZero eval(e, [eq1, D(eq1,t), eq2 , D(eq2,t)]), system)
 --R 
---RDaly Bug
---R   Anonymous user functions created with +-> that are processed in 
---R      interpret-code mode must have result target information 
---R      available. This information is not present so AXIOM cannot 
---R      proceed any further. This may be remedied by declaring the 
---R      function.
+--R
+--R   (165)  [0= 0,0= 0]
+--R                                       Type: List Equation Expression Integer
 --E 173
-
---S 174 of 201
-y(t) = simplify(%.particular) + C2 * %.basis.1
---R 
---R   There are 30 exposed and 3 unexposed library operations named elt 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                               )display op elt
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named elt 
---R      with argument type(s) 
---R                                   failed
---R                             Variable particular
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
---E 174
-
 )clear properties x y
  
 @
 \section{Operators}
 \subsection{Linear differential operator}
 <<*>>=
---S 175 of 201
+--S 174 of 200
 DD:= operator("D") :: Operator(Expression Integer)
 --R 
 --R
---R   (163)  D
+--R   (166)  D
 --R                                            Type: Operator Expression Integer
---E 175
+--E 174
 
---S 176 of 201
+--S 175 of 200
 evaluate(DD, e +-> D(e, x))$Operator(Expression Integer)
 --R 
 --R
---R   (164)  D
+--R   (167)  D
 --R                                            Type: Operator Expression Integer
---E 176
+--E 175
 
---S 177 of 201
+--S 176 of 200
 L:= (DD - 1) * (DD + 2)
 --R 
 --R
 --R                 2
---R   (165)  D 2 + D  - D - 2
+--R   (168)  D 2 + D  - D - 2
 --R                                            Type: Operator Expression Integer
---E 177
+--E 176
 
---S 178 of 201
+--S 177 of 200
 g:= operator('g)
 --R 
 --R
---R   (166)  g
+--R   (169)  g
 --R                                                          Type: BasicOperator
---E 178
+--E 177
 
---S 179 of 201
+--S 178 of 200
 L(f(x))
 --R 
 --R
 --R           ,,       ,
---R   (167)  f  (x) + f (x) - 2f(x)
+--R   (170)  f  (x) + f (x) - 2f(x)
 --R
 --R                                                     Type: Expression Integer
---E 179
+--E 178
 
---S 180 of 201
+--S 179 of 200
 subst(L(subst(g(y), y = x)), x = y)
 --R 
 --R
 --R           ,,       ,
---R   (168)  g  (y) + g (y) - 2g(y)
+--R   (171)  g  (y) + g (y) - 2g(y)
 --R
 --R                                                     Type: Expression Integer
---E 180
+--E 179
 
---S 181 of 201
+--S 180 of 200
 subst(L(subst(A * sin(z**2), z = x)), x = z)
 --R 
 --R
 --R                 2           2                    2
---R   (169)  (- 4A z  - 2A)sin(z ) + (2A z + 2A)cos(z )
+--R   (172)  (- 4A z  - 2A)sin(z ) + (2A z + 2A)cos(z )
 --R                                                     Type: Expression Integer
---E 181
+--E 180
 
 @
 \subsection{Truncated Taylor series operator}
 <<*>>=
---S 182 of 201
+--S 181 of 200
 T:= (f, xx, a) +-> subst((DD**0)(f(x)), x = a)/factorial(0) * (xx - a)**0 + _
                    subst((DD**1)(f(x)), x = a)/factorial(1) * (xx - a)**1 + _
                    subst((DD**2)(f(x)), x = a)/factorial(2) * (xx - a)**2
 --R 
 --R
---R   (170)
+--R   (173)
 --R     (f,xx,a)
 --R   +-> 
 --R               0                                 1
@@ -2762,70 +2702,70 @@ T:= (f, xx, a) +-> subst((DD**0)(f(x)), x = a)/factorial(0) * (xx - a)**0 + _
 --R       --------------------- (xx - a)
 --R            factorial(2)
 --R                                                      Type: AnonymousFunction
---E 182
+--E 181
 
---S 183 of 201
+--S 182 of 200
 T(f, x, a)
 --R 
 --R
 --R            2           2  ,,                ,
 --R          (x  - 2a x + a )f  (a) + (2x - 2a)f (a) + 2f(a)
 --R
---R   (171)  -----------------------------------------------
+--R   (174)  -----------------------------------------------
 --R                                 2
 --R                                                     Type: Expression Integer
---E 183
+--E 182
 
---S 184 of 201
+--S 183 of 200
 T(g, y, b)
 --R 
 --R
 --R            2           2  ,,                ,
 --R          (y  - 2b y + b )g  (b) + (2y - 2b)g (b) + 2g(b)
 --R
---R   (172)  -----------------------------------------------
+--R   (175)  -----------------------------------------------
 --R                                 2
 --R                                                     Type: Expression Integer
---E 184
+--E 183
 
---S 185 of 201
+--S 184 of 200
 Sin:= operator("sin") :: Operator(Expression Integer)
 --R 
 --R
---R   (173)  sin
+--R   (176)  sin
 --R                                            Type: Operator Expression Integer
---E 185
+--E 184
 
---S 186 of 201
+--S 185 of 200
 evaluate(Sin, x +-> sin(x))$Operator(Expression Integer)
 --R 
 --R
---R   (174)  sin
+--R   (177)  sin
 --R                                            Type: Operator Expression Integer
---E 186
+--E 185
 
---S 187 of 201
+--S 186 of 200
 T(Sin, z, c)
 --R 
 --R
 --R              2           2
 --R          (- z  + 2c z - c  + 2)sin(c) + (2z - 2c)cos(c)
---R   (175)  ----------------------------------------------
+--R   (178)  ----------------------------------------------
 --R                                 2
 --R                                                     Type: Expression Integer
---E 187
+--E 186
 
 @
 \section{Programming}
 Write a simple program to compute Legendre polynomials
 <<*>>=
---S 188 of 201
+--S 187 of 200
 p(n, x) == 1/(2**n*factorial(n)) * D((x**2 - 1)**n, x, n)
 --R 
 --R                                                                   Type: Void
---E 188
+--E 187
 
---S 189 of 201
+--S 188 of 200
 for i in 0..4 repeat {  output("");    output(concat(["p(", string(i), ", x) = "]));    output(p(i, x))}
 --R 
 --R   Compiling function p with type (NonNegativeInteger,Variable x) -> 
@@ -2852,40 +2792,40 @@ for i in 0..4 repeat {  output("");    output(concat(["p(", string(i), ", x) = "
 --R   -- x  - -- x  + -
 --R    8       4      8
 --R                                                                   Type: Void
---E 189
+--E 188
 
---S 190 of 201
+--S 189 of 200
 eval(p(4, x), x = 1)
 --R 
 --R   Compiling function p with type (PositiveInteger,Variable x) -> 
 --R      Polynomial Fraction Integer 
 --R
---R   (178)  1
+--R   (181)  1
 --R                                            Type: Polynomial Fraction Integer
---E 190
+--E 189
 
 @
 Now, perform the same computation using a recursive definition
 <<*>>=
---S 191 of 201
+--S 190 of 200
 pp(0, x) == 1
 --R 
 --R                                                                   Type: Void
---E 191
+--E 190
 
---S 192 of 201
+--S 191 of 200
 pp(1, x) == x
 --R 
 --R                                                                   Type: Void
---E 192
+--E 191
 
---S 193 of 201
+--S 192 of 200
 pp(n, x) == ((2*n - 1)*x*pp(n - 1, x) - (n - 1)*pp(n - 2, x))/n
 --R 
 --R                                                                   Type: Void
---E 193
+--E 192
 
---S 194 of 201
+--S 193 of 200
 for i in 0..4 repeat {   output("");   output(concat(["pp(", string(i), ", x) = "]));   output(pp(i, x))}
 --R 
 --R   Compiling function pp with type (Integer,Variable x) -> Polynomial 
@@ -2912,7 +2852,7 @@ for i in 0..4 repeat {   output("");   output(concat(["pp(", string(i), ", x) =
 --R   -- x  - -- x  + -
 --R    8       4      8
 --R                                                                   Type: Void
---E 194
+--E 193
 
 )clear properties p pp
 
@@ -2921,31 +2861,31 @@ for i in 0..4 repeat {   output("");   output(concat(["pp(", string(i), ", x) =
 \subsection{Horner's rule}
 This is important for numerical algorithms
 <<*>>=
---S 195 of 201
+--S 194 of 200
 a:= operator('a)
 --R 
 --R
---R   (183)  a
+--R   (186)  a
 --R                                                          Type: BasicOperator
---E 195
+--E 194
 
---S 196 of 201
+--S 195 of 200
 sum(a(i)*x**i, i = 1..5)
 --R 
 --R
 --R               5        4        3        2
---R   (184)  a(5)x  + a(4)x  + a(3)x  + a(2)x  + a(1)x
+--R   (187)  a(5)x  + a(4)x  + a(3)x  + a(2)x  + a(1)x
 --R                                                     Type: Expression Integer
---E 196
+--E 195
 
---S 197 of 201
+--S 196 of 200
 p:= factor(%)
 --R 
 --R
 --R               5        4        3        2
---R   (185)  a(5)x  + a(4)x  + a(3)x  + a(2)x  + a(1)x
+--R   (188)  a(5)x  + a(4)x  + a(3)x  + a(2)x  + a(1)x
 --R                                            Type: Factored Expression Integer
---E 197
+--E 196
 
 @
 Convert the result into FORTRAN syntax
@@ -2953,40 +2893,40 @@ Convert the result into FORTRAN syntax
 
 )set fortran ints2floats off
 
---S 198  of 201
+--S 197 of 200
 outputAsFortran('p = p)
 --R 
 --R      p=a(5)*x**5+a(4)*x**4+a(3)*x**3+a(2)*x*x+a(1)*x
 --R                                                                   Type: Void
---E 198
+--E 197
 
 @
 \section{Boolean Logic}
 \subsection{Simplify logical expressions}
 <<*>>=
---S 199 of 201
+--S 198 of 200
 true and false
 --R 
 --R
---R   (187)  false
+--R   (190)  false
 --R                                                                Type: Boolean
---E 199
+--E 198
 
---S 200 of 201
+--S 199 of 200
 x or (not x)
 --R 
 --R 
 --RDaly Bug
 --R   Argument number 1 to "or" must be a Boolean.
---E 200
+--E 199
 
---S 201 of 201
+--S 200 of 200
 x or y or (x and y)
 --R 
 --R 
 --RDaly Bug
 --R   Argument number 1 to "or" must be a Boolean.
---E 201
+--E 200
 )spool
 )lisp (bye)
  
diff --git a/src/input/exlap.input.pamphlet b/src/input/exlap.input.pamphlet
index 432b063..d290295 100644
--- a/src/input/exlap.input.pamphlet
+++ b/src/input/exlap.input.pamphlet
@@ -37,39 +37,31 @@ laplace((exp(a*t) - exp(b*t))/t, t, s)
 --E 2
 
 --S 3 of 6
-laplace(exp(a*t+b)*ei(c*t), t, s)
+laplace(exp(a*t+b)*Ei(c*t), t, s)
 --R 
---R   There are no library operations named ei 
---R      Use HyperDoc Browse or issue
---R                                 )what op ei
---R      to learn if there is any operation containing " ei " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ei 
---R      with argument type(s) 
---R                             Polynomial Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R          b    s + c - a
+--R        %e log(---------)
+--R                   c
+--R   (3)  -----------------
+--R              s - a
+--R                                                     Type: Expression Integer
 --E 3
 
 )clear all
 
 --S 4 of 6
-laplace(a*ci(b*t) + c*si(d*t), t, s)
+laplace(a*Ci(b*t) + c*Si(d*t), t, s)
 --R 
---R   There are no library operations named ci 
---R      Use HyperDoc Browse or issue
---R                                 )what op ci
---R      to learn if there is any operation containing " ci " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ci 
---R      with argument type(s) 
---R                             Polynomial Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R               2    2
+--R              s  + b             d
+--R        a log(-------) + 2c atan(-)
+--R                  2              s
+--R                 b
+--R   (1)  ---------------------------
+--R                     2s
+--R                                                     Type: Expression Integer
 --E 4
 
 )clear all
diff --git a/src/input/exsum.input.pamphlet b/src/input/exsum.input.pamphlet
index 7c2f9d0..df444c0 100644
--- a/src/input/exsum.input.pamphlet
+++ b/src/input/exsum.input.pamphlet
@@ -93,32 +93,19 @@ sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))
 )clear all
 
 --S 7 of 13
-[1..15]
+[i for i in 1..15]
 --R 
 --R
---R   (1)  [1..15]
---R                                           Type: List Segment PositiveInteger
+--R   (1)  [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
+--R                                                   Type: List PositiveInteger
 --E 7
 
 --S 8 of 13
-reduce(+,[1..15])
---R 
---R   There are 1 exposed and 3 unexposed library operations named reduce 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                             )display op reduce
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
+reduce(+,[i for i in 1..15])
 --R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      reduce with argument type(s) 
---R                                 Variable +
---R                        List Segment PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (2)  120
+--R                                                        Type: PositiveInteger
 --E 8
 
 )clear all
diff --git a/src/input/grpthry.input.pamphlet b/src/input/grpthry.input.pamphlet
index 544b278..b0296a4 100644
--- a/src/input/grpthry.input.pamphlet
+++ b/src/input/grpthry.input.pamphlet
@@ -19,7 +19,7 @@
 )set message auto off
 )clear all
  
---S 1 of 67
+--S 1 of 68
 x : PERM INT := [[1,3,5],[7,11,9]]
 --R 
 --R
@@ -27,7 +27,7 @@ x : PERM INT := [[1,3,5],[7,11,9]]
 --R                                                    Type: Permutation Integer
 --E 1
 
---S 2 of 67
+--S 2 of 68
 y : PERM INT := [[3,5,7,9]]
 --R 
 --R
@@ -35,7 +35,7 @@ y : PERM INT := [[3,5,7,9]]
 --R                                                    Type: Permutation Integer
 --E 2
 
---S 3 of 67
+--S 3 of 68
 z : PERM INT := [1,3,11]
 --R 
 --R
@@ -43,155 +43,113 @@ z : PERM INT := [1,3,11]
 --R                                                    Type: Permutation Integer
 --E 3
 
---S 4 of 67
-g1 : PERMGRPS INT := [ x , y ]
+--S 4 of 68
+g1 : PERMGRP INT := [ x , y ]
 --R 
---R 
---RDaly Bug
---R   Category, domain or package constructor PERMGRPS is not available.
+--R
+--R   (4)  <(1 3 5)(7 11 9),(3 5 7 9)>
+--R                                               Type: PermutationGroup Integer
 --E 4
 
---S 5 of 67
-g2 : PERMGRPS INT := [ x , z ]
+--S 5 of 68
+g2 : PERMGRP INT := [ x , z ]
 --R 
---R 
---RDaly Bug
---R   Category, domain or package constructor PERMGRPS is not available.
+--R
+--R   (5)  <(1 3 5)(7 11 9),(1 3 11)>
+--R                                               Type: PermutationGroup Integer
 --E 5
 
---S 6 of 67
-g3 : PERMGRPS INT := [ y , z ]
+--S 6 of 68
+g3 : PERMGRP INT := [ y , z ]
 --R 
---R 
---RDaly Bug
---R   Category, domain or package constructor PERMGRPS is not available.
+--R
+--R   (6)  <(3 5 7 9),(1 3 11)>
+--R                                               Type: PermutationGroup Integer
 --E 6
 
---S 7 of 67
+--S 7 of 68
 order g1
 --R 
---R   There are 9 exposed and 5 unexposed library operations named order 
---R      having 1 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                              )display op order
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named order
---R      with argument type(s) 
---R                                 Variable g1
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (7)  720
+--R                                                        Type: PositiveInteger
 --E 7
 
---S 8 of 67
+--S 8 of 68
 degree g3
 --R 
 --R
---R   (4)  g3
---R                                                Type: IndexedExponents Symbol
+--R   (8)  6
+--R                                                        Type: PositiveInteger
 --E 8
 
---S 9 of 67
+--S 9 of 68
 movedPoints g2
 --R 
---R   There are 2 exposed and 0 unexposed library operations named 
---R      movedPoints having 1 argument(s) but none was determined to be 
---R      applicable. Use HyperDoc Browse, or issue
---R                           )display op movedPoints
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      movedPoints with argument type(s) 
---R                                 Variable g2
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (9)  {1,3,5,7,9,11}
+--R                                                            Type: Set Integer
 --E 9
 
---S 10 of 67
+--S 10 of 68
 orbit (g1, 3)
 --R 
---R   There are 4 exposed and 0 unexposed library operations named orbit 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                              )display op orbit
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named orbit
---R      with argument type(s) 
---R                                 Variable g1
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (10)  {1,3,5,7,9,11}
+--R                                                            Type: Set Integer
 --E 10
 
---S 11 of 67
+--S 11 of 68
 orbits g3
 --R 
---R   There are 1 exposed and 0 unexposed library operations named orbits 
---R      having 1 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                             )display op orbits
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      orbits with argument type(s) 
---R                                 Variable g3
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (11)  {{1,3,5,7,9,11}}
+--R                                                        Type: Set Set Integer
 --E 11
 
---S 12 of 67
+--S 12 of 68
 member? ( y , g2 )
 --R 
---R   There are 2 exposed and 1 unexposed library operations named member?
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                             )display op member?
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      member? with argument type(s) 
---R                             Permutation Integer
---R                                 Variable g2
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (12)  false
+--R                                                                Type: Boolean
 --E 12
 
---S 13 of 67
-)sh PERMGRPS
+--S 13 of 68
+)sh PERMGRP
 --R 
---R   The )show system command is used to display information about types 
---R      or partial types. For example, )show Integer will show 
---R      information about Integer .
---R   
---R      PERMGRPS is not the name of a known type constructor. If you want
---R      to see information about any operations named PERMGRPS , issue
---R                        )display operations PERMGRPS 
+--R PermutationGroup S: SetCategory  is a domain constructor
+--R Abbreviation for PermutationGroup is PERMGRP 
+--R This constructor is exposed in this frame.
+--R Issue )edit permgrps.spad.pamphlet to see algebra source code for PERMGRP 
+--R
+--R------------------------------- Operations --------------------------------
+--R ?<? : (%,%) -> Boolean                ?<=? : (%,%) -> Boolean
+--R ?=? : (%,%) -> Boolean                base : % -> List S
+--R coerce : List Permutation S -> %      coerce : % -> List Permutation S
+--R coerce : % -> OutputForm              degree : % -> NonNegativeInteger
+--R hash : % -> SingleInteger             latex : % -> String
+--R movedPoints : % -> Set S              orbit : (%,List S) -> Set List S
+--R orbit : (%,Set S) -> Set Set S        orbit : (%,S) -> Set S
+--R orbits : % -> Set Set S               order : % -> NonNegativeInteger
+--R random : % -> Permutation S           ?~=? : (%,%) -> Boolean
+--R ?.? : (%,NonNegativeInteger) -> Permutation S
+--R generators : % -> List Permutation S
+--R initializeGroupForWordProblem : (%,Integer,Integer) -> Void
+--R initializeGroupForWordProblem : % -> Void
+--R member? : (Permutation S,%) -> Boolean
+--R permutationGroup : List Permutation S -> %
+--R random : (%,Integer) -> Permutation S
+--R strongGenerators : % -> List Permutation S
+--R wordInGenerators : (Permutation S,%) -> List NonNegativeInteger
+--R wordInStrongGenerators : (Permutation S,%) -> List NonNegativeInteger
+--R wordsForStrongGenerators : % -> List List NonNegativeInteger
+--R
 --E 13
 
 )clear all
 
---S 14 of 67
+--S 14 of 68
 ptn9 := partitions 9
 --R 
 --R
@@ -200,112 +158,1133 @@ ptn9 := partitions 9
 --R                                                    Type: Stream List Integer
 --E 14
 
---S 15 of 67
-map(dimIrrRepSym, ptn9)
---R 
---R   There are 68 exposed and 8 unexposed library operations named map 
---R      having 2 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                               )display op map
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named map 
---R      with argument type(s) 
---R                            Variable dimIrrRepSym
---R                             Stream List Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--S 15 of 68
+map(dimensionOfIrreducibleRepresentation, ptn9)
+--R 
+--R
+--R   (2)  [1,8,27,28,48,105,56,42,162,120,...]
+--R                                              Type: Stream NonNegativeInteger
 --E 15
 
---S 16 of 67
-yt := listYoungTableaus [4,2]; yt :: (LIST TABLEAU I)
---R 
+--S 16 of 68
+yt := listYoungTableaus [4,2]
 --R 
---RDaly Bug
---R   I is not a valid type.
+--R
+--R   (3)
+--R    +0  2  4  5+  +0  2  3  5+  +0  2  3  4+  +0  1  4  5+  +0  1  3  5+
+--R   [|          |, |          |, |          |, |          |, |          |,
+--R    +1  3  0  0+  +1  4  0  0+  +1  5  0  0+  +2  3  0  0+  +2  4  0  0+
+--R    +0  1  3  4+  +0  1  2  5+  +0  1  2  4+  +0  1  2  3+
+--R    |          |, |          |, |          |, |          |]
+--R    +2  5  0  0+  +3  4  0  0+  +3  5  0  0+  +4  5  0  0+
+--R                                                    Type: List Matrix Integer
 --E 16
 
---S 17 of 67
-r1 := irrRepSymNat([4,2],[1,2,4,5,3,6])
---R 
---R   There are no library operations named irrRepSymNat 
---R      Use HyperDoc Browse or issue
---R                            )what op irrRepSymNat
---R      to learn if there is any operation containing " irrRepSymNat " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      irrRepSymNat with argument type(s) 
---R                            List PositiveInteger
---R                            List PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--S 17 of 68
+r1 := irreducibleRepresentation([4,2],[1,2,4,5,3,6])
+--R 
+--R
+--R        + 0   - 1  - 1   0    0    0    0    0    1 +
+--R        |                                           |
+--R        |- 1   0    0    0    0    0    0    0    0 |
+--R        |                                           |
+--R        | 1    1    1    0    0    0    0    0    0 |
+--R        |                                           |
+--R        | 0    1    0    0    0    0    0    0   - 1|
+--R        |                                           |
+--R   (4)  | 0    0    0    0    0    0    1    0    0 |
+--R        |                                           |
+--R        | 0    0    0    0    1    0    0    0    0 |
+--R        |                                           |
+--R        | 1    0    0    0    0    0   - 1  - 1   0 |
+--R        |                                           |
+--R        |- 1  - 1  - 1  - 1  - 1  - 1   0    0    0 |
+--R        |                                           |
+--R        + 0    0    0    1    0    0    0    0    0 +
+--R                                                         Type: Matrix Integer
 --E 17
 
---S 18 of 67
-r2 := irrRepSymNat([4,2],[3,2,1,5,6,4])
---R 
---R   There are no library operations named irrRepSymNat 
---R      Use HyperDoc Browse or issue
---R                            )what op irrRepSymNat
---R      to learn if there is any operation containing " irrRepSymNat " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      irrRepSymNat with argument type(s) 
---R                            List PositiveInteger
---R                            List PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--S 18 of 68
+r2 := irreducibleRepresentation([4,2],[3,2,1,5,6,4])
+--R 
+--R
+--R        + 0    0   - 1   0    0    0    0   - 1   0 +
+--R        |                                           |
+--R        | 1    0    1    0   - 1   0   - 1   0    0 |
+--R        |                                           |
+--R        | 0    0    0    0    1    0    0    0    0 |
+--R        |                                           |
+--R        | 0    0    0    0    0    0    0    1    0 |
+--R        |                                           |
+--R   (5)  |- 1   0    0   - 1   0    0    0    0    0 |
+--R        |                                           |
+--R        | 0    0    0    0    0    0    1    0    0 |
+--R        |                                           |
+--R        | 0    0   - 1   0    0   - 1   0   - 1  - 1|
+--R        |                                           |
+--R        | 0    0    0    0    0    0    0    0    1 |
+--R        |                                           |
+--R        + 0   - 1   0    0   - 1   0   - 1   0    0 +
+--R                                                         Type: Matrix Integer
 --E 18
 
---S 19 of 67
-r3 := irrRepSymNat([4,2],[4,2,1,3,6,5])
---R 
---R   There are no library operations named irrRepSymNat 
---R      Use HyperDoc Browse or issue
---R                            )what op irrRepSymNat
---R      to learn if there is any operation containing " irrRepSymNat " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      irrRepSymNat with argument type(s) 
---R                            List PositiveInteger
---R                            List PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--S 19 of 68
+r3 := irreducibleRepresentation([4,2],[4,2,1,3,6,5])
+--R 
+--R
+--R        +0   0    0   0   1    0    1    0    1 +
+--R        |                                       |
+--R        |0   0    0   0   0    1    0    1    0 |
+--R        |                                       |
+--R        |0   0    0   1   0    0    0    0    0 |
+--R        |                                       |
+--R        |0  - 1   0   0  - 1   0   - 1   0    0 |
+--R        |                                       |
+--R   (6)  |0   0   - 1  0   0   - 1   0   - 1  - 1|
+--R        |                                       |
+--R        |1   1    1   0   0    0    0    0    0 |
+--R        |                                       |
+--R        |0   0    0   0   0    0    0    0    1 |
+--R        |                                       |
+--R        |0   0    0   0   1    0    0    0    0 |
+--R        |                                       |
+--R        +0   0    0   0   0    1    0    0    0 +
+--R                                                         Type: Matrix Integer
 --E 19
 
---S 20 of 67
+--S 20 of 68
 (r3 = r1*r2) :: Boolean
 --R 
 --R
---R   (2)  false
+--R   (7)  false
 --R                                                                Type: Boolean
 --E 20
 
---S 21 of 67
-irrRepSymNat [4,4,1]
+--S 21 of 68
+irreducibleRepresentation [4,4,1]
 --R 
 --R
---R   (3)  irrRepSymNat
---R                    4,4,1
---R                                                                 Type: Symbol
+--R   (8)
+--R   [
+--R   [
+--R     [- 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, 0, 0, 1, 0, - 1,
+--R      0, - 1, - 1, - 1, 0, 1, 1, 0, 0, 0, 0, - 1, 0, - 1, 1, - 1, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, 0, 0, 1, 1, 0,
+--R      1, 1, 0, - 1, - 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, - 1, 0, 1,
+--R      0, 0, 1, 1, 0, 0, - 1, 0, - 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, - 1,
+--R      - 1, 0, 0, 0, 0, 1, 0, 0, - 1, 0, - 1, - 1, 0, 0, 0, - 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      - 1, - 1, 0, 0, 1, 0, 0, 0, 0, 0, - 1, - 1, - 1, 1, - 1, - 1, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 1, 1, 0, 0, 0,
+--R      0, - 1, 0, - 1, 1, - 1, - 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 1, 0,
+--R      0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, - 1, 0, - 1,
+--R      0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, - 1, 0,
+--R      - 1, - 1, 0, 0, 0, - 1, 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
+--R      0, - 1, - 1, - 1, 1, - 1, - 1, - 1, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 1, - 1, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
+--R      0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
+--R      0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      1, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 1, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 1, - 1, - 1, - 1]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
+--R     ,
+--R
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+--R      0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 1]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
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+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
+--R     ]
+--R     ,
+--R
+--R   [
+--R     [- 1, 0, - 1, - 1, 0, 0, 0, - 1, 0, - 1, 1, 0, 0, 1, - 1, 1, 1, - 1, 0, 0,
+--R      0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0,
+--R      - 1, 0, - 1, 0, 0, - 1, - 1, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, - 1, - 1, 0, - 2, - 1, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 1,
+--R      1, 0]
+--R     ,
+--R
+--R     [1, 1, 1, 1, 0, 0, 0, 0, 0, 1, - 1, - 1, - 1, - 1, 0, 0, 0, 1, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,
+--R      1, 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0]
+--R     ,
+--R
+--R     [1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      1, - 1, 1, 1, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,
+--R      0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0]
+--R     ,
+--R
+--R     [- 1, - 1, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      1, - 1, - 1, 0, 0, 0, 1, 1, 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0,
+--R      0, 1, 1, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0,
+--R      - 1, - 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, - 1, - 1, - 1, 0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, - 1, - 1, - 1, 0, - 1, - 1, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, - 1,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, - 1, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, - 2, - 1, - 1, 1, 0, 1, 1, 0, 1,
+--R      1, 0, 1, 0, 0, - 1, - 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0]
+--R     ,
+--R
+--R     [- 1, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, - 1, - 1, - 1, - 1,
+--R      0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [- 1, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, - 1, 0, - 1, 0, 0, 0,
+--R      0, 0, - 1, 0, - 1, 0, 0, - 1, 0, - 1, 1, 1, - 1, 0, 0, 0, - 1, 0, 0, - 1,
+--R      0, 0, - 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+--R     ,
+--R
+--R     [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 1, 1, 0, 0, - 1, - 1,
+--R      0, 0, 0, 0, 1, 1, 1, - 1, - 1, - 1, - 1, 0, 0, 0, 1, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, 1, 1, 1, 1,
+--R      0, 1, 1, 0, - 1, - 1, - 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 1,
+--R      1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, - 1, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0,
+--R      0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, - 1, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 1, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, - 1, - 1, 0,
+--R      0, 0, 0, 0, 1, - 1, 0, 0, 0, - 1, - 1, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 1, 1,
+--R      1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0,
+--R      0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0,
+--R      0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [- 1, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, - 1, 0, - 1,
+--R      0, 0, - 1, - 1, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, - 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0,
+--R      1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [- 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 1, 1,
+--R      0, 0, 0, 0, - 1, - 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1,
+--R      - 1, - 1, 0, - 1, - 1, 0, 0, 0, 0, - 1, - 1, - 1, - 1, 0, - 1, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, - 1, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, - 1, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, - 1, - 1, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0,
+--R      1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      - 1, - 1, - 1, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      - 1, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0,
+--R      1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0,
+--R      - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, - 1, - 1, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, - 1, - 1, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0,
+--R      0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0,
+--R      0, - 1, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0]
+--R     ,
+--R
+--R     [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+--R      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1,
+--R      0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, - 1, 0]
+--R     ]
+--R     ]
+--R                                                    Type: List Matrix Integer
 --E 21
 
 )clear all
 
---S 22 of 67
+--S 22 of 68
 permutationRepresentation [2,3,1,4,6,5,11,10,7,8,9]
 --R 
 --R
@@ -333,7 +1312,7 @@ permutationRepresentation [2,3,1,4,6,5,11,10,7,8,9]
 --R                                                         Type: Matrix Integer
 --E 22
 
---S 23 of 67
+--S 23 of 68
 gm2 := createGenericMatrix 2
 --R 
 --R
@@ -345,7 +1324,7 @@ gm2 := createGenericMatrix 2
 --R                                              Type: Matrix Polynomial Integer
 --E 23
 
---S 24 of 67
+--S 24 of 68
 symmetricTensors (gm2,2)
 --R 
 --R
@@ -362,7 +1341,7 @@ symmetricTensors (gm2,2)
 --R                                              Type: Matrix Polynomial Integer
 --E 24
 
---S 25 of 67
+--S 25 of 68
 gm3 := createGenericMatrix 3
 --R 
 --R
@@ -377,7 +1356,7 @@ gm3 := createGenericMatrix 3
 --R                                              Type: Matrix Polynomial Integer
 --E 25
 
---S 26 of 67
+--S 26 of 68
 antisymmetricTensors (gm3,2)
 --R 
 --R
@@ -392,7 +1371,7 @@ antisymmetricTensors (gm3,2)
 --R                                              Type: Matrix Polynomial Integer
 --E 26
 
---S 27 of 67
+--S 27 of 68
 tensorProduct(gm2,gm2)
 --R 
 --R
@@ -412,7 +1391,7 @@ tensorProduct(gm2,gm2)
 --R                                              Type: Matrix Polynomial Integer
 --E 27
 
---S 28 of 67
+--S 28 of 68
 )sh REP1
 --R 
 --R RepresentationPackage1 R: Ring  is a package constructor
@@ -439,207 +1418,402 @@ tensorProduct(gm2,gm2)
 
 )clear all
 
---S 29 of 67
-r0 := irrRepSymNat [2,2,2,1,1]; r28 := meatAxe (r0::(LIST MATRIX PF 2))
---R 
+--S 29 of 68
+r0 := irreducibleRepresentation [2,2,2,1,1];
 --R 
---RDaly Bug
---R   Cannot convert from type Symbol to List Matrix PrimeField 2 for 
---R      value
---R   irrRepSymNat
---R               2,2,2,1,1
 --R
+--R                                                    Type: List Matrix Integer
 --E 29
 
---S 30 of 67
-areEquivalent? (r28.1, r28.2)
+--S 30 of 68
+r28 := meatAxe (r0::(LIST MATRIX PF 2))
 --R 
---R   There are no library operations named r28 
---R      Use HyperDoc Browse or issue
---R                                )what op r28
---R      to learn if there is any operation containing " r28 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named r28 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is singular
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R   (2)
+--R   [
+--R      +0  1  1  1  1  1  1  0  0  1  1  1  0  0+
+--R      |                                        |
+--R      |1  0  1  1  1  0  0  1  1  1  0  0  1  1|
+--R      |                                        |
+--R      |1  1  0  1  0  1  0  0  1  0  1  0  0  1|
+--R      |                                        |
+--R      |1  1  1  0  0  0  1  1  0  0  0  1  1  0|
+--R      |                                        |
+--R      |1  1  0  0  0  1  1  1  1  1  1  1  1  1|
+--R      |                                        |
+--R      |1  0  1  0  1  0  1  0  1  1  1  1  0  1|
+--R      |                                        |
+--R      |1  0  0  1  1  1  0  1  0  1  1  1  1  0|
+--R     [|                                        |,
+--R      |0  1  1  0  1  1  0  1  0  1  1  0  0  0|
+--R      |                                        |
+--R      |0  1  0  1  1  0  1  0  1  1  0  1  0  0|
+--R      |                                        |
+--R      |1  1  0  0  1  0  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |1  0  1  0  0  1  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |1  0  0  1  0  0  1  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |0  1  1  0  0  0  0  1  0  0  0  0  0  0|
+--R      |                                        |
+--R      +0  1  0  1  0  0  0  0  1  0  0  0  0  0+
+--R      +1  1  1  1  0  0  0  0  0  0  0  0  0  0+
+--R      |                                        |
+--R      |1  1  1  0  0  0  1  0  0  1  1  0  0  0|
+--R      |                                        |
+--R      |1  1  1  0  0  0  0  0  1  1  0  0  1  0|
+--R      |                                        |
+--R      |1  1  1  0  0  0  0  1  1  0  1  0  1  0|
+--R      |                                        |
+--R      |1  1  1  0  0  0  0  1  0  1  1  1  1  0|
+--R      |                                        |
+--R      |1  1  1  0  0  1  0  0  0  1  1  0  1  1|
+--R      |                                        |
+--R      |1  1  1  0  1  0  0  0  0  1  1  0  0  1|
+--R      |                                        |]
+--R      |1  1  0  0  0  0  0  0  0  0  1  1  1  1|
+--R      |                                        |
+--R      |1  0  1  0  0  0  0  0  0  1  0  1  0  1|
+--R      |                                        |
+--R      |0  0  0  1  0  0  1  0  0  1  1  1  1  0|
+--R      |                                        |
+--R      |0  0  0  1  0  0  0  0  1  1  1  0  1  1|
+--R      |                                        |
+--R      |0  0  0  1  0  0  0  1  1  1  1  0  0  1|
+--R      |                                        |
+--R      |0  0  0  0  0  0  1  0  1  0  1  1  1  1|
+--R      |                                        |
+--R      +0  0  0  0  0  0  1  1  1  1  0  1  0  1+
+--R     ,
+--R
+--R      +1  0  0  0  0  0  0  0  1  1  1  1  1  1+
+--R      |                                        |
+--R      |0  1  0  0  0  0  0  0  1  1  1  0  0  0|
+--R      |                                        |
+--R      |0  0  1  0  0  1  1  0  1  0  0  1  0  0|
+--R      |                                        |
+--R      |0  0  0  1  0  1  0  1  0  1  0  0  1  0|
+--R      |                                        |
+--R      |0  0  0  0  1  0  1  1  1  1  0  0  0  1|
+--R      |                                        |
+--R      |0  0  0  0  0  1  1  1  1  1  0  1  1  0|
+--R      |                                        |
+--R      |0  0  0  0  0  1  1  1  1  0  1  1  0  1|
+--R     [|                                        |,
+--R      |0  0  0  0  0  1  1  1  0  1  1  0  1  1|
+--R      |                                        |
+--R      |0  0  0  0  0  1  1  0  1  1  1  1  0  0|
+--R      |                                        |
+--R      |0  0  0  0  0  1  0  1  1  1  1  0  1  0|
+--R      |                                        |
+--R      |0  0  0  0  0  0  1  1  1  1  1  1  1  0|
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  0  0  0  0  0  1  1|
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  0  0  0  0  1  0  1|
+--R      |                                        |
+--R      +0  0  0  0  0  0  0  0  0  0  0  0  0  1+
+--R      +0  0  1  1  1  1  1  0  1  0  0  0  0  0+
+--R      |                                        |
+--R      |0  0  1  0  0  0  0  0  0  0  0  0  1  1|
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  1  0  1  0  0  1  0|
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  1  0  0  1  0  0  1|
+--R      |                                        |
+--R      |0  0  1  0  0  0  0  0  0  1  1  0  1  1|
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  1  1  0  0  1  0  0|
+--R      |                                        |
+--R      |0  0  0  0  0  0  1  0  0  1  0  1  0  0|
+--R      |                                        |]
+--R      |1  1  0  0  0  1  0  0  0  0  1  1  0  0|
+--R      |                                        |
+--R      |0  0  1  1  0  0  1  0  1  0  0  0  1  0|
+--R      |                                        |
+--R      |1  0  1  0  1  1  0  0  1  0  0  0  0  1|
+--R      |                                        |
+--R      |1  0  1  1  1  0  0  0  1  0  0  0  1  1|
+--R      |                                        |
+--R      |0  0  1  1  0  0  1  1  1  1  0  1  1  0|
+--R      |                                        |
+--R      |0  1  1  0  1  1  0  1  1  0  1  1  0  1|
+--R      |                                        |
+--R      +0  1  1  1  1  0  0  0  1  1  1  1  1  1+
+--R     ]
+--R                                          Type: List List Matrix PrimeField 2
 --E 30
 
---S 31 of 67
-meatAxe r28.2
+--S 31 of 68
+areEquivalent? (r28.1, r28.2)
 --R 
---R   There are no library operations named r28 
---R      Use HyperDoc Browse or issue
---R                                )what op r28
---R      to learn if there is any operation containing " r28 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named r28 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Dimensions of kernels differ
+--R
+--R   Representations are not equivalent.
+--R
+--R   (3)  [0]
+--R                                                    Type: Matrix PrimeField 2
 --E 31
 
---S 32 of 67
-isAbsolutelyIrreducible? r28.2
+--S 32 of 68
+meatAxe r28.2
 --R 
---R   There are no library operations named r28 
---R      Use HyperDoc Browse or issue
---R                                )what op r28
---R      to learn if there is any operation containing " r28 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named r28 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is irreducible, but we don't know
+--R       whether it is absolutely irreducible
+--R
+--R   (4)
+--R   [
+--R      +1  0  0  0  0  0  0  0  0  0  0  0  0  0+
+--R      |                                        |
+--R      |0  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |0  0  1  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |0  0  0  0  1  0  0  0  0  0  0  0  0  0|
+--R      |                                        |
+--R      |0  0  1  1  0  1  1  1  1  1  0  0  0  0|
+--R      |                                        |
+--R      |0  0  1  0  1  1  1  1  1  0  1  0  0  0|
+--R     [|                                        |,
+--R      |0  0  0  1  1  1  1  1  0  1  1  0  0  0|
+--R      |                                        |
+--R      |1  1  1  0  1  1  1  0  1  1  1  0  0  0|
+--R      |                                        |
+--R      |1  1  0  1  1  1  0  1  1  1  1  0  0  0|
+--R      |                                        |
+--R      |1  1  0  0  0  0  1  1  1  1  1  0  0  0|
+--R      |                                        |
+--R      |1  0  1  0  0  1  1  0  1  0  1  0  1  0|
+--R      |                                        |
+--R      |1  0  0  1  0  1  0  1  0  1  1  1  0  0|
+--R      |                                        |
+--R      +1  0  0  0  1  0  1  1  0  0  0  1  1  1+
+--R      +0  0  0  0  0  0  0  1  0  1  1  0  0  0+
+--R      |                                        |
+--R      |0  0  0  0  0  0  0  1  0  0  0  0  1  1|
+--R      |                                        |
+--R      |1  1  0  0  1  0  0  0  1  1  1  1  1  1|
+--R      |                                        |
+--R      |1  0  0  0  0  0  0  0  1  0  1  1  0  1|
+--R      |                                        |
+--R      |1  0  0  0  0  0  0  0  0  1  1  0  1  1|
+--R      |                                        |
+--R      |1  0  0  0  0  0  0  1  0  1  0  0  1  0|
+--R      |                                        |
+--R      |1  0  0  0  0  0  1  0  1  0  0  1  0  0|
+--R      |                                        |]
+--R      |0  0  1  1  0  1  0  0  0  0  0  1  1  0|
+--R      |                                        |
+--R      |1  0  0  0  0  1  0  0  1  1  1  1  1  1|
+--R      |                                        |
+--R      |0  0  1  0  1  0  1  0  0  0  0  1  0  1|
+--R      |                                        |
+--R      |0  0  0  1  1  0  0  1  0  0  0  0  1  1|
+--R      |                                        |
+--R      |0  0  0  0  0  1  1  1  0  0  0  1  1  1|
+--R      |                                        |
+--R      |0  1  1  0  1  0  0  0  1  0  1  1  0  1|
+--R      |                                        |
+--R      +0  1  0  1  1  0  0  0  0  1  1  0  1  1+
+--R     ]
+--R                                          Type: List List Matrix PrimeField 2
 --E 32
 
---S 33 of 67
-ma := meatAxe r28.1
+--S 33 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? r28.2
 --R 
---R   There are no library operations named r28 
---R      Use HyperDoc Browse or issue
---R                                )what op r28
---R      to learn if there is any operation containing " r28 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named r28 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (5)  true
+--R                                                                Type: Boolean
 --E 33
 
---S 34 of 67
-isAbsolutelyIrreducible? ma.1
+--S 34 of 68
+ma := meatAxe r28.1
 --R 
---R   There are no library operations named ma 
---R      Use HyperDoc Browse or issue
---R                                 )what op ma
---R      to learn if there is any operation containing " ma " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ma 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R   (6)
+--R     +0  0  0  0  1  0  1  1+ +1  1  1  1  1  1  0  0+
+--R     |                      | |                      |
+--R     |0  0  0  0  0  1  0  1| |1  0  0  1  1  0  1  0|
+--R     |                      | |                      |
+--R     |0  0  0  0  0  0  1  1| |0  0  1  1  0  0  1  0|
+--R     |                      | |                      |
+--R     |0  0  0  0  0  0  0  1| |1  1  0  1  1  1  1  1|
+--R   [[|                      |,|                      |],
+--R     |1  0  1  0  0  0  0  0| |1  1  1  1  0  0  1  0|
+--R     |                      | |                      |
+--R     |0  1  0  1  0  0  0  0| |1  0  0  1  1  1  1  1|
+--R     |                      | |                      |
+--R     |0  0  1  1  0  0  0  0| |0  1  1  0  1  0  1  1|
+--R     |                      | |                      |
+--R     +0  0  0  1  0  0  0  0+ +1  0  0  1  0  1  0  1+
+--R     +0  1  1  0  0  1+ +1  1  0  0  0  0+
+--R     |                | |                |
+--R     |1  0  1  0  0  1| |1  0  1  1  0  0|
+--R     |                | |                |
+--R     |1  1  0  0  0  1| |1  0  0  1  0  1|
+--R    [|                |,|                |]]
+--R     |0  0  0  1  0  0| |1  0  1  1  1  0|
+--R     |                | |                |
+--R     |0  0  0  0  1  0| |1  0  0  0  1  1|
+--R     |                | |                |
+--R     +1  1  1  0  0  0+ +0  1  1  1  0  1+
+--R                                          Type: List List Matrix PrimeField 2
 --E 34
 
---S 35 of 67
-isAbsolutelyIrreducible? ma.2
+--S 35 of 68
+isAbsolutelyIrreducible? ma.1
 --R 
---R   There are no library operations named ma 
---R      Use HyperDoc Browse or issue
---R                                 )what op ma
---R      to learn if there is any operation containing " ma " in its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named ma 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (7)  true
+--R                                                                Type: Boolean
 --E 35
 
+--S 36 of 68
+isAbsolutelyIrreducible? ma.2
+--R 
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (8)  true
+--R                                                                Type: Boolean
+--E 36
+
 )clear all
 
---S 36 of 67
+--S 37 of 68
 px : PERM PF 29 := cycles [[1,3,5],[7,11,9]]
 --R 
 --R
 --R   (1)  (1 3 5)(7 11 9)
 --R                                              Type: Permutation PrimeField 29
---E 36
+--E 37
 
---S 37 of 67
+--S 38 of 68
 py : PERM PF 29 := cycles [[3,5,7,9]]
 --R 
 --R
 --R   (2)  (3 5 7 9)
 --R                                              Type: Permutation PrimeField 29
---E 37
+--E 38
 
---S 38 of 67
+--S 39 of 68
 pz : PERM PF 29 := cycle [1,3,11]
 --R 
 --R
 --R   (3)  (1 3 11)
 --R                                              Type: Permutation PrimeField 29
---E 38
+--E 39
 
---S 39 of 67
+--S 40 of 68
 px * pz
 --R 
 --R
 --R   (4)  (1 5)(3 9 7 11)
 --R                                              Type: Permutation PrimeField 29
---E 39
+--E 40
 
---S 40 of 67
+--S 41 of 68
 py ** 3
 --R 
 --R
 --R   (5)  (3 9 7 5)
 --R                                              Type: Permutation PrimeField 29
---E 40
+--E 41
 
---S 41 of 67
+--S 42 of 68
 inv px
 --R 
 --R
 --R   (6)  (1 5 3)(7 9 11)
 --R                                              Type: Permutation PrimeField 29
---E 41
+--E 42
 
---S 42 of 67
+--S 43 of 68
 order px
 --R 
 --R
 --R   (7)  3
 --R                                                        Type: PositiveInteger
---E 42
+--E 43
 
---S 43 of 67
+--S 44 of 68
 movedPoints py
 --R 
 --R
 --R   (8)  {3,5,7,9}
 --R                                                      Type: Set PrimeField 29
---E 43
+--E 44
 
---S 44 of 67
+--S 45 of 68
 orbit ( pz , 3 )
 --R 
 --R
 --R   (9)  {3,11,1}
 --R                                                      Type: Set PrimeField 29
---E 44
+--E 45
 
---S 45 of 67
+--S 46 of 68
 eval ( py , 7 )
 --R 
 --R
 --R   (10)  9
 --R                                                          Type: PrimeField 29
---E 45
+--E 46
 
---S 46 of 67
+--S 47 of 68
 )sh PERM
 --R 
 --R Permutation S: SetCategory  is a domain constructor
@@ -679,19 +1853,19 @@ eval ( py , 7 )
 --R min : (%,%) -> % if S has FINITE or S has ORDSET
 --R numberOfCycles : % -> NonNegativeInteger
 --R
---E
+--E 47
 
 )clear all
 
---S 47 of 67
+--S 48 of 68
 genA6 : List PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]
 --R 
 --R
 --R   (1)  [(1 2 3),(2 3 4 5 6)]
 --R                                               Type: List Permutation Integer
---E 47
+--E 48
 
---S 48 of 67
+--S 49 of 68
 pRA6 := permutationRepresentation (genA6,6)
 --R 
 --R
@@ -707,9 +1881,9 @@ pRA6 := permutationRepresentation (genA6,6)
 --R         |                | |                |
 --R         +0  0  0  0  0  1+ +0  0  0  0  1  0+
 --R                                                    Type: List Matrix Integer
---E 48
+--E 49
 
---S 49 of 67
+--S 50 of 68
 sp0 := meatAxe (pRA6::(List Matrix PF 2))
 --R 
 --R   Fingerprint element in generated algebra is singular
@@ -728,9 +1902,9 @@ sp0 := meatAxe (pRA6::(List Matrix PF 2))
 --R          |             | |             |
 --R          +0  0  0  0  1+ +0  0  0  1  0+
 --R                                          Type: List List Matrix PrimeField 2
---E 49
+--E 50
 
---S 50 of 67
+--S 51 of 68
 sp1 := meatAxe sp0.1
 --R 
 --R   Fingerprint element in generated algebra is singular
@@ -754,9 +1928,9 @@ sp1 := meatAxe sp0.1
 --R                    |          | |          |
 --R                    +0  0  0  1+ +1  1  1  1+
 --R                                          Type: List List Matrix PrimeField 2
---E 50
+--E 51
 
---S 51 of 67 random generation, failure ok.
+--S 52 of 68 random generation, FAILURE OK.
 isAbsolutelyIrreducible? sp1.2
 --R 
 --R   Random element in generated algebra has
@@ -769,260 +1943,600 @@ isAbsolutelyIrreducible? sp1.2
 --R
 --I   (5)  true
 --R                                                                Type: Boolean
---E 51
-
---S 52 of 67
-d2211 := irrRepSymNat ([2,2,1,1],genA6)
---R 
---R   There are no library operations named irrRepSymNat 
---R      Use HyperDoc Browse or issue
---R                            )what op irrRepSymNat
---R      to learn if there is any operation containing " irrRepSymNat " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      irrRepSymNat with argument type(s) 
---R                            List PositiveInteger
---R                          List Permutation Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
 --E 52
 
---S 53 of 67
-d2211m2 := (d2211::(List Matrix PF 2)); sp2 := meatAxe d2211m2
+--S 53 of 68
+d2211 := irreducibleRepresentation ([2,2,1,1],genA6)
 --R 
---R 
---RDaly Bug
---R   Cannot convert from type Variable d2211 to List Matrix PrimeField 2 
---R      for value
---R   d2211
 --R
+--R   (6)
+--R    +1  0  0  - 1   1    0    0    0    0 + + 0    0   1   0   0  0   1   0  0+
+--R    |                                     | |                                 |
+--R    |0  1  0   1    0    1    0    0    0 | | 0    0   0   0   1  0  - 1  0  0|
+--R    |                                     | |                                 |
+--R    |0  0  1   0    1   - 1   0    0    0 | | 0    0   0   0   0  1   1   0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0  - 1   0    0   - 1   0    0 | | 0    0   0   0   0  0   1   1  0|
+--R    |                                     | |                                 |
+--R   [|0  0  0   0   - 1   0    0   - 1   0 |,| 0    0   0   0   0  0  - 1  0  1|]
+--R    |                                     | |                                 |
+--R    |0  0  0   0    0   - 1   0    0   - 1| | 0    0   0   0   0  0   1   0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0   1    0    0    0    0    0 | |- 1   0   0   0   0  0  - 1  0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0   0    1    0    0    0    0 | | 0   - 1  0   0   0  0   1   0  0|
+--R    |                                     | |                                 |
+--R    +0  0  0   0    0    1    0    0    0 + + 0    0   0  - 1  0  0  - 1  0  0+
+--R                                                    Type: List Matrix Integer
 --E 53
 
---S 54 of 67
-isAbsolutelyIrreducible? sp2.1
+--S 54 of 68
+d2211m2 := (d2211::(List Matrix PF 2)); sp2 := meatAxe d2211m2
 --R 
---R   There are no library operations named sp2 
---R      Use HyperDoc Browse or issue
---R                                )what op sp2
---R      to learn if there is any operation containing " sp2 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp2 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is singular
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R                                      +1  0  0  0  0+ +1  1  1  0  0+
+--R          +1  0  1  1+ +0  0  1  0+   |             | |             |
+--R          |          | |          |   |0  1  1  1  1| |0  0  1  1  1|
+--R          |0  1  0  1| |1  1  1  1|   |             | |             |
+--R   (7)  [[|          |,|          |],[|0  1  1  0  0|,|1  0  0  1  0|]]
+--R          |1  1  0  0| |1  0  1  1|   |             | |             |
+--R          |          | |          |   |0  1  0  1  0| |0  0  1  0  1|
+--R          +0  1  0  0+ +0  1  0  1+   |             | |             |
+--R                                      +0  1  1  1  0+ +1  0  0  1  1+
+--R                                          Type: List List Matrix PrimeField 2
 --E 54
 
---S 55 of 67
-areEquivalent? (sp2.1, sp1.2)
+--S 55 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp2.1
 --R 
---R   There are no library operations named sp2 
---R      Use HyperDoc Browse or issue
---R                                )what op sp2
---R      to learn if there is any operation containing " sp2 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp2 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (8)  true
+--R                                                                Type: Boolean
 --E 55
 
---S 56 of 67
-dA6d16 := tensorProduct(sp2.1,sp1.2); meatAxe dA6d16
+--S 56 of 68 random generation, FAILURE OK.
+areEquivalent? (sp2.1, sp1.2)
 --R 
---R   There are no library operations named sp2 
---R      Use HyperDoc Browse or issue
---R                                )what op sp2
---R      to learn if there is any operation containing " sp2 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp2 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Dimensions of kernels differ
+--R
+--R   Representations are not equivalent.
+--R
+--R   (9)  [0]
+--R                                                    Type: Matrix PrimeField 2
 --E 56
 
---S 57 of 67
-isAbsolutelyIrreducible? dA6d16
+--S 57 of 68
+dA6d16 := tensorProduct(sp2.1,sp1.2); meatAxe dA6d16
 --R 
---R   There are 1 exposed and 0 unexposed library operations named 
---R      isAbsolutelyIrreducible? having 1 argument(s) but none was 
---R      determined to be applicable. Use HyperDoc Browse, or issue
---R                    )display op isAbsolutelyIrreducible?
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      isAbsolutelyIrreducible? with argument type(s) 
---R                               Variable dA6d16
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is irreducible, but we don't know
+--R       whether it is absolutely irreducible
+--R
+--R   (10)
+--R   [
+--R      +0  0  1  0  0  0  0  0  0  0  1  0  0  0  0  0+
+--R      |                                              |
+--R      |1  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  1  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  1  0  0  0  0  0  0  0  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0|
+--R      |                                              |
+--R      |0  0  0  0  1  0  0  0  1  0  0  0  1  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  1  0  0  0  1  0  0  0  1  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  1  0  0  0  1  0  0  0  1|
+--R     [|                                              |,
+--R      |0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  1  0  0  0  1  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      +0  0  0  1  0  0  0  1  0  0  0  0  0  0  0  0+
+--R      +0  0  0  0  0  1  1  1  0  1  1  1  0  0  0  0+
+--R      |                                              |
+--R      |0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  1  0  1  1  1  0  1  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  1  1  0  1  1  1  0  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  1  1  1  0  0  0  0  0  1  1  1|
+--R      |                                              |
+--R      |0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1|
+--R      |                                              |
+--R      |0  0  0  0  1  0  1  1  0  0  0  0  1  0  1  1|
+--R      |                                              |
+--R      |0  0  0  0  1  1  0  1  0  0  0  0  1  1  0  1|
+--R      |                                              |]
+--R      |0  1  1  1  0  1  1  1  0  1  1  1  0  0  0  0|
+--R      |                                              |
+--R      |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
+--R      |                                              |
+--R      |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
+--R      |                                              |
+--R      |1  1  0  1  1  1  0  1  1  1  0  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  1  1  1  0  1  1  1  0  1  1  1|
+--R      |                                              |
+--R      |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
+--R      |                                              |
+--R      |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
+--R      |                                              |
+--R      +0  0  0  0  1  1  0  1  1  1  0  1  1  1  0  1+
+--R     ]
+--R                                          Type: List List Matrix PrimeField 2
 --E 57
 
---S 58 of 67
-sp3 := meatAxe (dA6d16 :: (List Matrix FF(2,2)))
---R 
+--S 58 of 68
+isAbsolutelyIrreducible? dA6d16
 --R 
---RDaly Bug
---R   Cannot convert from type Variable dA6d16 to List Matrix FiniteField(
---R      2,2) for value
---R   dA6d16
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   We have not found a one-dimensional kernel so far,
+--R     as we do a random search you could try again
 --R
+--R   (11)  false
+--R                                                                Type: Boolean
 --E 58
 
---S 59 of 67
-isAbsolutelyIrreducible? sp3.1
+--S 59 of 68
+sp3 := meatAxe (dA6d16 :: (List Matrix FF(2,2)))
 --R 
---R   There are no library operations named sp3 
---R      Use HyperDoc Browse or issue
---R                                )what op sp3
---R      to learn if there is any operation containing " sp3 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp3 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R   (12)
+--R   [
+--R      +%A + 1  %A + 1    0       %A      1       %A      %A    %A + 1+
+--R      |                                                              |
+--R      |  %A      0     %A + 1  %A + 1    1     %A + 1  %A + 1    %A  |
+--R      |                                                              |
+--R      |  %A    %A + 1    %A      0       1       %A    %A + 1    0   |
+--R      |                                                              |
+--R      |  0       %A    %A + 1    %A      0       1       1       0   |
+--R     [|                                                              |,
+--R      |  %A      %A    %A + 1    1     %A + 1    %A      0       %A  |
+--R      |                                                              |
+--R      |%A + 1    %A    %A + 1    1       %A      0       %A    %A + 1|
+--R      |                                                              |
+--R      |  1       1       1       0     %A + 1  %A + 1    %A      0   |
+--R      |                                                              |
+--R      +  0     %A + 1    %A      0       0       %A    %A + 1  %A + 1+
+--R      +  0     %A + 1  %A + 1    %A    1     1       0       %A  +
+--R      |                                                          |
+--R      |%A + 1  %A + 1    1       0     1   %A + 1    1     %A + 1|
+--R      |                                                          |
+--R      |  %A      0       1       1     %A  %A + 1  %A + 1    0   |
+--R      |                                                          |
+--R      |  %A      1       0       %A    %A    0       1       %A  |
+--R      |                                                          |]
+--R      |  1       1       0     %A + 1  0     1       1       0   |
+--R      |                                                          |
+--R      |  1       %A      1       0     1     0       0       %A  |
+--R      |                                                          |
+--R      |%A + 1    0       1       1     0     %A    %A + 1    1   |
+--R      |                                                          |
+--R      +%A + 1    %A      %A    %A + 1  0   %A + 1    %A      0   +
+--R     ,
+--R
+--R      +%A + 1    %A      %A      0       %A      1     %A + 1    0   +
+--R      |                                                              |
+--R      |%A + 1    1       0     %A + 1    1     %A + 1    1       %A  |
+--R      |                                                              |
+--R      |  1       1       %A      %A      1       %A      1     %A + 1|
+--R      |                                                              |
+--R      |  1       0       1     %A + 1  %A + 1    0       %A      1   |
+--R     [|                                                              |,
+--R      |  1       1       1       0     %A + 1  %A + 1    %A      0   |
+--R      |                                                              |
+--R      |%A + 1    %A      %A      1     %A + 1    1       1     %A + 1|
+--R      |                                                              |
+--R      |%A + 1  %A + 1    %A      1       0       1       %A      %A  |
+--R      |                                                              |
+--R      +  %A      0     %A + 1    0       1       0       1       %A  +
+--R      +  1     1     %A    %A + 1    0       %A    %A + 1  %A + 1+
+--R      |                                                          |
+--R      |%A + 1  0     0       1     %A + 1    1       1     %A + 1|
+--R      |                                                          |
+--R      |  %A    0     1       0     %A + 1    0     %A + 1    1   |
+--R      |                                                          |
+--R      |  1     1   %A + 1    %A      %A      %A      1       0   |
+--R      |                                                          |]
+--R      |  1     %A    0       1       1       %A      1       0   |
+--R      |                                                          |
+--R      |  1     0     1     %A + 1    0     %A + 1    1     %A + 1|
+--R      |                                                          |
+--R      |  0     1   %A + 1    1       1     %A + 1  %A + 1    1   |
+--R      |                                                          |
+--R      +  %A    %A  %A + 1  %A + 1    %A      %A      0       1   +
+--R     ]
+--R                                      Type: List List Matrix FiniteField(2,2)
 --E 59
 
---S 60 of 67
-isAbsolutelyIrreducible? sp3.2
+--S 60 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp3.1
 --R 
---R   There are no library operations named sp3 
---R      Use HyperDoc Browse or issue
---R                                )what op sp3
---R      to learn if there is any operation containing " sp3 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp3 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (13)  true
+--R                                                                Type: Boolean
 --E 60
 
---S 61 of 67
-areEquivalent? (sp3.1,sp3.2)
+--S 61 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp3.2
 --R 
---R   There are no library operations named sp3 
---R      Use HyperDoc Browse or issue
---R                                )what op sp3
---R      to learn if there is any operation containing " sp3 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp3 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (14)  true
+--R                                                                Type: Boolean
 --E 61
 
---S 62 of 67
+--S 62 of 68 random generation, FAILURE OK.
+areEquivalent? (sp3.1,sp3.2)
+--R 
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     There is no isomorphism, as the only possible one
+--R       fails to do the necessary base change
+--R
+--R   Representations are not equivalent.
+--R
+--R   (15)  [0]
+--R                                                Type: Matrix FiniteField(2,2)
+--E 62
+
+--S 63 of 68
 sp0.2
 --R 
 --R
---R   (6)  [[1],[1]]
+--R   (16)  [[1],[1]]
 --R                                               Type: List Matrix PrimeField 2
---E 62
+--E 63
 
---S 63 of 67
+--S 64 of 68
 sp1.2
 --R 
 --R
---R         +0  1  0  0+ +0  1  1  1+
---R         |          | |          |
---R         |0  0  1  0| |1  1  0  1|
---R   (7)  [|          |,|          |]
---R         |1  0  0  0| |1  1  1  0|
---R         |          | |          |
---R         +0  0  0  1+ +1  1  1  1+
+--R          +0  1  0  0+ +0  1  1  1+
+--R          |          | |          |
+--R          |0  0  1  0| |1  1  0  1|
+--R   (17)  [|          |,|          |]
+--R          |1  0  0  0| |1  1  1  0|
+--R          |          | |          |
+--R          +0  0  0  1+ +1  1  1  1+
 --R                                               Type: List Matrix PrimeField 2
---E 63
+--E 64
 
---S 64 of 67
+--S 65 of 68
 sp2.1
 --R 
---R   There are no library operations named sp2 
---R      Use HyperDoc Browse or issue
---R                                )what op sp2
---R      to learn if there is any operation containing " sp2 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp2 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
---E 64
+--R
+--R          +1  0  1  1+ +0  0  1  0+
+--R          |          | |          |
+--R          |0  1  0  1| |1  1  1  1|
+--R   (18)  [|          |,|          |]
+--R          |1  1  0  0| |1  0  1  1|
+--R          |          | |          |
+--R          +0  1  0  0+ +0  1  0  1+
+--R                                               Type: List Matrix PrimeField 2
+--E 65
 
---S 65 of 67
+--S 66 of 68
 sp3.1
 --R 
---R   There are no library operations named sp3 
---R      Use HyperDoc Browse or issue
---R                                )what op sp3
---R      to learn if there is any operation containing " sp3 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp3 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
---E 65
+--R
+--R   (19)
+--R    +%A + 1  %A + 1    0       %A      1       %A      %A    %A + 1+
+--R    |                                                              |
+--R    |  %A      0     %A + 1  %A + 1    1     %A + 1  %A + 1    %A  |
+--R    |                                                              |
+--R    |  %A    %A + 1    %A      0       1       %A    %A + 1    0   |
+--R    |                                                              |
+--R    |  0       %A    %A + 1    %A      0       1       1       0   |
+--R   [|                                                              |,
+--R    |  %A      %A    %A + 1    1     %A + 1    %A      0       %A  |
+--R    |                                                              |
+--R    |%A + 1    %A    %A + 1    1       %A      0       %A    %A + 1|
+--R    |                                                              |
+--R    |  1       1       1       0     %A + 1  %A + 1    %A      0   |
+--R    |                                                              |
+--R    +  0     %A + 1    %A      0       0       %A    %A + 1  %A + 1+
+--R    +  0     %A + 1  %A + 1    %A    1     1       0       %A  +
+--R    |                                                          |
+--R    |%A + 1  %A + 1    1       0     1   %A + 1    1     %A + 1|
+--R    |                                                          |
+--R    |  %A      0       1       1     %A  %A + 1  %A + 1    0   |
+--R    |                                                          |
+--R    |  %A      1       0       %A    %A    0       1       %A  |
+--R    |                                                          |]
+--R    |  1       1       0     %A + 1  0     1       1       0   |
+--R    |                                                          |
+--R    |  1       %A      1       0     1     0       0       %A  |
+--R    |                                                          |
+--R    |%A + 1    0       1       1     0     %A    %A + 1    1   |
+--R    |                                                          |
+--R    +%A + 1    %A      %A    %A + 1  0   %A + 1    %A      0   +
+--R                                           Type: List Matrix FiniteField(2,2)
+--E 66
 
---S 66 of 67
+--S 67 of 68
 sp3.2
 --R 
---R   There are no library operations named sp3 
---R      Use HyperDoc Browse or issue
---R                                )what op sp3
---R      to learn if there is any operation containing " sp3 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp3 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
---E 66
+--R
+--R   (20)
+--R    +%A + 1    %A      %A      0       %A      1     %A + 1    0   +
+--R    |                                                              |
+--R    |%A + 1    1       0     %A + 1    1     %A + 1    1       %A  |
+--R    |                                                              |
+--R    |  1       1       %A      %A      1       %A      1     %A + 1|
+--R    |                                                              |
+--R    |  1       0       1     %A + 1  %A + 1    0       %A      1   |
+--R   [|                                                              |,
+--R    |  1       1       1       0     %A + 1  %A + 1    %A      0   |
+--R    |                                                              |
+--R    |%A + 1    %A      %A      1     %A + 1    1       1     %A + 1|
+--R    |                                                              |
+--R    |%A + 1  %A + 1    %A      1       0       1       %A      %A  |
+--R    |                                                              |
+--R    +  %A      0     %A + 1    0       1       0       1       %A  +
+--R    +  1     1     %A    %A + 1    0       %A    %A + 1  %A + 1+
+--R    |                                                          |
+--R    |%A + 1  0     0       1     %A + 1    1       1     %A + 1|
+--R    |                                                          |
+--R    |  %A    0     1       0     %A + 1    0     %A + 1    1   |
+--R    |                                                          |
+--R    |  1     1   %A + 1    %A      %A      %A      1       0   |
+--R    |                                                          |]
+--R    |  1     %A    0       1       1       %A      1       0   |
+--R    |                                                          |
+--R    |  1     0     1     %A + 1    0     %A + 1    1     %A + 1|
+--R    |                                                          |
+--R    |  0     1   %A + 1    1       1     %A + 1  %A + 1    1   |
+--R    |                                                          |
+--R    +  %A    %A  %A + 1  %A + 1    %A      %A      0       1   +
+--R                                           Type: List Matrix FiniteField(2,2)
+--E 67
 
---S 67 of 67
+--S 68 of 68
 dA6d16
 --R 
 --R
---R   (8)  dA6d16
---R                                                        Type: Variable dA6d16
---E 67
+--R   (21)
+--R    +0  1  0  0  0  0  0  0  0  1  0  0  0  1  0  0+
+--R    |                                              |
+--R    |0  0  1  0  0  0  0  0  0  0  1  0  0  0  1  0|
+--R    |                                              |
+--R    |1  0  0  0  0  0  0  0  1  0  0  0  1  0  0  0|
+--R    |                                              |
+--R    |0  0  0  1  0  0  0  0  0  0  0  1  0  0  0  1|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  1  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  1  0  0  0  0  0  0  0  1  0|
+--R    |                                              |
+--R    |0  0  0  0  1  0  0  0  0  0  0  0  1  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  1|
+--R   [|                                              |,
+--R    |0  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  1  0  0  0  1  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    +0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0+
+--R    +0  0  0  0  0  0  0  0  0  1  1  1  0  0  0  0+
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  1  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1|
+--R    |                                              |
+--R    |1  1  0  1  1  1  0  1  1  1  0  1  1  1  0  1|
+--R    |                                              |
+--R    |1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |]
+--R    |0  1  1  1  0  0  0  0  0  1  1  1  0  1  1  1|
+--R    |                                              |
+--R    |1  1  0  1  0  0  0  0  1  1  0  1  1  1  0  1|
+--R    |                                              |
+--R    |1  1  1  0  0  0  0  0  1  1  1  0  1  1  1  0|
+--R    |                                              |
+--R    |1  1  1  1  0  0  0  0  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  1  1  1  0  0  0  0  0  1  1  1|
+--R    |                                              |
+--R    |0  0  0  0  1  1  0  1  0  0  0  0  1  1  0  1|
+--R    |                                              |
+--R    |0  0  0  0  1  1  1  0  0  0  0  0  1  1  1  0|
+--R    |                                              |
+--R    +0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1+
+--R                                               Type: List Matrix PrimeField 2
+--E 68
 )spool 
 )lisp (bye)
  
diff --git a/src/input/knot2.input.pamphlet b/src/input/knot2.input.pamphlet
index a793b46..7973989 100644
--- a/src/input/knot2.input.pamphlet
+++ b/src/input/knot2.input.pamphlet
@@ -63,57 +63,49 @@ l := lcm(p, q) quo p
 --E 4
 
 --S 5 of 8
-maxRange := (odd? l => l * %pi::SF; 2 * l * %pi::SF)  
+maxRange := (odd? l => l * %pi; 2 * l * %pi)  
 --R 
 --R
---R   (5)  15.707963267948966
---R                                                            Type: DoubleFloat
+--R   (5)  5%pi
+--R                                                                     Type: Pi
 --E 5
 
 --S 6 of 8
 theRange := 0..maxRange
 --R 
 --R
---R   (6)  0...15.707963267948966
---R                                                    Type: Segment DoubleFloat
+--R   (6)  0..(5%pi)
+--R                                                             Type: Segment Pi
 --E 6
 
 @
 Create the knot
-<<*>>=
---S 7 of 8
+\begin{verbatim}
 knot:TUBE := tubePlot(sin t * cos(PQ*t),cos t * cos(PQ*t),cos t * sin(PQ*t),
                       f, theRange, 0.1::SF, 6, "open" )
+\end{verbatim}
+<<*>>=
+--S 7 of 8
+v:=draw(curve(sin t * cos(PQ*t),cos t * cos(PQ*t),cos t * sin(PQ*t)), _
+        t=theRange, tubeRadius==0.1)
 --R 
---R 
---RDaly Bug
---R   Although TubePlot is the name of a constructor, a full type must be 
---R      specified in the context you have used it. Issue )show TubePlot 
---R      for more information.
+--I   Compiling function %B with type DoubleFloat -> DoubleFloat 
+--I   Compiling function %D with type DoubleFloat -> DoubleFloat 
+--I   Compiling function %F with type DoubleFloat -> DoubleFloat 
+--R   Transmitting data...
+--R
+--R   (7)  ThreeDimensionalViewport: "DCOS((3*t)/5)*DSIN(t)"
+--R                                               Type: ThreeDimensionalViewport
 --E 7
 
 @
-Make a viewport out of it
+close the viewport
+\begin{verbatim}
+makeViewport3D(knot, concat ["knot",p::String,q::String])$VIEW3D
+\end{verbatim}
 <<*>>=
 --S 8 of 8
-makeViewport3D(knot, concat ["knot",p::String,q::String])$VIEW3D
---R 
---R   There are 2 exposed and 0 unexposed library operations named 
---R      makeViewport3D having 2 argument(s) but none was determined to be
---R      applicable. Use HyperDoc Browse, or issue
---R                         )display op makeViewport3D
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      makeViewport3D with argument type(s) 
---R                                   Symbol
---R                                   String
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+close(v)
 --E 8
 )spool 
 )lisp (bye)
diff --git a/src/input/repa6.input.pamphlet b/src/input/repa6.input.pamphlet
index 352a022..099c75d 100644
--- a/src/input/repa6.input.pamphlet
+++ b/src/input/repa6.input.pamphlet
@@ -39,7 +39,7 @@ representations of the alternating group A6.
 We generate A6 by the permutations threecycle x=(1,2,3)
 and the 5-cycle y=(2,3,4,5,6)
 <<*>>=
---S 1  of 33
+--S 1 of 33
 genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
 --R 
 --R
@@ -50,7 +50,7 @@ genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
 @ 
 pRA6 is the permutation representation over the Integers...
 <<*>>=
---S 2  of 33
+--S 2 of 33
 pRA6 := permutationRepresentation (genA6, 6)
 --R 
 --R
@@ -71,7 +71,7 @@ pRA6 := permutationRepresentation (genA6, 6)
 @
 And pRA6m2 is the permutation representation over PrimeField 2:
 <<*>>=
---S 3  of 33
+--S 3 of 33
 pRA6m2 : List Matrix PrimeField 2 := pRA6
 --R 
 --R
@@ -117,7 +117,7 @@ sp0 := meatAxe pRA6m2
 We have found the trivial module as a factormodule
 and a 5-dimensional submodule.
 <<*>>=
---S 5  of 33
+--S 5 of 33
 dA6d1 := sp0.2
 --R 
 --R
@@ -128,7 +128,7 @@ dA6d1 := sp0.2
 @ 
 Try to split again...
 <<*>>=
---S 6  of 33
+--S 6 of 33
 sp1 := meatAxe sp0.1
 --R 
 --R   Fingerprint element in generated algebra is singular
@@ -158,7 +158,7 @@ sp1 := meatAxe sp0.1
 And find a 4-dimensional submodule, say dA6d4a, and the
 trivial one again.
 <<*>>=
---S 7  of 33
+--S 7 of 33
 dA6d4a := sp1.2
 --R 
 --R
@@ -175,13 +175,13 @@ dA6d4a := sp1.2
 @
 Now we want to test, whether dA6d4a is absolutely irreducible...
 <<*>>=
---S 8  of 33 random input, ok to fail
+--S 8 of 33 random input, FAILURE OK
 isAbsolutelyIrreducible? dA6d4a
 --R 
 --R   Random element in generated algebra does
 --R     not have a one-dimensional kernel
---R   Random element in generated algebra has
---R     one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
 --R   Random element in generated algebra has
 --R     one-dimensional kernel
 --R     We know that all the cyclic submodules generated by all
@@ -204,7 +204,7 @@ What is the degree of the representation belonging to partition
 [2,2,1,1]?
 <<*>>=
 -- lambda : PRTITION := partition [2,2,1,1]
---S 9  of 33
+--S 9 of 33
 lambda := [2,2,1,1]
 --R 
 --R
@@ -213,21 +213,11 @@ lambda := [2,2,1,1]
 --E 9
 
 --S 10 of 33
-dimIrrRepSym lambda
---R 
---R   There are no library operations named dimIrrRepSym 
---R      Use HyperDoc Browse or issue
---R                            )what op dimIrrRepSym
---R      to learn if there is any operation containing " dimIrrRepSym " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      dimIrrRepSym with argument type(s) 
---R                            List PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+dimensionOfIrreducibleRepresentation lambda
+--R 
+--R
+--R   (10)  9
+--R                                                        Type: PositiveInteger
 --E 10
 
 @
@@ -235,22 +225,28 @@ Now create the restriction to A6:
 <<*>>=
 
 --S 11 of 33
-d2211  := irrRepSymNat(lambda, genA6)
---R 
---R   There are no library operations named irrRepSymNat 
---R      Use HyperDoc Browse or issue
---R                            )what op irrRepSymNat
---R      to learn if there is any operation containing " irrRepSymNat " in
---R      its name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      irrRepSymNat with argument type(s) 
---R                            List PositiveInteger
---R                          List Permutation Integer
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+d2211  := irreducibleRepresentation(lambda, genA6)
+--R 
+--R
+--R   (11)
+--R    +1  0  0  - 1   1    0    0    0    0 + + 0    0   1   0   0  0   1   0  0+
+--R    |                                     | |                                 |
+--R    |0  1  0   1    0    1    0    0    0 | | 0    0   0   0   1  0  - 1  0  0|
+--R    |                                     | |                                 |
+--R    |0  0  1   0    1   - 1   0    0    0 | | 0    0   0   0   0  1   1   0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0  - 1   0    0   - 1   0    0 | | 0    0   0   0   0  0   1   1  0|
+--R    |                                     | |                                 |
+--R   [|0  0  0   0   - 1   0    0   - 1   0 |,| 0    0   0   0   0  0  - 1  0  1|]
+--R    |                                     | |                                 |
+--R    |0  0  0   0    0   - 1   0    0   - 1| | 0    0   0   0   0  0   1   0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0   1    0    0    0    0    0 | |- 1   0   0   0   0  0  - 1  0  0|
+--R    |                                     | |                                 |
+--R    |0  0  0   0    1    0    0    0    0 | | 0   - 1  0   0   0  0   1   0  0|
+--R    |                                     | |                                 |
+--R    +0  0  0   0    0    1    0    0    0 + + 0    0   0  - 1  0  0  - 1  0  0+
+--R                                                    Type: List Matrix Integer
 --E 11
 
 @
@@ -259,13 +255,25 @@ And d2211m2 is the representation over PrimeField 2:
 --S 12 of 33
 d2211m2 : List Matrix PrimeField 2 := d2211
 --R 
---R 
---RDaly Bug
---R   Cannot convert right-hand side of assignment
---R   d2211
 --R
---R      to an object of the type List Matrix PrimeField 2 of the 
---R      left-hand side.
+--R          +1  0  0  1  1  0  0  0  0+ +0  0  1  0  0  0  1  0  0+
+--R          |                         | |                         |
+--R          |0  1  0  1  0  1  0  0  0| |0  0  0  0  1  0  1  0  0|
+--R          |                         | |                         |
+--R          |0  0  1  0  1  1  0  0  0| |0  0  0  0  0  1  1  0  0|
+--R          |                         | |                         |
+--R          |0  0  0  1  0  0  1  0  0| |0  0  0  0  0  0  1  1  0|
+--R          |                         | |                         |
+--R   (12)  [|0  0  0  0  1  0  0  1  0|,|0  0  0  0  0  0  1  0  1|]
+--R          |                         | |                         |
+--R          |0  0  0  0  0  1  0  0  1| |0  0  0  0  0  0  1  0  0|
+--R          |                         | |                         |
+--R          |0  0  0  1  0  0  0  0  0| |1  0  0  0  0  0  1  0  0|
+--R          |                         | |                         |
+--R          |0  0  0  0  1  0  0  0  0| |0  1  0  0  0  0  1  0  0|
+--R          |                         | |                         |
+--R          +0  0  0  0  0  1  0  0  0+ +0  0  0  1  0  0  1  0  0+
+--R                                               Type: List Matrix PrimeField 2
 --E 12
 
 @ 
@@ -274,10 +282,22 @@ And split it:
 --S 13 of 33
 sp2 := meatAxe d2211m2
 --R 
---R 
---RDaly Bug
---R   d2211m2 is declared as being in List Matrix PrimeField 2 but has not
---R      been given a value.
+--R   Fingerprint element in generated algebra is singular
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R                                       +1  0  0  0  0+ +1  1  1  0  0+
+--R           +1  0  1  1+ +0  0  1  0+   |             | |             |
+--R           |          | |          |   |0  1  1  1  1| |0  0  1  1  1|
+--R           |0  1  0  1| |1  1  1  1|   |             | |             |
+--R   (13)  [[|          |,|          |],[|0  1  1  0  0|,|1  0  0  1  0|]]
+--R           |1  1  0  0| |1  0  1  1|   |             | |             |
+--R           |          | |          |   |0  1  0  1  0| |0  0  1  0  1|
+--R           +0  1  0  0+ +0  1  0  1+   |             | |             |
+--R                                       +0  1  1  1  0+ +1  0  0  1  1+
+--R                                          Type: List List Matrix PrimeField 2
 --E 13
 
 @
@@ -285,69 +305,52 @@ A 5 and a 4-dimensional one.
  
 we take the 4-dimensional one, say dA6d4b:
 <<*>>=
---S 14  of 33
+--S 14 of 33
 dA6d4b := sp2.1
 --R 
---R   There are no library operations named sp2 
---R      Use HyperDoc Browse or issue
---R                                )what op sp2
---R      to learn if there is any operation containing " sp2 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp2 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R          +1  0  1  1+ +0  0  1  0+
+--R          |          | |          |
+--R          |0  1  0  1| |1  1  1  1|
+--R   (14)  [|          |,|          |]
+--R          |1  1  0  0| |1  0  1  1|
+--R          |          | |          |
+--R          +0  1  0  0+ +0  1  0  1+
+--R                                               Type: List Matrix PrimeField 2
 --E 14
 
 @
 This is absolutely irreducible, too ...
 <<*>>=
---S 15  of 33
+--S 15 of 33 random generation, FAILURE OK.
 isAbsolutelyIrreducible? dA6d4b
 --R 
---R   There are 1 exposed and 0 unexposed library operations named 
---R      isAbsolutelyIrreducible? having 1 argument(s) but none was 
---R      determined to be applicable. Use HyperDoc Browse, or issue
---R                    )display op isAbsolutelyIrreducible?
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      isAbsolutelyIrreducible? with argument type(s) 
---R                               Variable dA6d4b
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (15)  true
+--R                                                                Type: Boolean
 --E 15
 
 @ 
 And dA6d4a and dA6d4b are not equivalent:
 <<*>>=
---S 16 of 33
+--S 16 of 33 random generation, FAILURE OK.
 areEquivalent? ( dA6d4a , dA6d4b )
 --R 
---R   There are 1 exposed and 0 unexposed library operations named 
---R      areEquivalent? having 2 argument(s) but none was determined to be
---R      applicable. Use HyperDoc Browse, or issue
---R                         )display op areEquivalent?
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      areEquivalent? with argument type(s) 
---R                          List Matrix PrimeField 2
---R                               Variable dA6d4b
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Dimensions of kernels differ
+--R
+--R   Representations are not equivalent.
+--R
+--R   (16)  [0]
+--R                                                    Type: Matrix PrimeField 2
 --E 16
 
 @ 
@@ -355,25 +358,74 @@ So the third irreducible representation is found.
  
 Now construct a new representation with the help of the tensorproduct
 <<*>>=
---S 17  of 33
+--S 17 of 33
 dA6d16 := tensorProduct ( dA6d4a , dA6d4b )
 --R 
---R   There are 2 exposed and 0 unexposed library operations named 
---R      tensorProduct having 2 argument(s) but none was determined to be 
---R      applicable. Use HyperDoc Browse, or issue
---R                          )display op tensorProduct
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      tensorProduct with argument type(s) 
---R                          List Matrix PrimeField 2
---R                               Variable dA6d4b
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (17)
+--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
+--R   [|                                              |,
+--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
+--R    |                                              |
+--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
+--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
+--R    |                                              |
+--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
+--R    |                                              |]
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
+--R                                               Type: List Matrix PrimeField 2
 --E 17
 
 @ 
@@ -382,149 +434,508 @@ And try to split it...
 --S 18 of 33
 sp3 := meatAxe dA6d16
 --R 
---R   There are 1 exposed and 0 unexposed library operations named meatAxe
---R      having 1 argument(s) but none was determined to be applicable. 
---R      Use HyperDoc Browse, or issue
---R                             )display op meatAxe
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      meatAxe with argument type(s) 
---R                               Variable dA6d16
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is irreducible, but we don't know
+--R       whether it is absolutely irreducible
+--R
+--R   (18)
+--R   [
+--R      +0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0+
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  0  1  1  1  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
+--R      |                                              |
+--R      |1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R     [|                                              |,
+--R      |0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1|
+--R      |                                              |
+--R      |0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0|
+--R      |                                              |
+--R      +0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0+
+--R      +0  0  0  0  0  1  1  0  0  1  1  0  0  1  1  0+
+--R      |                                              |
+--R      |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
+--R      |                                              |
+--R      |0  0  0  0  1  1  1  0  1  1  1  0  1  1  1  0|
+--R      |                                              |
+--R      |0  0  0  0  0  1  1  1  0  1  1  1  0  1  1  1|
+--R      |                                              |
+--R      |0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0|
+--R      |                                              |
+--R      |0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1|
+--R      |                                              |
+--R      |1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0|
+--R      |                                              |
+--R      |0  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1|
+--R      |                                              |]
+--R      |0  1  1  0  0  0  0  0  0  1  1  0  0  1  1  0|
+--R      |                                              |
+--R      |0  1  0  1  0  0  0  0  0  1  0  1  0  1  0  1|
+--R      |                                              |
+--R      |1  1  1  0  0  0  0  0  1  1  1  0  1  1  1  0|
+--R      |                                              |
+--R      |0  1  1  1  0  0  0  0  0  1  1  1  0  1  1  1|
+--R      |                                              |
+--R      |0  1  1  0  0  1  1  0  0  0  0  0  0  1  1  0|
+--R      |                                              |
+--R      |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
+--R      |                                              |
+--R      |1  1  1  0  1  1  1  0  0  0  0  0  1  1  1  0|
+--R      |                                              |
+--R      +0  1  1  1  0  1  1  1  0  0  0  0  0  1  1  1+
+--R     ]
+--R                                          Type: List List Matrix PrimeField 2
 --E 18
 
 @ 
 The representation is irreducible, but may be not
 absolutely irreducible.
 <<*>>=
---S 19  of 33
+--S 19 of 33
 isAbsolutelyIrreducible? dA6d16
 --R 
---R   There are 1 exposed and 0 unexposed library operations named 
---R      isAbsolutelyIrreducible? having 1 argument(s) but none was 
---R      determined to be applicable. Use HyperDoc Browse, or issue
---R                    )display op isAbsolutelyIrreducible?
---R      to learn more about the available operations. Perhaps 
---R      package-calling the operation or using coercions on the arguments
---R      will allow you to apply the operation.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named 
---R      isAbsolutelyIrreducible? with argument type(s) 
---R                               Variable dA6d16
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   We have not found a one-dimensional kernel so far,
+--R     as we do a random search you could try again
+--R
+--R   (19)  false
+--R                                                                Type: Boolean
 --E 19
 
 @ 
 So let's try the same over the field with 4 elements:
 <<*>>=
---S 20  of 33
+--S 20 of 33
 gf4 := FiniteField(2,2)
 --R 
 --R
---R   (10)  FiniteField(2,2)
+--R   (20)  FiniteField(2,2)
 --R                                                                 Type: Domain
 --E 20
 
---S 21  of 33
+--S 21 of 33
 dA6d16gf4 : List Matrix gf4 := dA6d16
 --R 
---R 
---RDaly Bug
---R   Cannot convert right-hand side of assignment
---R   dA6d16
 --R
---R      to an object of the type List Matrix FiniteField(2,2) of the 
---R      left-hand side.
+--R   (21)
+--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
+--R   [|                                              |,
+--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
+--R    |                                              |
+--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
+--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
+--R    |                                              |
+--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
+--R    |                                              |]
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
+--R                                           Type: List Matrix FiniteField(2,2)
 --E 21
 
 --S 22 of 33
 sp4 := meatAxe dA6d16gf4
 --R 
---R 
---RDaly Bug
---R   dA6d16gf4 is declared as being in List Matrix FiniteField(2,2) but 
---R      has not been given a value.
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is non-singular
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R   Fingerprint element in generated algebra is singular
+--R     The generated cyclic submodule was not proper
+--R     The generated cyclic submodule was not proper
+--R     A proper cyclic submodule is found.
+--R     Transition matrix computed
+--R     The inverse of the transition matrix computed
+--R     Now transform the matrices
+--R
+--R   (22)
+--R   [
+--R      +  %A    %A + 1    0       %A      1     %A + 1    0     0 +
+--R      |                                                          |
+--R      |  0       0       %A    %A + 1    %A      %A      0     0 |
+--R      |                                                          |
+--R      |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
+--R      |                                                          |
+--R      |  %A    %A + 1    %A      1       %A      0       0     0 |
+--R     [|                                                          |,
+--R      |%A + 1    1       1       1       0       0     %A + 1  %A|
+--R      |                                                          |
+--R      |  0       0     %A + 1    1       0       0       %A    0 |
+--R      |                                                          |
+--R      |  1       0       1       1       0       0       0     0 |
+--R      |                                                          |
+--R      +  1       1       0       0       0       0       0     0 +
+--R      +  1       0       %A      0       1       1       %A    %A + 1+
+--R      |                                                              |
+--R      |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
+--R      |                                                              |
+--R      |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
+--R      |                                                              |
+--R      |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
+--R      |                                                              |]
+--R      |  1       0     %A + 1    0       1       1       %A      %A  |
+--R      |                                                              |
+--R      |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
+--R      |                                                              |
+--R      |  0       0       1       0       0       1       0       1   |
+--R      |                                                              |
+--R      +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
+--R     ,
+--R
+--R      +0     1       1     %A + 1  0  0  0  0+
+--R      |                                      |
+--R      |1     1     %A + 1    0     0  0  0  0|
+--R      |                                      |
+--R      |%A    0       0       0     0  0  0  0|
+--R      |                                      |
+--R      |1     %A      0       0     0  0  0  0|
+--R     [|                                      |,
+--R      |%A  %A + 1    1       1     1  0  1  1|
+--R      |                                      |
+--R      |0     0       %A      1     0  1  0  1|
+--R      |                                      |
+--R      |%A    1       0       1     1  1  0  0|
+--R      |                                      |
+--R      +1     %A    %A + 1    %A    0  1  0  0+
+--R      +%A + 1    1       %A      0       0     %A + 1    0       1   +
+--R      |                                                              |
+--R      |  0       %A      1       1       1       0     %A + 1    %A  |
+--R      |                                                              |
+--R      |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
+--R      |                                                              |
+--R      |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
+--R      |                                                              |]
+--R      |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
+--R      |                                                              |
+--R      |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
+--R      |                                                              |
+--R      |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
+--R      |                                                              |
+--R      +  %A      %A      %A      1       %A      %A      1     %A + 1+
+--R     ]
+--R                                      Type: List List Matrix FiniteField(2,2)
 --E 22
 
 @
 Now we find two 8-dimensional ones, dA6d8a and dA6d8b.
 <<*>>=
---S 23  of 33
+--S 23 of 33
 dA6d8a : List Matrix gf4  := sp4.1
 --R 
---R   There are no library operations named sp4 
---R      Use HyperDoc Browse or issue
---R                                )what op sp4
---R      to learn if there is any operation containing " sp4 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp4 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (23)
+--R    +  %A    %A + 1    0       %A      1     %A + 1    0     0 +
+--R    |                                                          |
+--R    |  0       0       %A    %A + 1    %A      %A      0     0 |
+--R    |                                                          |
+--R    |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
+--R    |                                                          |
+--R    |  %A    %A + 1    %A      1       %A      0       0     0 |
+--R   [|                                                          |,
+--R    |%A + 1    1       1       1       0       0     %A + 1  %A|
+--R    |                                                          |
+--R    |  0       0     %A + 1    1       0       0       %A    0 |
+--R    |                                                          |
+--R    |  1       0       1       1       0       0       0     0 |
+--R    |                                                          |
+--R    +  1       1       0       0       0       0       0     0 +
+--R    +  1       0       %A      0       1       1       %A    %A + 1+
+--R    |                                                              |
+--R    |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
+--R    |                                                              |
+--R    |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
+--R    |                                                              |
+--R    |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
+--R    |                                                              |]
+--R    |  1       0     %A + 1    0       1       1       %A      %A  |
+--R    |                                                              |
+--R    |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
+--R    |                                                              |
+--R    |  0       0       1       0       0       1       0       1   |
+--R    |                                                              |
+--R    +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
+--R                                           Type: List Matrix FiniteField(2,2)
 --E 23
 
 --S 24 of 33
 dA6d8b : List Matrix gf4  := sp4.2
 --R 
---R   There are no library operations named sp4 
---R      Use HyperDoc Browse or issue
---R                                )what op sp4
---R      to learn if there is any operation containing " sp4 " in its 
---R      name.
---R 
---RDaly Bug
---R   Cannot find a definition or applicable library operation named sp4 
---R      with argument type(s) 
---R                               PositiveInteger
---R      
---R      Perhaps you should use "@" to indicate the required return type, 
---R      or "$" to specify which version of the function you need.
+--R
+--R   (24)
+--R    +0     1       1     %A + 1  0  0  0  0+
+--R    |                                      |
+--R    |1     1     %A + 1    0     0  0  0  0|
+--R    |                                      |
+--R    |%A    0       0       0     0  0  0  0|
+--R    |                                      |
+--R    |1     %A      0       0     0  0  0  0|
+--R   [|                                      |,
+--R    |%A  %A + 1    1       1     1  0  1  1|
+--R    |                                      |
+--R    |0     0       %A      1     0  1  0  1|
+--R    |                                      |
+--R    |%A    1       0       1     1  1  0  0|
+--R    |                                      |
+--R    +1     %A    %A + 1    %A    0  1  0  0+
+--R    +%A + 1    1       %A      0       0     %A + 1    0       1   +
+--R    |                                                              |
+--R    |  0       %A      1       1       1       0     %A + 1    %A  |
+--R    |                                                              |
+--R    |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
+--R    |                                                              |
+--R    |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
+--R    |                                                              |]
+--R    |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
+--R    |                                                              |
+--R    |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
+--R    |                                                              |
+--R    |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
+--R    |                                                              |
+--R    +  %A      %A      %A      1       %A      %A      1     %A + 1+
+--R                                           Type: List Matrix FiniteField(2,2)
 --E 24
 
 @
 Both are absolutely irreducible...
 <<*>>=
---S 25  of 33
+--S 25 of 33 random generation, FAILURE OK.
 isAbsolutelyIrreducible? dA6d8a
 --R 
---R 
---RDaly Bug
---R   dA6d8a is declared as being in List Matrix FiniteField(2,2) but has 
---R      not been given a value.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (25)  true
+--R                                                                Type: Boolean
 --E 25
 
---S 26 of 33
+--S 26 of 33 random generation, FAILURE OK.
 isAbsolutelyIrreducible? dA6d8b
 --R 
---R 
---RDaly Bug
---R   dA6d8b is declared as being in List Matrix FiniteField(2,2) but has 
---R      not been given a value.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     We know that all the cyclic submodules generated by all
+--R       non-trivial element of the singular matrix under view are
+--R       not proper, hence Norton's irreducibility test can be done:
+--R     The generated cyclic submodule was not proper
+--R     Representation is absolutely irreducible
+--R
+--R   (26)  true
+--R                                                                Type: Boolean
 --E 26
 
 @
 And they are not equivalent...
 <<*>>=
---S 27  of 33
+--S 27 of 33 random generation, FAILURE OK.
 areEquivalent? ( dA6d8a, dA6d8b )
 --R 
---R 
---RDaly Bug
---R   dA6d8a is declared as being in List Matrix FiniteField(2,2) but has 
---R      not been given a value.
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra does
+--R     not have a one-dimensional kernel
+--R   Random element in generated algebra has
+--R     one-dimensional kernel
+--R     There is no isomorphism, as the only possible one
+--R       fails to do the necessary base change
+--R
+--R   Representations are not equivalent.
+--R
+--R   (27)  [0]
+--R                                                Type: Matrix FiniteField(2,2)
 --E 27
 
 @ 
@@ -535,11 +946,11 @@ The theory tells us that there are no more irreducible ones.
 Here again are all absolutely irreducible 2-modular
 representations of A6
 <<*>>=
---S 28  of 33
+--S 28 of 33
 dA6d1
 --R 
 --R
---R   (11)  [[1],[1]]
+--R   (28)  [[1],[1]]
 --R                                               Type: List Matrix PrimeField 2
 --E 28
 
@@ -550,7 +961,7 @@ dA6d4a
 --R          +0  1  0  0+ +0  1  1  1+
 --R          |          | |          |
 --R          |0  0  1  0| |1  1  0  1|
---R   (12)  [|          |,|          |]
+--R   (29)  [|          |,|          |]
 --R          |1  0  0  0| |1  1  1  0|
 --R          |          | |          |
 --R          +0  0  0  1+ +1  1  1  1+
@@ -561,26 +972,90 @@ dA6d4a
 dA6d4b
 --R 
 --R
---R   (13)  dA6d4b
---R                                                        Type: Variable dA6d4b
+--R          +1  0  1  1+ +0  0  1  0+
+--R          |          | |          |
+--R          |0  1  0  1| |1  1  1  1|
+--R   (30)  [|          |,|          |]
+--R          |1  1  0  0| |1  0  1  1|
+--R          |          | |          |
+--R          +0  1  0  0+ +0  1  0  1+
+--R                                               Type: List Matrix PrimeField 2
 --E 30
 
 --S 31 of 33
 dA6d8a
 --R 
---R 
---RDaly Bug
---R   dA6d8a is declared as being in List Matrix FiniteField(2,2) but has 
---R      not been given a value.
+--R
+--R   (31)
+--R    +  %A    %A + 1    0       %A      1     %A + 1    0     0 +
+--R    |                                                          |
+--R    |  0       0       %A    %A + 1    %A      %A      0     0 |
+--R    |                                                          |
+--R    |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
+--R    |                                                          |
+--R    |  %A    %A + 1    %A      1       %A      0       0     0 |
+--R   [|                                                          |,
+--R    |%A + 1    1       1       1       0       0     %A + 1  %A|
+--R    |                                                          |
+--R    |  0       0     %A + 1    1       0       0       %A    0 |
+--R    |                                                          |
+--R    |  1       0       1       1       0       0       0     0 |
+--R    |                                                          |
+--R    +  1       1       0       0       0       0       0     0 +
+--R    +  1       0       %A      0       1       1       %A    %A + 1+
+--R    |                                                              |
+--R    |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
+--R    |                                                              |
+--R    |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
+--R    |                                                              |
+--R    |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
+--R    |                                                              |]
+--R    |  1       0     %A + 1    0       1       1       %A      %A  |
+--R    |                                                              |
+--R    |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
+--R    |                                                              |
+--R    |  0       0       1       0       0       1       0       1   |
+--R    |                                                              |
+--R    +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
+--R                                           Type: List Matrix FiniteField(2,2)
 --E 31
 
 --S 32 of 33
 dA6d8b
 --R 
---R 
---RDaly Bug
---R   dA6d8b is declared as being in List Matrix FiniteField(2,2) but has 
---R      not been given a value.
+--R
+--R   (32)
+--R    +0     1       1     %A + 1  0  0  0  0+
+--R    |                                      |
+--R    |1     1     %A + 1    0     0  0  0  0|
+--R    |                                      |
+--R    |%A    0       0       0     0  0  0  0|
+--R    |                                      |
+--R    |1     %A      0       0     0  0  0  0|
+--R   [|                                      |,
+--R    |%A  %A + 1    1       1     1  0  1  1|
+--R    |                                      |
+--R    |0     0       %A      1     0  1  0  1|
+--R    |                                      |
+--R    |%A    1       0       1     1  1  0  0|
+--R    |                                      |
+--R    +1     %A    %A + 1    %A    0  1  0  0+
+--R    +%A + 1    1       %A      0       0     %A + 1    0       1   +
+--R    |                                                              |
+--R    |  0       %A      1       1       1       0     %A + 1    %A  |
+--R    |                                                              |
+--R    |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
+--R    |                                                              |
+--R    |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
+--R    |                                                              |]
+--R    |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
+--R    |                                                              |
+--R    |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
+--R    |                                                              |
+--R    |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
+--R    |                                                              |
+--R    +  %A      %A      %A      1       %A      %A      1     %A + 1+
+--R                                           Type: List Matrix FiniteField(2,2)
 --E 32
 
 @ 
@@ -591,8 +1066,70 @@ representations of A6 over PrimeField 2
 dA6d16
 --R 
 --R
---R   (14)  dA6d16
---R                                                        Type: Variable dA6d16
+--R   (33)
+--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
+--R   [|                                              |,
+--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
+--R    |                                              |
+--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
+--R    |                                              |
+--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
+--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
+--R    |                                              |
+--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
+--R    |                                              |]
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
+--R    |                                              |
+--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
+--R    |                                              |
+--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
+--R    |                                              |
+--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
+--R    |                                              |
+--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|
+--R    |                                              |
+--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
+--R                                               Type: List Matrix PrimeField 2
 --E 33
 )spool 
 )lisp (bye)



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Subject: [Axiom-developer] 20080107.01.tpd.patch (7089/343)
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This patch by Martin Rubey fixes the ordering of symbols in a 
free abelian group. There also is a regression test file. So

a:=a::FreeAbelianGroup Symbol
b:=b::FreeAbelianGroup Symbol
z:=0::FreeAbelianGroup Symbol

a < -b ==> false
-b < z ==> true
z < a  ==> true

Tim

=======================================================================
diff --git a/changelog b/changelog
index a4b3797..9c87000 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080107 tpd src/input/Makefile add free to test bug 7089/343
+20080107 tpd src/input/free.input added to test bug 7089/343
+20080107 mxr src/input/free.spad fix FreeAbelianGroup bug 7089/343
 20080106 tpd src/algebra/combfunc.spad \mathbb -> \bf
 20080106 tpd src/algebra/op.spad revert dgamma change, reject patch
 20080106 tpd src/scripts/tex/axiom.sty add binom for combfunc
diff --git a/src/algebra/free.spad.pamphlet b/src/algebra/free.spad.pamphlet
index 95e1771..8300350 100644
--- a/src/algebra/free.spad.pamphlet
+++ b/src/algebra/free.spad.pamphlet
@@ -531,21 +531,21 @@ FreeAbelianGroup(S:SetCategory): Exports == Implementation where
       inmax l ==
         mx := first l
         for t in rest l repeat
-          if t.gen > mx.gen then mx := t
+          if mx.gen < t.gen then mx := t
         mx
 
+      -- lexicographic order
       a < b ==
         zero? a  =>
           zero? b => false
-          (inmax terms b).exp > 0
+          0 < (inmax terms b).exp
         ta := inmax terms a
         zero? b => ta.exp < 0
-        ta := inmax terms a
         tb := inmax terms b
-        ta.gen < tb.gen => true
-        ta.gen > tb.gen => false
+        ta.gen < tb.gen => 0 < tb.exp
+        tb.gen < ta.gen => ta.exp < 0
         ta.exp < tb.exp => true
-        ta.exp > tb.exp => false
+        tb.exp < ta.exp => false
         lc := ta.exp * ta.gen
         (a - lc) < (b - lc)
 
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index df8457c..f533ad3 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -312,8 +312,8 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     fib.regress       file.regress     float1.regress   float2.regress \
     float.regress     fname1.regress   fname.regress    fnla.regress \
     fns.regress       fparfrac.regress fparfrc.regress  fr1.regress \
-    fr2.regress       frac.regress     fr.regress       galois.regress \
-    gamma.regress \
+    fr2.regress       frac.regress     fr.regress       free.regress \
+    galois.regress    gamma.regress \
     gbf.regress       genups.regress   gonshor.regress  grpthry.regress \
     gstbl.regress     heap.regress     heat.regress     help.regress \
     herm.regress      hexadec.regress  ico.regress      ideal.regress \
@@ -649,7 +649,8 @@ FILES2=${OUT}/arith.input    ${OUT}/bugs.input \
        ${OUT}/exlimit.input  \
        ${OUT}/expr.input     ${OUT}/exprode.input    ${OUT}/exseries.input \
        ${OUT}/exsum.input    ${OUT}/fns.input        \
-       ${OUT}/function.input ${OUT}/galois.input     ${OUT}/gamma.input \
+       ${OUT}/function.input ${OUT}/free.input       \
+       ${OUT}/galois.input   ${OUT}/gamma.input \
        ${OUT}/grpthry.input \
        ${OUT}/help.input       ${OUT}/intef2.input \
        ${OUT}/intmix2.input  ${OUT}/knot2.input      ${OUT}/linalg.input \
@@ -806,6 +807,7 @@ DOCFILES= \
   ${DOC}/fparfrac.input.dvi    ${DOC}/fparfrc.input.dvi    \
   ${DOC}/fr1.input.dvi         ${DOC}/fr2.input.dvi        \
   ${DOC}/frac.input.dvi        ${DOC}/fr.input.dvi         \
+  ${DOC}/free.input.dvi        \
   ${DOC}/function.input.dvi    ${DOC}/galois.input.dvi     \
   ${DOC}/gamma.input.dvi \
   ${DOC}/gary1.input.dvi       ${DOC}/gbf.input.dvi        \
diff --git a/src/input/free.input.pamphlet b/src/input/free.input.pamphlet
new file mode 100644
index 0000000..a65daf6
--- /dev/null
+++ b/src/input/free.input.pamphlet
@@ -0,0 +1,83 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input free.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+We expect that $-b < z < a < b$
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool free.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 8
+Z2:=FreeAbelianGroup Symbol
+--R
+--R   (1)  FreeAbelianGroup Symbol
+--R                                                                 Type: Domain
+--E 1
+
+--S 2 of 8
+a:=a::FreeAbelianGroup Symbol
+--R
+--R   (2)  a
+--R                                                Type: FreeAbelianGroup Symbol
+--E 2
+
+--S 3 of 8
+b:=b::FreeAbelianGroup Symbol
+--R
+--R   (3)  b
+--R                                                Type: FreeAbelianGroup Symbol
+--E 3
+
+--S 4 of 8
+z:=0::FreeAbelianGroup Symbol
+--R
+--R   (4)  0
+--R                                                Type: FreeAbelianGroup Symbol
+--E 4
+
+--S 5 of 8
+a < -b
+--R
+--R   (5)  false
+--R                                                                Type: Boolean
+--E 5
+
+--S 6 of 8
+-b < z
+--R
+--R   (6)  true
+--R                                                                Type: Boolean
+--E 6
+
+--S 7 of 8
+z < a
+--R
+--R   (7)  true
+--R                                                                Type: Boolean
+--E 7
+
+--S 8 of 8
+a < b
+--R
+--R   (8)  true
+--R                                                                Type: Boolean
+--E 8
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}



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To: daly@axiom-developer.org
Subject: Re: [Axiom-developer] 20080107.01.tpd.patch (7089/343)
References: <200801070550.m075oxE3018210@axiom-developer.org>
From: Martin Rubey <martin.rubey@univie.ac.at>
Date: 07 Jan 2008 06:57:37 +0100
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daly@axiom-developer.org writes:

> This patch by Martin Rubey fixes the ordering of symbols in a free abelian
> group.

Sorry for nitpicking again: are you sure this patch is mine?  I do not think
so, but the bug report on MathAction is unsigned (or rather: got lost during
the transition), so maybe you should attribute it to 

"the axiom/fricas/open-axiom team"


Martin




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Add a regression test file for the besselk change -- Tim

============================================================================
diff --git a/changelog b/changelog
index 41ba008..693db6a 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080107 tpd src/input/Makefile besselk.input regression test added
+20080107 tpd src/input/besselk.input regression test added
 20080107 tpd src/input/gamma.input missing file added
 20080107 tpd src/input/pfaffian.input missing file added
 20080107 tpd src/input/Makefile add free to test bug 7089/343
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index f533ad3..d727a41 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -285,7 +285,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     allfact.regress   antoine.regress \
     arith.regress     array1.regress   array2.regress \
     arrows.regress    assign.regress   atansqrt.regress \
-    asec.regress      bags.regress      bbtree.regress \
+    asec.regress      bags.regress     bbtree.regress   besselk.regress \
     binary.regress    bop.regress      bstree.regress   bouquet.regress \
     bug100.regress    bug101.regress \
     bug103.regress    bug10069.regress \
@@ -504,7 +504,8 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/array1.input   ${OUT}/array2.input     ${OUT}/arrows.input \
        ${OUT}/assign.input   ${OUT}/atansqrt.input   ${OUT}/axiom.input \
        ${OUT}/asec.input \
-       ${OUT}/bags.input     ${OUT}/bbtree.input     ${OUT}/bern.input \
+       ${OUT}/bags.input     ${OUT}/bbtree.input     ${OUT}/besselk.input \
+       ${OUT}/bern.input \
        ${OUT}/bernpoly.input ${OUT}/binary.input     ${OUT}/bop.input \
        ${OUT}/bouquet.input  ${OUT}/bstree.input     ${OUT}/bug6357.input \
        ${OUT}/bug9057.input  ${OUT}/bug100.input     ${OUT}/bug101.input \
@@ -686,7 +687,8 @@ DOCFILES= \
   ${DOC}/asec.input.dvi \
   ${DOC}/assign.input.dvi      ${DOC}/atansqrt.input.dvi   \
   ${DOC}/axiom.input.dvi       ${DOC}/bags.input.dvi       \
-  ${DOC}/bbtree.input.dvi      ${DOC}/bern.input.dvi       \
+  ${DOC}/bbtree.input.dvi      ${DOC}/besselk.input.dvi    \
+  ${DOC}/bern.input.dvi       \
   ${DOC}/bernpoly.input.dvi    ${DOC}/binary.input.dvi     \
   ${DOC}/bop.input.dvi         ${DOC}/bouquet.input.dvi    \
   ${DOC}/bstree.input.dvi      ${DOC}/bug10069.input.dvi   \
diff --git a/src/input/besselk.input.pamphlet b/src/input/besselk.input.pamphlet
new file mode 100644
index 0000000..0cd01ea
--- /dev/null
+++ b/src/input/besselk.input.pamphlet
@@ -0,0 +1,57 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input besselk.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool besselk.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 4
+D(besselK(a,x),x)
+--R
+--R        - besselK(a + 1,x) - besselK(a - 1,x)
+--R   (1)  -------------------------------------
+--R                          2
+--R                                                     Type: Expression Integer
+--E 1
+
+--S 2 of 4
+D(besselK(a,x),a)
+--R
+--R   (2)  besselK  (a,x)
+--R               ,1
+--R                                                     Type: Expression Integer
+--E 2
+
+--S 3 of 4
+integrate(D(besselK(a,x),a),a)
+--R
+--R   (3)  besselK(a,x)
+--R                                          Type: Union(Expression Integer,...)
+--E 3
+
+--S 4 of 4
+limit(D(besselK(a,x),a),a=1/2)
+--R
+--R   (4)  "failed"
+--R                                                    Type: Union("failed",...)
+--E 4
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}
+



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> Sorry for nitpicking again: are you sure this patch is mine?
I think this was me (anonymously) before I subscribed to this list.

Franz



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Subject: Re: [Axiom-developer] mathml sums and products patch
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Martin Rubey wrote:
>     Thus, it seems to me that
>
>       plexOps     := ["SIGMA","SIGMA2","PI", "PI2", "INTSIGN","INDEFINTEGRAL"]
>       plexPrecs   := [    700,     800, 700,   800,       700]
>
>     is possibly a mistake: why should SIGMA and SIGMA2 have different precs?
>     Maybe it should rather read
>
>       plexOps     := ["SIGMA","SIGMA2","PI", "PI2", "INTSIGN","INDEFINTEGRAL"]
>       plexPrecs   := [    700,     700, 800,   800,       700]
>
>     However, I do not understand why "-", "+" and "/" have the same precedence,
>     either...
>
>   

Martin,

I'm thinking about a response to the questions you posed, but in the
meantime what is an example of a command that produces SIGMA
rather than SIGMA2?

Arthur



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This started with a change in integer.spad.pamphlet from
 PositiveInteger: Join(AbelianSemiGroup,OrderedSet,Monoid) with
to
 PositiveInteger: Join(OrderedAbelianSemiGroup,Monoid) with

This had the interesting side-effect of changing complex values from

  a + -bi to a - bi

Many of the regression test files were update to reflect this change.


This change also backed out the Gamma change. Gamma used to print as

     _ 3
    |  - 
       4

but the prior patch prints is as:

     - 3
    |  - 
       4

which looks like a negative 3 so the patch was withdrawn.

The change also highlighted some missed gensyms that needed to be
ignored.

These are all minor matters and combined into one patch.

I will be away at a funeral for the week.

Tim

===================================================================
diff --git a/changelog b/changelog
index 7c5d8fe..a4b3797 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,26 @@
+20080106 tpd src/algebra/combfunc.spad \mathbb -> \bf
+20080106 tpd src/algebra/op.spad revert dgamma change, reject patch
+20080106 tpd src/scripts/tex/axiom.sty add binom for combfunc
+20080106 tpd src/doc/axiom.sty add binom for combfunc
+20080105 tpd src/input/Makefile add pfaffian
+20080105 tpd src/input/pfaffian.input created
+20080105 tpd src/algebra/acplot.spad fix output form of negative numbers
+20080105 tpd src/algebra/sf.spad fix output form of negative numbers
+20080105 tpd src/input/summation.input add mathml output
+20080105 tpd src/input/r20bugs.input change spacing in S 17
+20080105 tpd src/input/pmint.input rewrite
+20080105 tpd src/input/ndftip.input fix missing blank lines
+20080105 tpd src/input/mathml.input new mathml sub/sup handling
+20080105 tpd src/input/mapleok.input ignore gensyms
+20080105 tpd src/input/kamke3.input mark random generation failure ok
+20080105 tpd src/input/is.input type-declare the function f
+20080105 tpd src/input/intg0.input ignore gensyms
+20080105 tpd src/input/ico.input mark random generation failure ok
+20080105 tpd src/input/grpthry.input mark random generation failure ok
+20080105 tpd src/input/elemnum.input a + -bi now becomes a - bi
+20080105 tpd src/input/dfloat.input fix output form of negative numbers
+20080105 tpd src/input/classtalk.input ignore gensyms
+20080105 tpd src/algebra/integer.spad transparent(?) category change
 20080104 tpd changelog correct the author of besselk patches
 20080104 mxr src/input/repa6.input fix function names (7093)
 20080104 mxr src/input/knot2.input fix function names (7093)
diff --git a/src/algebra/acplot.spad.pamphlet b/src/algebra/acplot.spad.pamphlet
index 8a1ad4a..a39e3b6 100644
--- a/src/algebra/acplot.spad.pamphlet
+++ b/src/algebra/acplot.spad.pamphlet
@@ -228,8 +228,8 @@ makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT
 --R                       1         1      1         1
 --R        y + x = 0,   - - <= x <= -,   - - <= y <= -
 --R                       2         2      2         2
---R                        [0.5,-0.5]
---R                        [-0.5,0.5]
+--R                        [0.5,- 0.5]
+--R                        [- 0.5,0.5]
 --R                                                Type: PlaneAlgebraicCurvePlot
 --E 1
 )spool
diff --git a/src/algebra/combfunc.spad.pamphlet b/src/algebra/combfunc.spad.pamphlet
index 9da05e9..24dd91a 100644
--- a/src/algebra/combfunc.spad.pamphlet
+++ b/src/algebra/combfunc.spad.pamphlet
@@ -110,7 +110,7 @@ $$\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}.$$
 
 Alternatively, one can use the formula
 $$ \binom{n}{k}=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}, $$
-and leave the case where $k\in\mathbb Z$, $n\in\mathbb Z$ and $k \leq n < 0$
+and leave the case where $k\in {\bf Z}$, $n\in {\bf Z}$ and $k \leq n < 0$
 undefined, since the limit does not exist in this case:
 
 Since we then have that $n-k+1\geq 1$, $\Gamma(n-k+1)$ is finite. So it is
@@ -123,7 +123,7 @@ $$\lim_{k_0\to k} \lim_{n_0\to n}\frac{\Gamma(n_0+1)}{\Gamma(k_0+1)}$$
 does not exist, since for non-integral $k_0$, $\Gamma(k_0+1)$ is finite while
 $\Gamma(n_0+1)$ is unbounded.
 
-However, since for $k\in\mathbb Z$, $n\in\mathbb Z$ and $0 < k < n$ both
+However, since for $k\in {\bf Z}$, $n\in {\bf Z}$ and $0 < k < n$ both
 definitions agree, one could also combine them. This is what, for example,
 Mathematica does. It seems that MuPAD sets [[binomial(n,n)=1]] for all
 arguments [[n]], and returns [[binomial(-2, n)]] unevaluated. Provisos may help
diff --git a/src/algebra/integer.spad.pamphlet b/src/algebra/integer.spad.pamphlet
index 16ac727..c415a4e 100644
--- a/src/algebra/integer.spad.pamphlet
+++ b/src/algebra/integer.spad.pamphlet
@@ -1323,7 +1323,7 @@ Note that this code is not included in the generated catdef.spad file.
 ++ Keywords: positive integer
 ++ Description: \spadtype{PositiveInteger} provides functions for
 ++   positive integers.
-PositiveInteger: Join(AbelianSemiGroup,OrderedSet,Monoid) with
+PositiveInteger: Join(OrderedAbelianSemiGroup,Monoid) with
             gcd: (%,%) -> %
               ++ gcd(a,b) computes the greatest common divisor of two
               ++ positive integers \spad{a} and b.
diff --git a/src/algebra/op.spad.pamphlet b/src/algebra/op.spad.pamphlet
index 51d4d6c..cd18222 100644
--- a/src/algebra/op.spad.pamphlet
+++ b/src/algebra/op.spad.pamphlet
@@ -749,23 +749,7 @@ CommonOperators(): Exports == Implementation where
     dpi l    == "%pi"::Symbol::O
     dfact x  == postfix("!"::Symbol::O, (ATOM(x)$Lisp => x; paren x))
     dquote l == prefix(quote(first(l)::O), rest l)
-@
-It is certainly an abuse of OutputForm to produce a Gamma as done below.
-Originally, it was even worse (Issue~\#6):
-\begin{verbatim}
     dgamma l == prefix(hconcat("|"::Symbol::O, overbar(" "::Symbol::O)), l)
-\end{verbatim}
-which was TeXed to
-$${|{\overline{\ }}}
-\left(
-{x}
-\right).
-$$
-
-The right thing would be to introduce Greek letters in OutputForm, but
-that should be coordinated with the new mathml package
-<<package COMMONOP CommonOperators>>=
-    dgamma l == prefix(super("|"::Symbol::O, "-"::Symbol::O), l)
     setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)
 
     dexp x ==
diff --git a/src/algebra/sf.spad.pamphlet b/src/algebra/sf.spad.pamphlet
index 571ccd0..1517c92 100644
--- a/src/algebra/sf.spad.pamphlet
+++ b/src/algebra/sf.spad.pamphlet
@@ -843,7 +843,7 @@ avg [3.4,9.7,-6.8]
 cos(3.1415926)$DoubleFloat
 --R 
 --R
---R   (9)  -0.99999999999999856
+--R   (9)  - 0.99999999999999856
 --R                                                            Type: DoubleFloat
 --E 9
 
@@ -851,7 +851,7 @@ cos(3.1415926)$DoubleFloat
 cos(3.1415926 :: DoubleFloat)
 --R 
 --R
---R   (10)  -0.99999999999999856
+--R   (10)  - 0.99999999999999856
 --R                                                            Type: DoubleFloat
 --E 10
 )spool
diff --git a/src/doc/axiom.sty.pamphlet b/src/doc/axiom.sty.pamphlet
index cce87ad..a8ffdba 100644
--- a/src/doc/axiom.sty.pamphlet
+++ b/src/doc/axiom.sty.pamphlet
@@ -1613,6 +1613,8 @@ macro. There must be a more elegant way of coding the tests below\ldots
 % We use providecommand so it doesn't clash with amsmath package
 \providecommand{\eth}[1]{{#1}-th}
 
+% add the binom function for combfunc
+\providecommand{\binom}[2]{\left(\begin{array}{c}#1\\#2\end{array}\right)}
 
 @ 
 \subsubsection{Support for English}
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 8b79668..df8457c 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -313,6 +313,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     float.regress     fname1.regress   fname.regress    fnla.regress \
     fns.regress       fparfrac.regress fparfrc.regress  fr1.regress \
     fr2.regress       frac.regress     fr.regress       galois.regress \
+    gamma.regress \
     gbf.regress       genups.regress   gonshor.regress  grpthry.regress \
     gstbl.regress     heap.regress     heat.regress     help.regress \
     herm.regress      hexadec.regress  ico.regress      ideal.regress \
@@ -342,7 +343,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     padic.regress     parabola.regress pascal1.regress  pascal.regress \
     patch51.regress   page.regress \
     patmatch.regress  pat.regress      perman.regress   perm.regress \
-    pfr1.regress      pfr.regress      pmint.regress \
+    pfaffian.regress  pfr1.regress     pfr.regress      pmint.regress \
     poly1.regress     polycoer.regress poly.regress     psgenfcn.regress \
     quat1.regress     quat.regress     r20abugs.regress r20bugs.regress \
     r21bugsbig.regress r21bugs.regress radff.regress    radix.regress \
@@ -599,7 +600,7 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/pascal.input   \
        ${OUT}/patch51.input \
        ${OUT}/patmatch.input ${OUT}/perman.input \
-       ${OUT}/perm.input     \
+       ${OUT}/perm.input     ${OUT}/pfaffian.input \
        ${OUT}/pfr.input      ${OUT}/pfr1.input \
        ${OUT}/pinch.input    ${OUT}/plotfile.input   ${OUT}/pollevel.input \
        ${OUT}/pmint.input    ${OUT}/polycoer.input \
@@ -648,7 +649,7 @@ FILES2=${OUT}/arith.input    ${OUT}/bugs.input \
        ${OUT}/exlimit.input  \
        ${OUT}/expr.input     ${OUT}/exprode.input    ${OUT}/exseries.input \
        ${OUT}/exsum.input    ${OUT}/fns.input        \
-       ${OUT}/function.input ${OUT}/galois.input     \
+       ${OUT}/function.input ${OUT}/galois.input     ${OUT}/gamma.input \
        ${OUT}/grpthry.input \
        ${OUT}/help.input       ${OUT}/intef2.input \
        ${OUT}/intmix2.input  ${OUT}/knot2.input      ${OUT}/linalg.input \
@@ -806,6 +807,7 @@ DOCFILES= \
   ${DOC}/fr1.input.dvi         ${DOC}/fr2.input.dvi        \
   ${DOC}/frac.input.dvi        ${DOC}/fr.input.dvi         \
   ${DOC}/function.input.dvi    ${DOC}/galois.input.dvi     \
+  ${DOC}/gamma.input.dvi \
   ${DOC}/gary1.input.dvi       ${DOC}/gbf.input.dvi        \
   ${DOC}/genups.input.dvi      ${DOC}/gnarly1.input.dvi    \
   ${DOC}/gonshor.input.dvi     ${DOC}/graphics.input.dvi   \
@@ -881,7 +883,7 @@ DOCFILES= \
   ${DOC}/pat.input.dvi         ${DOC}/patch51.input.dvi   \
   ${DOC}/patmatch.input.dvi   \
   ${DOC}/pdecomp0.as.dvi       ${DOC}/perman.input.dvi     \
-  ${DOC}/perm.input.dvi        \
+  ${DOC}/perm.input.dvi        ${DOC}/pfaffian.input.dvi   \
   ${DOC}/pfr1.input.dvi       \
   ${DOC}/pfr.input.dvi         ${DOC}/pinch.input.dvi      \
   ${DOC}/plotfile.input.dvi    ${DOC}/plotlist.input.dvi   \
diff --git a/src/input/classtalk.input.pamphlet b/src/input/classtalk.input.pamphlet
index 7f89df1..2f3eca3 100644
--- a/src/input/classtalk.input.pamphlet
+++ b/src/input/classtalk.input.pamphlet
@@ -705,7 +705,7 @@ solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])
 --S 70
 solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])
 --R
---R   (2)  [[x= %W - 2,y= - 2%W + 2,z= %W]]
+--I   (2)  [[x= %W - 2,y= - 2%W + 2,z= %W]]
 --R                         Type: List List Equation Fraction Polynomial Integer
 --E 70
 
diff --git a/src/input/dfloat.input.pamphlet b/src/input/dfloat.input.pamphlet
index f3e3468..75eeb5f 100644
--- a/src/input/dfloat.input.pamphlet
+++ b/src/input/dfloat.input.pamphlet
@@ -86,7 +86,7 @@ avg [3.4,9.7,-6.8]
 cos(3.1415926)$DoubleFloat
 --R 
 --R
---R   (9)  -0.99999999999999856
+--R   (9)  - 0.99999999999999856
 --R                                                            Type: DoubleFloat
 --E 9
 
@@ -94,7 +94,7 @@ cos(3.1415926)$DoubleFloat
 cos(3.1415926 :: DoubleFloat)
 --R 
 --R
---R   (10)  -0.99999999999999856
+--R   (10)  - 0.99999999999999856
 --R                                                            Type: DoubleFloat
 --E 10
 )spool
diff --git a/src/input/elemnum.input.pamphlet b/src/input/elemnum.input.pamphlet
index 0581410..137b32d 100644
--- a/src/input/elemnum.input.pamphlet
+++ b/src/input/elemnum.input.pamphlet
@@ -176,7 +176,7 @@ x := 0.7::DoubleFloat
 --R 
 --R
 --R   (18)
---R   [0.69999999999999996, 0.69999999999999996, 0.69999999999999996,
+--R   [0.69999999999999996, 0.70000000000000007, 0.69999999999999996,
 --R    0.69999999999999996]
 --R                                                       Type: List DoubleFloat
 --E 18
@@ -255,7 +255,7 @@ x := 1.1::DoubleFloat
 --R 
 --R
 --R   (26)
---R   [1.1000000000000001, 1.1000000000000001, 1.1000000000000001,
+--R   [1.1000000000000003, 1.1000000000000001, 1.1000000000000001,
 --R    1.1000000000000001, 1.0999999999999999, 1.1000000000000001]
 --R                                                       Type: List DoubleFloat
 --E 26
@@ -351,25 +351,24 @@ qtest(sa, sb, 1)
 --R      PositiveInteger) -> List Complex DoubleFloat 
 --R
 --R   (35)
---R   [1.1102230246251565E-16,
---R    2.2204460492503131E-16 + -4.4408920985006262E-16 %i,
---R    -2.2204460492503131E-16,
---R    -4.4408920985006262E-16 + -2.2204460492503131E-16 %i,
---R    4.4408920985006262E-16, -1.1102230246251565E-16,
---R    -2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
+--R   [1.1102230246251565E-16, 2.2204460492503131E-16 - 4.4408920985006262E-16 %i,
+--R    - 2.2204460492503131E-16,
+--R    - 4.4408920985006262E-16 - 2.2204460492503131E-16 %i,
+--R    4.4408920985006262E-16, - 1.1102230246251565E-16,
+--R    - 2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
 --R    1.1102230246251565E-16,
---R    -2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
+--R    - 2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
 --R    2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
---R    1.1102230246251565E-16 + -2.2204460492503131E-16 %i, 0.,
---R    -6.6613381477509392E-16 %i, -1.1102230246251565E-16,
---R    -4.4408920985006262E-16 + -4.4408920985006262E-16 %i,
---R    3.3306690738754696E-16 + -2.2204460492503131E-16 %i,
---R    -3.3306690738754696E-16,
---R    -4.4408920985006262E-16 + -4.4408920985006262E-16 %i,
---R    3.3306690738754696E-16 + -2.2204460492503131E-16 %i,
---R    -5.5511151231257827E-16, -2.2204460492503131E-16, -1.1102230246251565E-16,
---R    1.1102230246251565E-16, 0., -1.1102230246251565E-16,
---R    -1.1102230246251565E-16]
+--R    1.1102230246251565E-16 - 2.2204460492503131E-16 %i, 0.,
+--R    - 6.6613381477509392E-16 %i, - 1.1102230246251565E-16,
+--R    - 4.4408920985006262E-16 - 4.4408920985006262E-16 %i,
+--R    3.3306690738754696E-16 - 2.2204460492503131E-16 %i,
+--R    - 3.3306690738754696E-16,
+--R    - 4.4408920985006262E-16 - 4.4408920985006262E-16 %i,
+--R    3.3306690738754696E-16 - 2.2204460492503131E-16 %i,
+--R    - 5.5511151231257827E-16, - 2.2204460492503131E-16,
+--R    - 1.1102230246251565E-16, 1.1102230246251565E-16, 0.,
+--R    - 1.1102230246251565E-16, - 1.1102230246251565E-16]
 --R                                               Type: List Complex DoubleFloat
 --E 35
 
@@ -414,25 +413,24 @@ qtest(sa, sb, 2)
 --R 
 --R
 --R   (39)
---R   [-1.1102230246251565E-16,
---R    -2.2204460492503131E-16 + -4.4408920985006262E-16 %i,
---R    -3.3306690738754696E-16,
---R    4.4408920985006262E-16 + -2.2204460492503131E-16 %i,
---R    -4.4408920985006262E-16,
---R    1.1102230246251565E-16 + -2.2204460492503131E-16 %i,
---R    3.3306690738754696E-16 + -2.2204460492503131E-16 %i,
---R    -1.1102230246251565E-16,
---R    4.4408920985006262E-16 + -2.2204460492503131E-16 %i,
---R    -2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
---R    2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
---R    -1.1102230246251565E-16 + -2.2204460492503131E-16 %i,
---R    2.2204460492503131E-16 + -2.2204460492503131E-16 %i, 0.,
---R    4.4408920985006262E-16 + -4.4408920985006262E-16 %i, 0.,
---R    3.3306690738754696E-16,
---R    4.4408920985006262E-16 + -4.4408920985006262E-16 %i, 0.,
---R    8.8817841970012523E-16, 2.2204460492503131E-16, 1.1102230246251565E-16,
---R    3.3306690738754696E-16, -1.1102230246251565E-16, 1.1102230246251565E-16,
---R    2.2204460492503131E-16 + -2.2204460492503131E-16 %i]
+--R   [- 1.1102230246251565E-16,
+--R    - 2.2204460492503131E-16 - 4.4408920985006262E-16 %i,
+--R    - 3.3306690738754696E-16,
+--R    4.4408920985006262E-16 - 2.2204460492503131E-16 %i,
+--R    - 4.4408920985006262E-16,
+--R    1.1102230246251565E-16 - 2.2204460492503131E-16 %i,
+--R    3.3306690738754696E-16 - 2.2204460492503131E-16 %i,
+--R    - 1.1102230246251565E-16,
+--R    4.4408920985006262E-16 - 2.2204460492503131E-16 %i,
+--R    - 2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
+--R    2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
+--R    - 1.1102230246251565E-16 - 2.2204460492503131E-16 %i,
+--R    2.2204460492503131E-16 - 2.2204460492503131E-16 %i, 0.,
+--R    4.4408920985006262E-16 - 4.4408920985006262E-16 %i, 0.,
+--R    3.3306690738754696E-16, 4.4408920985006262E-16 - 4.4408920985006262E-16 %i,
+--R    0., 8.8817841970012523E-16, 2.2204460492503131E-16, 1.1102230246251565E-16,
+--R    3.3306690738754696E-16, - 1.1102230246251565E-16, 1.1102230246251565E-16,
+--R    2.2204460492503131E-16 - 2.2204460492503131E-16 %i]
 --R                                               Type: List Complex DoubleFloat
 --E 39
 
@@ -472,20 +470,20 @@ qtest(sa, sb, 3)
 --R 
 --R
 --R   (43)
---R   [-1.1102230246251565E-16, 0.,
---R    -2.2204460492503131E-16 + 4.4408920985006262E-16 %i,
---R    -2.2204460492503131E-16, 1.1102230246251565E-16, -4.4408920985006262E-16,
---R    -1.1102230246251565E-16, -1.1102230246251565E-16,
+--R   [- 1.1102230246251565E-16, 0.,
+--R    - 2.2204460492503131E-16 + 4.4408920985006262E-16 %i,
+--R    - 2.2204460492503131E-16, 1.1102230246251565E-16, - 4.4408920985006262E-16,
+--R    - 1.1102230246251565E-16, - 1.1102230246251565E-16,
 --R    4.4408920985006262E-16 + 2.2204460492503131E-16 %i,
---R    -2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
+--R    - 2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
 --R    2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
---R    -1.1102230246251565E-16 + 2.2204460492503131E-16 %i,
+--R    - 1.1102230246251565E-16 + 2.2204460492503131E-16 %i,
 --R    2.2204460492503131E-16 + 2.2204460492503131E-16 %i, 0., 0.,
---R    -3.3306690738754696E-16 + 2.2204460492503131E-16 %i,
+--R    - 3.3306690738754696E-16 + 2.2204460492503131E-16 %i,
 --R    2.2204460492503131E-16 %i, 0.,
---R    -3.3306690738754696E-16 + 2.2204460492503131E-16 %i,
+--R    - 3.3306690738754696E-16 + 2.2204460492503131E-16 %i,
 --R    8.8817841970012523E-16, 2.2204460492503131E-16, 1.1102230246251565E-16,
---R    3.3306690738754696E-16, -1.1102230246251565E-16, 1.1102230246251565E-16,
+--R    3.3306690738754696E-16, - 1.1102230246251565E-16, 1.1102230246251565E-16,
 --R    2.2204460492503131E-16 + 2.2204460492503131E-16 %i]
 --R                                               Type: List Complex DoubleFloat
 --E 43
@@ -529,16 +527,16 @@ qtest(sa, sb, 4)
 --R   (47)
 --R   [1.1102230246251565E-16, 0.,
 --R    3.3306690738754696E-16 + 4.4408920985006262E-16 %i, 2.2204460492503131E-16,
---R    -1.1102230246251565E-16,
---R    2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
+--R    - 1.1102230246251565E-16,
+--R    2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
 --R    2.2204460492503131E-16 + 2.2204460492503131E-16 %i, 1.1102230246251565E-16,
---R    -2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
---R    2.2204460492503131E-16 + -2.2204460492503131E-16 %i,
+--R    - 2.2204460492503131E-16 + 2.2204460492503131E-16 %i,
+--R    2.2204460492503131E-16 - 2.2204460492503131E-16 %i,
 --R    1.1102230246251565E-16 + 2.2204460492503131E-16 %i, 0.,
---R    6.6613381477509392E-16 %i, -1.1102230246251565E-16, 0., 0.,
---R    2.2204460492503131E-16 %i, 0., 0., -6.6613381477509392E-16,
---R    -2.2204460492503131E-16, -1.1102230246251565E-16, 1.1102230246251565E-16,
---R    0., -1.1102230246251565E-16, -1.1102230246251565E-16]
+--R    6.6613381477509392E-16 %i, - 1.1102230246251565E-16, 0., 0.,
+--R    2.2204460492503131E-16 %i, 0., 0., - 6.6613381477509392E-16,
+--R    - 2.2204460492503131E-16, - 1.1102230246251565E-16, 1.1102230246251565E-16,
+--R    0., - 1.1102230246251565E-16, - 1.1102230246251565E-16]
 --R                                               Type: List Complex DoubleFloat
 --E 47
 
diff --git a/src/input/grpthry.input.pamphlet b/src/input/grpthry.input.pamphlet
index b0296a4..e713c5c 100644
--- a/src/input/grpthry.input.pamphlet
+++ b/src/input/grpthry.input.pamphlet
@@ -1697,7 +1697,7 @@ ma := meatAxe r28.1
 --R                                          Type: List List Matrix PrimeField 2
 --E 34
 
---S 35 of 68
+--S 35 of 68 random generation, FAILURE OK.
 isAbsolutelyIrreducible? ma.1
 --R 
 --R   Random element in generated algebra does
diff --git a/src/input/ico.input.pamphlet b/src/input/ico.input.pamphlet
index 1643180..f096f56 100644
--- a/src/input/ico.input.pamphlet
+++ b/src/input/ico.input.pamphlet
@@ -344,7 +344,7 @@ p2:=point([real ca2,imag ca2,0.5])@Point(SF)
 p3:=point([real ca3,imag ca3,0.5])@Point(SF)
 --R 
 --R
---R   (34)  [-0.80901699437494745,0.58778525229247314,0.5]
+--R   (34)  [- 0.80901699437494745,0.58778525229247314,0.5]
 --R                                                      Type: Point DoubleFloat
 --E 35
 
@@ -352,7 +352,7 @@ p3:=point([real ca3,imag ca3,0.5])@Point(SF)
 p4:=point([real ca4,imag ca4,0.5])@Point(SF)
 --R 
 --R
---R   (35)  [-0.80901699437494745,-0.58778525229247314,0.5]
+--R   (35)  [- 0.80901699437494745,- 0.58778525229247314,0.5]
 --R                                                      Type: Point DoubleFloat
 --E 36
 
@@ -360,7 +360,7 @@ p4:=point([real ca4,imag ca4,0.5])@Point(SF)
 p5:=point([real ca5,imag ca5,0.5])@Point(SF)
 --R 
 --R
---R   (36)  [0.30901699437494745,-0.95105651629515353,0.5]
+--R   (36)  [0.30901699437494745,- 0.95105651629515353,0.5]
 --R                                                      Type: Point DoubleFloat
 --E 37
 
@@ -368,7 +368,7 @@ p5:=point([real ca5,imag ca5,0.5])@Point(SF)
 p6:=point([real cb1,imag cb1,-0.5])@Point(SF)
 --R 
 --R
---R   (37)  [0.80901699437494745,0.58778525229247314,-0.5]
+--R   (37)  [0.80901699437494745,0.58778525229247314,- 0.5]
 --R                                                      Type: Point DoubleFloat
 --E 38
 
@@ -376,7 +376,7 @@ p6:=point([real cb1,imag cb1,-0.5])@Point(SF)
 p7:=point([real cb2,imag cb2,-0.5])@Point(SF)
 --R 
 --R
---R   (38)  [-0.30901699437494745,0.95105651629515353,-0.5]
+--R   (38)  [- 0.30901699437494745,0.95105651629515353,- 0.5]
 --R                                                      Type: Point DoubleFloat
 --E 39
 
@@ -384,7 +384,7 @@ p7:=point([real cb2,imag cb2,-0.5])@Point(SF)
 p8:=point([real cb3,imag cb3,-0.5])@Point(SF)
 --R 
 --R
---R   (39)  [-1.,0.,-0.5]
+--R   (39)  [- 1.,0.,- 0.5]
 --R                                                      Type: Point DoubleFloat
 --E 40
 
@@ -392,7 +392,7 @@ p8:=point([real cb3,imag cb3,-0.5])@Point(SF)
 p9:=point([real cb4,imag cb4,-0.5])@Point(SF)
 --R 
 --R
---R   (40)  [-0.30901699437494745,-0.95105651629515353,-0.5]
+--R   (40)  [- 0.30901699437494745,- 0.95105651629515353,- 0.5]
 --R                                                      Type: Point DoubleFloat
 --E 41
 
@@ -400,7 +400,7 @@ p9:=point([real cb4,imag cb4,-0.5])@Point(SF)
 p10:=point([real cb5,imag cb5,-0.5])@Point(SF)
 --R 
 --R
---R   (41)  [0.80901699437494745,-0.58778525229247314,-0.5]
+--R   (41)  [0.80901699437494745,- 0.58778525229247314,- 0.5]
 --R                                                      Type: Point DoubleFloat
 --E 42
 
@@ -408,7 +408,7 @@ p10:=point([real cb5,imag cb5,-0.5])@Point(SF)
 p11:=point([0,0,-u-1/2])@Point(SF)
 --R 
 --R
---R   (42)  [0.,0.,-1.1180339887498949]
+--R   (42)  [0.,0.,- 1.1180339887498949]
 --R                                                      Type: Point DoubleFloat
 --E 43
 
diff --git a/src/input/intg0.input.pamphlet b/src/input/intg0.input.pamphlet
index ea802ef..bc5a258 100644
--- a/src/input/intg0.input.pamphlet
+++ b/src/input/intg0.input.pamphlet
@@ -32,7 +32,7 @@ y := sqrt(a * x + b)
 --E 1
 
 --S 2 of 25
-integrate(%,x)
+integrate(y,x)
 --R 
 --R
 --R                    +-------+
@@ -43,7 +43,7 @@ integrate(%,x)
 --E 2
 
 --S 3 of 25
-x * y
+t1:=x * y
 --R 
 --R
 --R          +-------+
@@ -52,7 +52,7 @@ x * y
 --E 3
 
 --S 4 of 25
-integrate(%,x)
+integrate(t1,x)
 --R 
 --R
 --R           2 2              2  +-------+
@@ -74,7 +74,7 @@ z := sqrt(a**2 - x**2)
 --E 5
 
 --S 6 of 25
-1 / z
+t2:=1 / z
 --R 
 --R
 --R              1
@@ -86,7 +86,7 @@ z := sqrt(a**2 - x**2)
 --E 6
 
 --S 7 of 25
-integrate(%,x)
+integrate(t2,x)
 --R 
 --R
 --R                 +---------+
@@ -98,7 +98,7 @@ integrate(%,x)
 --E 7
 
 --S 8 of 25
-x**2 * z
+t3:=x**2 * z
 --R 
 --R
 --R           +---------+
@@ -108,7 +108,7 @@ x**2 * z
 --E 8
 
 --S 9 of 25
-integrate(%,x)
+integrate(t3,x)
 --R 
 --R
 --R   (9)
@@ -133,7 +133,7 @@ integrate(%,x)
 --E 9
 
 --S 10 of 25
-x**3 / (a+b*x)**(1/3)
+t4:=x**3 / (a+b*x)**(1/3)
 --R 
 --R
 --R              3
@@ -145,7 +145,7 @@ x**3 / (a+b*x)**(1/3)
 --E 10
 
 --S 11 of 25
-integrate(%,x)
+integrate(t4,x)
 --R 
 --R
 --R              3 3         2 2       2          3 3+-------+2
@@ -157,7 +157,7 @@ integrate(%,x)
 --E 11
 
 --S 12 of 25
-1 / (x**3 * (a+b*x)**(1/3))
+t5:=1 / (x**3 * (a+b*x)**(1/3))
 --R 
 --R
 --R               1
@@ -168,7 +168,7 @@ integrate(%,x)
 --E 12
 
 --S 13 of 25
-integrate(%,x)
+integrate(t5,x)
 --R 
 --R
 --R   (13)
@@ -192,7 +192,7 @@ integrate(%,x)
 Examples of transcendentals over a curve of genus 0
 <<*>>=
 --S 14 of 25
-x / (y + y**2) + log(y + 1)
+t6:=x / (y + y**2) + log(y + 1)
 --R 
 --R
 --R           +-------+                +-------+
@@ -204,7 +204,7 @@ x / (y + y**2) + log(y + 1)
 --E 14
 
 --S 15 of 25
-integrate(%,x)
+integrate(t6,x)
 --R 
 --R
 --R   (15)
@@ -220,7 +220,7 @@ integrate(%,x)
 --E 15
 
 --S 16 of 25
-(2 + 1/sqrt(x)) * cos(x + sqrt x)
+t7:=(2 + 1/sqrt(x)) * cos(x + sqrt x)
 --R 
 --R
 --R            +-+          +-+
@@ -232,7 +232,7 @@ integrate(%,x)
 --E 16
 
 --S 17 of 25
-integrate(%,x)
+integrate(t7,x)
 --R 
 --R
 --R               +-+
@@ -244,7 +244,7 @@ integrate(%,x)
 One which does not have an elementary integral
 <<*>>=
 --S 18 of 25
-log(1 + y) / x
+t8:=log(1 + y) / x
 --R 
 --R
 --R              +-------+
@@ -255,13 +255,13 @@ log(1 + y) / x
 --E 18
 
 --S 19 of 25
-integrate(%,x)
+integrate(t8,x)
 --R 
 --R
 --R            x      +--------+
---R          ++  log(\|b + %K a  + 1)
---R   (19)   |   -------------------- d%K
---R         ++            %K
+--I          ++  log(\|b + %K a  + 1)
+--I   (19)   |   -------------------- d%K
+--I         ++            %K
 --R                                          Type: Union(Expression Integer,...)
 --E 19
 
@@ -269,7 +269,7 @@ integrate(%,x)
 Integrals of hyperbolic trigs are returned in terms of hyperb. trigs.
 <<*>>=
 --S 20 of 25
-(sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b))))
+t9:=(sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b))))
 --R 
 --R
 --R                   +-----+          +-----+
@@ -281,7 +281,7 @@ Integrals of hyperbolic trigs are returned in terms of hyperb. trigs.
 --E 20
 
 --S 21 of 25
-integrate(%,x)
+integrate(t9,x)
 --R 
 --R
 --R                              +-----+
@@ -296,7 +296,7 @@ integrate(%,x)
 Examples of genus 0 functions presented as transcendentals
 <<*>>=
 --S 22 of 25
-tan(atan(x)/2)
+t10:=tan(atan(x)/2)
 --R 
 --R
 --R             atan(x)
@@ -306,7 +306,7 @@ tan(atan(x)/2)
 --E 22
 
 --S 23 of 25
-integrate(%,x)
+integrate(t10,x)
 --R 
 --R
 --R   (23)
@@ -328,7 +328,7 @@ integrate(%,x)
 --E 23
 
 --S 24 of 25
-tan(atan(x)/3)
+t11:=tan(atan(x)/3)
 --R 
 --R
 --R             atan(x)
@@ -338,7 +338,7 @@ tan(atan(x)/3)
 --E 24
 
 --S 25 of 25
-integrate(%,x)
+integrate(t11,x)
 --R 
 --R
 --R                   atan(x) 2             atan(x) 2           atan(x)
diff --git a/src/input/is.input.pamphlet b/src/input/is.input.pamphlet
index 8ee12d6..7dd1ade 100644
--- a/src/input/is.input.pamphlet
+++ b/src/input/is.input.pamphlet
@@ -37,32 +37,37 @@ but we code using the pattern matching syntax:
 The 'integer' is to return an integer instead of a polynomial
 \end{verbatim}
 <<*>>=
---S 1 of 4
+--S 1 of 5
+f: INT -> INT
+--R                                                                   Type: Void
+--E 1
+
+--S 2 of 5
  f n ==
    not empty?(u := Is(n, 2*m%)) => integer eval(m%, u)
    3 * n + 1
 --R 
 --R                                                                   Type: Void
---E 1
+--E 2
 
 )set stream showall on
  
 @
 The function g generates the stream of the repeated applications of f
 <<*>>=
---S 2 of 4
+--S 3 of 5
 g(n:INT):STREAM(INT) == generate(f, n)
 --R 
 --R   Function declaration g : Integer -> Stream Integer has been added to
 --R      workspace.
 --R                                                                   Type: Void
---E 2
+--E 3
 
 @
 There is a conjecture that g(n) gets into the cycle (1 4 2)
 for every n
 <<*>>=
---S 3 of 4
+--S 4 of 5
 s := g 27
 --R 
 --R   Compiling function f with type Integer -> Integer 
@@ -70,9 +75,9 @@ s := g 27
 --R
 --R   (3)  [27,82,41,124,62,31,94,47,142,71,...]
 --R                                                         Type: Stream Integer
---E 3
+--E 4
 
---S 4 of 4
+--S 5 of 5
 extend(s, 150)
 --R 
 --R
@@ -88,7 +93,7 @@ extend(s, 150)
 --R    10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4,
 --R    2, 7, 22, 11, 34, 17, 52, 26, ...]
 --R                                                         Type: Stream Integer
---E 4
+--E 5
 )spool 
 )lisp (bye)
  
diff --git a/src/input/kamke3.input.pamphlet b/src/input/kamke3.input.pamphlet
index 2d9ac94..fc280ab 100644
--- a/src/input/kamke3.input.pamphlet
+++ b/src/input/kamke3.input.pamphlet
@@ -1249,7 +1249,7 @@ ode180 := (a*x**2+b*x+c)*(x*D(y(x),x)-y(x)) - y(x)**2 + x**2
 --R                                                     Type: Expression Integer
 --E 84
 
---S 85 of 139
+--S 85 of 139  random generation, FAILURE OK.
 yx:=solve(ode180,y,x)
 --R 
 --R   WARNING (genufact): No known algorithm to factor
diff --git a/src/input/mapleok.input.pamphlet b/src/input/mapleok.input.pamphlet
index c468cb3..1fb777d 100644
--- a/src/input/mapleok.input.pamphlet
+++ b/src/input/mapleok.input.pamphlet
@@ -662,50 +662,50 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R         ROOT
 --R                 +-----------------------------------------+
 --R                 |         2                          2          +-+
---R                \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R              + 
 --R                    +-+
---R                - 4\|2 %%BC0
+--I                - 4\|2 %%BC0
 --R           /
 --R                +-+
 --R              2\|2
 --R      *
 --R         log
 --R                              +-+            +-+             +-+           +-+
---R                      ((24576\|2 %%BC0 - 768\|2 )%%BC1 - 768\|2 %%BC0 - 48\|2 )
+--I                      ((24576\|2 %%BC0 - 768\|2 )%%BC1 - 768\|2 %%BC0 - 48\|2 )
 --R                   *
 --R                       +-----------------------------------------+
 --R                       |         2                          2
---R                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                  + 
 --R                                             2               2
---R                    (196608%%BC0 - 6144)%%BC1  + (196608%%BC0  + 384)%%BC1
+--I                    (196608%%BC0 - 6144)%%BC1  + (196608%%BC0  + 384)%%BC1
 --R                  + 
 --R                               2
---R                    - 6144%%BC0  + 384%%BC0 + 48
+--I                    - 6144%%BC0  + 384%%BC0 + 48
 --R               *
 --R                  ROOT
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                       + 
 --R                             +-+          +-+
---R                         - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                         - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                    /
 --R                         +-+
 --R                       2\|2
 --R              + 
---R                  ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
+--I                  ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
 --R               *
 --R                   +-----------------------------------------+
 --R                   |         2                          2
---R                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R              + 
 --R                         +-+             +-+      2
---R                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R              + 
 --R                         +-+     2      +-+              +-+     2      +-+
---R                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
+--I                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
 --R              + 
 --R                   +-+
 --R                60\|2
@@ -717,55 +717,55 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R            ROOT
 --R                    +-----------------------------------------+
 --R                    |         2                          2          +-+
---R                   \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                   \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R                 + 
 --R                       +-+
---R                   - 4\|2 %%BC0
+--I                   - 4\|2 %%BC0
 --R              /
 --R                   +-+
 --R                 2\|2
 --R         *
 --R            log
---R                         ((24576%%BC0 - 768)%%BC1 - 768%%BC0 - 48)
+--I                         ((24576%%BC0 - 768)%%BC1 - 768%%BC0 - 48)
 --R                      *
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                     + 
 --R                              +-+             +-+      2
---R                       (98304\|2 %%BC0 - 3072\|2 )%%BC1
+--I                       (98304\|2 %%BC0 - 3072\|2 )%%BC1
 --R                     + 
 --R                              +-+     2       +-+              +-+     2
---R                       (98304\|2 %%BC0  + 192\|2 )%%BC1 - 3072\|2 %%BC0
+--I                       (98304\|2 %%BC0  + 192\|2 )%%BC1 - 3072\|2 %%BC0
 --R                     + 
 --R                           +-+           +-+
---R                       192\|2 %%BC0 + 24\|2
+--I                       192\|2 %%BC0 + 24\|2
 --R                  *
 --R                     ROOT
 --R                             +-----------------------------------------+
 --R                             |         2                          2
---R                            \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                            \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                          + 
 --R                                +-+          +-+
---R                            - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                            - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                       /
 --R                            +-+
 --R                          2\|2
 --R                 + 
---R                     ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
+--I                     ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
 --R                  *
 --R                      +-----------------------------------------+
 --R                      |         2                          2
---R                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                 + 
 --R                            +-+             +-+      2
---R                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R                 + 
 --R                            +-+     2      +-+              +-+     2
---R                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
+--I                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
 --R                 + 
 --R                      +-+           +-+
---R                   48\|2 %%BC0 + 42\|2
+--I                   48\|2 %%BC0 + 42\|2
 --R              /
 --R                  +-+
 --R                 \|2
@@ -774,56 +774,56 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R            ROOT
 --R                      +-----------------------------------------+
 --R                      |         2                          2          +-+
---R                   - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                   - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R                 + 
 --R                       +-+
---R                   - 4\|2 %%BC0
+--I                   - 4\|2 %%BC0
 --R              /
 --R                   +-+
 --R                 2\|2
 --R         *
 --R            log
 --R                                    +-+            +-+             +-+
---R                             (24576\|2 %%BC0 - 768\|2 )%%BC1 - 768\|2 %%BC0
+--I                             (24576\|2 %%BC0 - 768\|2 )%%BC1 - 768\|2 %%BC0
 --R                           + 
 --R                                  +-+
 --R                             - 48\|2
 --R                      *
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                     + 
 --R                                                  2
---R                       (- 196608%%BC0 + 6144)%%BC1
+--I                       (- 196608%%BC0 + 6144)%%BC1
 --R                     + 
 --R                                     2                        2
---R                       (- 196608%%BC0  - 384)%%BC1 + 6144%%BC0  - 384%%BC0 - 48
+--I                       (- 196608%%BC0  - 384)%%BC1 + 6144%%BC0  - 384%%BC0 - 48
 --R                  *
 --R                     ROOT
 --R                               +-----------------------------------------+
 --R                               |         2                          2
---R                            - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                            - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                          + 
 --R                                +-+          +-+
---R                            - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                            - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                       /
 --R                            +-+
 --R                          2\|2
 --R                 + 
---R                     ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
+--I                     ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
 --R                  *
 --R                      +-----------------------------------------+
 --R                      |         2                          2
---R                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                 + 
 --R                            +-+             +-+      2
---R                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R                 + 
 --R                            +-+     2      +-+              +-+     2
---R                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
+--I                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
 --R                 + 
 --R                      +-+           +-+
---R                   48\|2 %%BC0 + 60\|2
+--I                   48\|2 %%BC0 + 60\|2
 --R              /
 --R                  +-+
 --R                 \|2
@@ -831,52 +831,52 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R         ROOT
 --R                   +-----------------------------------------+
 --R                   |         2                          2          +-+
---R                - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R              + 
 --R                    +-+
---R                - 4\|2 %%BC0
+--I                - 4\|2 %%BC0
 --R           /
 --R                +-+
 --R              2\|2
 --R      *
 --R         log
---R                      ((24576%%BC0 - 768)%%BC1 - 768%%BC0 - 48)
+--I                      ((24576%%BC0 - 768)%%BC1 - 768%%BC0 - 48)
 --R                   *
 --R                       +-----------------------------------------+
 --R                       |         2                          2
---R                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                  + 
 --R                             +-+             +-+      2
---R                    (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
+--I                    (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
 --R                  + 
 --R                             +-+     2       +-+              +-+     2
---R                    (- 98304\|2 %%BC0  - 192\|2 )%%BC1 + 3072\|2 %%BC0
+--I                    (- 98304\|2 %%BC0  - 192\|2 )%%BC1 + 3072\|2 %%BC0
 --R                  + 
 --R                          +-+           +-+
---R                    - 192\|2 %%BC0 - 24\|2
+--I                    - 192\|2 %%BC0 - 24\|2
 --R               *
 --R                  ROOT
 --R                            +-----------------------------------------+
 --R                            |         2                          2
---R                         - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                       + 
 --R                             +-+          +-+
---R                         - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                         - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                    /
 --R                         +-+
 --R                       2\|2
 --R              + 
---R                  ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
+--I                  ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
 --R               *
 --R                   +-----------------------------------------+
 --R                   |         2                          2
---R                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R              + 
 --R                         +-+             +-+      2
---R                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R              + 
 --R                         +-+     2      +-+              +-+     2      +-+
---R                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
+--I                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
 --R              + 
 --R                   +-+
 --R                42\|2
@@ -888,55 +888,55 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R            ROOT
 --R                      +-----------------------------------------+
 --R                      |         2                          2          +-+
---R                   - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                   - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R                 + 
 --R                       +-+
---R                   - 4\|2 %%BC0
+--I                   - 4\|2 %%BC0
 --R              /
 --R                   +-+
 --R                 2\|2
 --R         *
 --R            log
---R                         ((- 24576%%BC0 + 768)%%BC1 + 768%%BC0 + 48)
+--I                         ((- 24576%%BC0 + 768)%%BC1 + 768%%BC0 + 48)
 --R                      *
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                     + 
 --R                              +-+             +-+      2
---R                       (98304\|2 %%BC0 - 3072\|2 )%%BC1
+--I                       (98304\|2 %%BC0 - 3072\|2 )%%BC1
 --R                     + 
 --R                              +-+     2       +-+              +-+     2
---R                       (98304\|2 %%BC0  + 192\|2 )%%BC1 - 3072\|2 %%BC0
+--I                       (98304\|2 %%BC0  + 192\|2 )%%BC1 - 3072\|2 %%BC0
 --R                     + 
 --R                           +-+           +-+
---R                       192\|2 %%BC0 + 24\|2
+--I                       192\|2 %%BC0 + 24\|2
 --R                  *
 --R                     ROOT
 --R                               +-----------------------------------------+
 --R                               |         2                          2
---R                            - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                            - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                          + 
 --R                                +-+          +-+
---R                            - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                            - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                       /
 --R                            +-+
 --R                          2\|2
 --R                 + 
---R                     ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
+--I                     ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
 --R                  *
 --R                      +-----------------------------------------+
 --R                      |         2                          2
---R                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                 + 
 --R                            +-+             +-+      2
---R                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R                 + 
 --R                            +-+     2      +-+              +-+     2
---R                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
+--I                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
 --R                 + 
 --R                      +-+           +-+
---R                   48\|2 %%BC0 + 42\|2
+--I                   48\|2 %%BC0 + 42\|2
 --R              /
 --R                  +-+
 --R                 \|2
@@ -944,53 +944,53 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R         ROOT
 --R                   +-----------------------------------------+
 --R                   |         2                          2          +-+
---R                - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R              + 
 --R                    +-+
---R                - 4\|2 %%BC0
+--I                - 4\|2 %%BC0
 --R           /
 --R                +-+
 --R              2\|2
 --R      *
 --R         log
 --R                                   +-+            +-+             +-+
---R                          (- 24576\|2 %%BC0 + 768\|2 )%%BC1 + 768\|2 %%BC0
+--I                          (- 24576\|2 %%BC0 + 768\|2 )%%BC1 + 768\|2 %%BC0
 --R                        + 
 --R                             +-+
 --R                          48\|2
 --R                   *
 --R                       +-----------------------------------------+
 --R                       |         2                          2
---R                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                  + 
 --R                                             2               2
---R                    (196608%%BC0 - 6144)%%BC1  + (196608%%BC0  + 384)%%BC1
+--I                    (196608%%BC0 - 6144)%%BC1  + (196608%%BC0  + 384)%%BC1
 --R                  + 
 --R                               2
---R                    - 6144%%BC0  + 384%%BC0 + 48
+--I                    - 6144%%BC0  + 384%%BC0 + 48
 --R               *
 --R                  ROOT
 --R                            +-----------------------------------------+
 --R                            |         2                          2
---R                         - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         - \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                       + 
 --R                             +-+          +-+
---R                         - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                         - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                    /
 --R                         +-+
 --R                       2\|2
 --R              + 
---R                  ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
+--I                  ((3072%%BC0 - 384)%%BC1 - 384%%BC0 + 12)
 --R               *
 --R                   +-----------------------------------------+
 --R                   |         2                          2
---R                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R              + 
 --R                         +-+             +-+      2
---R                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R              + 
 --R                         +-+     2      +-+              +-+     2      +-+
---R                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
+--I                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
 --R              + 
 --R                   +-+
 --R                60\|2
@@ -1001,52 +1001,52 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R         ROOT
 --R                 +-----------------------------------------+
 --R                 |         2                          2          +-+
---R                \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R              + 
 --R                    +-+
---R                - 4\|2 %%BC0
+--I                - 4\|2 %%BC0
 --R           /
 --R                +-+
 --R              2\|2
 --R      *
 --R         log
---R                      ((- 24576%%BC0 + 768)%%BC1 + 768%%BC0 + 48)
+--I                      ((- 24576%%BC0 + 768)%%BC1 + 768%%BC0 + 48)
 --R                   *
 --R                       +-----------------------------------------+
 --R                       |         2                          2
---R                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                      \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                  + 
 --R                             +-+             +-+      2
---R                    (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
+--I                    (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
 --R                  + 
 --R                             +-+     2       +-+              +-+     2
---R                    (- 98304\|2 %%BC0  - 192\|2 )%%BC1 + 3072\|2 %%BC0
+--I                    (- 98304\|2 %%BC0  - 192\|2 )%%BC1 + 3072\|2 %%BC0
 --R                  + 
 --R                          +-+           +-+
---R                    - 192\|2 %%BC0 - 24\|2
+--I                    - 192\|2 %%BC0 - 24\|2
 --R               *
 --R                  ROOT
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                       + 
 --R                             +-+          +-+
---R                         - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                         - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                    /
 --R                         +-+
 --R                       2\|2
 --R              + 
---R                  ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
+--I                  ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
 --R               *
 --R                   +-----------------------------------------+
 --R                   |         2                          2
---R                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                  \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R              + 
 --R                         +-+             +-+      2
---R                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R              + 
 --R                         +-+     2      +-+              +-+     2      +-+
---R                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
+--I                (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0  + 48\|2 %%BC0
 --R              + 
 --R                   +-+
 --R                42\|2
@@ -1058,189 +1058,189 @@ in1185a:=integrate((z+1)^(1/2)/(1+z^4), z= 0..1,"noPole")
 --R            ROOT
 --R                    +-----------------------------------------+
 --R                    |         2                          2          +-+
---R                   \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
+--I                   \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1  - 4\|2 %%BC1
 --R                 + 
 --R                       +-+
---R                   - 4\|2 %%BC0
+--I                   - 4\|2 %%BC0
 --R              /
 --R                   +-+
 --R                 2\|2
 --R         *
 --R            log
 --R                                    +-+            +-+             +-+
---R                           (- 24576\|2 %%BC0 + 768\|2 )%%BC1 + 768\|2 %%BC0
+--I                           (- 24576\|2 %%BC0 + 768\|2 )%%BC1 + 768\|2 %%BC0
 --R                         + 
 --R                              +-+
 --R                           48\|2
 --R                      *
 --R                          +-----------------------------------------+
 --R                          |         2                          2
---R                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                         \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                     + 
 --R                                                  2
---R                       (- 196608%%BC0 + 6144)%%BC1
+--I                       (- 196608%%BC0 + 6144)%%BC1
 --R                     + 
 --R                                     2                        2
---R                       (- 196608%%BC0  - 384)%%BC1 + 6144%%BC0  - 384%%BC0 - 48
+--I                       (- 196608%%BC0  - 384)%%BC1 + 6144%%BC0  - 384%%BC0 - 48
 --R                  *
 --R                     ROOT
 --R                             +-----------------------------------------+
 --R                             |         2                          2
---R                            \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                            \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                          + 
 --R                                +-+          +-+
---R                            - 4\|2 %%BC1 - 4\|2 %%BC0
+--I                            - 4\|2 %%BC1 - 4\|2 %%BC0
 --R                       /
 --R                            +-+
 --R                          2\|2
 --R                 + 
---R                     ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
+--I                     ((- 3072%%BC0 + 384)%%BC1 + 384%%BC0 - 12)
 --R                  *
 --R                      +-----------------------------------------+
 --R                      |         2                          2
---R                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
+--I                     \|- 96%%BC1  - 64%%BC0 %%BC1 - 96%%BC0  - 1
 --R                 + 
 --R                            +-+             +-+      2
---R                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
+--I                   (- 12288\|2 %%BC0 + 1536\|2 )%%BC1
 --R                 + 
 --R                            +-+     2      +-+              +-+     2
---R                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
+--I                   (- 12288\|2 %%BC0  + 48\|2 )%%BC1 + 1536\|2 %%BC0
 --R                 + 
 --R                      +-+           +-+
---R                   48\|2 %%BC0 + 60\|2
+--I                   48\|2 %%BC0 + 60\|2
 --R              /
 --R                  +-+
 --R                 \|2
 --R     + 
 --R       -
 --R             +------+
---R            \|4%%BC1
+--I            \|4%%BC1
 --R         *
 --R            log
 --R                            +-+             +-+      2
---R                     (98304\|2 %%BC0 - 3072\|2 )%%BC1
+--I                     (98304\|2 %%BC0 - 3072\|2 )%%BC1
 --R                   + 
 --R                            +-+     2       +-+               +-+     3
---R                     (98304\|2 %%BC0  + 192\|2 )%%BC1 + 98304\|2 %%BC0
+--I                     (98304\|2 %%BC0  + 192\|2 )%%BC1 + 98304\|2 %%BC0
 --R                   + 
 --R                         +-+           +-+
---R                     768\|2 %%BC0 - 36\|2
+--I                     768\|2 %%BC0 - 36\|2
 --R                *
 --R                    +------+
---R                   \|4%%BC1
+--I                   \|4%%BC1
 --R               + 
 --R                                         2              2
---R                 (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1
+--I                 (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1
 --R               + 
 --R                           3
---R                 12288%%BC0  + 96%%BC0 + 18
+--I                 12288%%BC0  + 96%%BC0 + 18
 --R     + 
 --R          +------+
---R         \|4%%BC1
+--I         \|4%%BC1
 --R      *
 --R         log
 --R                                          2              2
---R                  (98304%%BC0 - 3072)%%BC1  + (98304%%BC0  + 192)%%BC1
+--I                  (98304%%BC0 - 3072)%%BC1  + (98304%%BC0  + 192)%%BC1
 --R                + 
 --R                            3
---R                  98304%%BC0  + 768%%BC0 - 36
+--I                  98304%%BC0  + 768%%BC0 - 36
 --R             *
 --R                 +------+
---R                \|4%%BC1
+--I                \|4%%BC1
 --R            + 
 --R                                      2              2                        3
---R              (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1 + 12288%%BC0
+--I              (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1 + 12288%%BC0
 --R            + 
---R              96%%BC0 + 9
+--I              96%%BC0 + 9
 --R     + 
 --R       -
 --R             +------+
---R            \|4%%BC1
+--I            \|4%%BC1
 --R         *
 --R            log
 --R                                               2                2
---R                     (- 98304%%BC0 + 3072)%%BC1  + (- 98304%%BC0  - 192)%%BC1
+--I                     (- 98304%%BC0 + 3072)%%BC1  + (- 98304%%BC0  - 192)%%BC1
 --R                   + 
 --R                                 3
---R                     - 98304%%BC0  - 768%%BC0 + 36
+--I                     - 98304%%BC0  - 768%%BC0 + 36
 --R                *
 --R                    +------+
---R                   \|4%%BC1
+--I                   \|4%%BC1
 --R               + 
 --R                                         2              2
---R                 (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1
+--I                 (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1
 --R               + 
 --R                           3
---R                 12288%%BC0  + 96%%BC0 + 9
+--I                 12288%%BC0  + 96%%BC0 + 9
 --R     + 
 --R          +------+
---R         \|4%%BC1
+--I         \|4%%BC1
 --R      *
 --R         log
 --R                           +-+             +-+      2
---R                  (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
+--I                  (- 98304\|2 %%BC0 + 3072\|2 )%%BC1
 --R                + 
 --R                           +-+     2       +-+               +-+     3
---R                  (- 98304\|2 %%BC0  - 192\|2 )%%BC1 - 98304\|2 %%BC0
+--I                  (- 98304\|2 %%BC0  - 192\|2 )%%BC1 - 98304\|2 %%BC0
 --R                + 
 --R                        +-+           +-+
---R                  - 768\|2 %%BC0 + 36\|2
+--I                  - 768\|2 %%BC0 + 36\|2
 --R             *
 --R                 +------+
---R                \|4%%BC1
+--I                \|4%%BC1
 --R            + 
 --R                                      2              2                        3
---R              (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1 + 12288%%BC0
+--I              (12288%%BC0 - 1536)%%BC1  + (12288%%BC0  - 48)%%BC1 + 12288%%BC0
 --R            + 
---R              96%%BC0 + 18
+--I              96%%BC0 + 18
 --R     + 
 --R          +------+
---R         \|4%%BC0
+--I         \|4%%BC0
 --R      *
 --R         log
 --R                       +-+     3        +-+     2       +-+           +-+
---R                (98304\|2 %%BC0  + 3072\|2 %%BC0  + 576\|2 %%BC0 - 60\|2 )
+--I                (98304\|2 %%BC0  + 3072\|2 %%BC0  + 576\|2 %%BC0 - 60\|2 )
 --R             *
 --R                 +------+
---R                \|4%%BC0
+--I                \|4%%BC0
 --R            + 
 --R                          3            2
---R              - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 30
+--I              - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 30
 --R     + 
 --R       -
 --R             +------+
---R            \|4%%BC0
+--I            \|4%%BC0
 --R         *
 --R            log
 --R                            3            2                  +------+
---R                 (98304%%BC0  + 3072%%BC0  + 576%%BC0 - 60)\|4%%BC0
+--I                 (98304%%BC0  + 3072%%BC0  + 576%%BC0 - 60)\|4%%BC0
 --R               + 
 --R                             3            2
---R                 - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 21
+--I                 - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 21
 --R     + 
 --R          +------+
---R         \|4%%BC0
+--I         \|4%%BC0
 --R      *
 --R         log
 --R                           3            2                  +------+
---R              (- 98304%%BC0  - 3072%%BC0  - 576%%BC0 + 60)\|4%%BC0
+--I              (- 98304%%BC0  - 3072%%BC0  - 576%%BC0 + 60)\|4%%BC0
 --R            + 
 --R                          3            2
---R              - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 21
+--I              - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 21
 --R     + 
 --R       -
 --R             +------+
---R            \|4%%BC0
+--I            \|4%%BC0
 --R         *
 --R            log
 --R                            +-+     3        +-+     2       +-+           +-+
---R                   (- 98304\|2 %%BC0  - 3072\|2 %%BC0  - 576\|2 %%BC0 + 60\|2 )
+--I                   (- 98304\|2 %%BC0  - 3072\|2 %%BC0  - 576\|2 %%BC0 + 60\|2 )
 --R                *
 --R                    +------+
---R                   \|4%%BC0
+--I                   \|4%%BC0
 --R               + 
 --R                             3            2
---R                 - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 30
+--I                 - 12288%%BC0  - 1536%%BC0  - 144%%BC0 + 30
 --R  /
 --R     4
 --R                    Type: Union(f1: OrderedCompletion Expression Integer,...)
@@ -2116,7 +2116,7 @@ in13:=integrate(cos(z)^2*sin(z)^n, z)
 --R
 --R            z
 --R          ++         2       n
---R   (48)   |   cos(%K) sin(%K) d%K
+--I   (48)   |   cos(%K) sin(%K) d%K
 --R         ++
 --R                                          Type: Union(Expression Integer,...)
 --E 49
@@ -2818,8 +2818,8 @@ in160:=integrate((-z^2-z)^sin(z), z)
 --R 
 --R
 --R            z
---R          ++       2      sin(%K)
---R   (79)   |   (- %K  - %K)       d%K
+--I          ++       2      sin(%K)
+--I   (79)   |   (- %K  - %K)       d%K
 --R         ++
 --R                                          Type: Union(Expression Integer,...)
 --E 80
@@ -3006,10 +3006,10 @@ in161:=integrate((-z^2)^(1/3), z)
 in163:=integrate(log(z)*abs(exp(z)/z), z)
 --R 
 --R
---R            z              %K
+--I            z              %K
 --R          ++             %e
---R   (98)   |   log(%K)abs(----)d%K
---R         ++               %K
+--I   (98)   |   log(%K)abs(----)d%K
+--I         ++               %K
 --R                                          Type: Union(Expression Integer,...)
 --E 100
 
@@ -3018,8 +3018,8 @@ in166:=integrate((1+z)^z, z)
 --R 
 --R
 --R            z
---R          ++          %K
---R   (99)   |   (%K + 1)  d%K
+--I          ++          %K
+--I   (99)   |   (%K + 1)  d%K
 --R         ++
 --R                                          Type: Union(Expression Integer,...)
 --E 101
diff --git a/src/input/mathml.input.pamphlet b/src/input/mathml.input.pamphlet
index 461f32c..0cbcf7b 100644
--- a/src/input/mathml.input.pamphlet
+++ b/src/input/mathml.input.pamphlet
@@ -209,15 +209,17 @@ y:=operator 'y
 
 --S 14 of 21
 D(y(x,z),[x,x,z,x])
+--R 
 --R
 --R   (14)  y        (x,z)
 --R          ,1,1,2,1
 --R<math xmlns="http://www.w3.org/1998/Math/MathML" mathsize="big" display="block">
---R<mfrac><mrow><msup><mo>&#x02202;</mo><mn>4</mn></msup><mi>y</mi></mrow><mrow><mo>&#x02202;</mo><msup><mi>x</mi><mn>3</mn></msup><mo>&#x02202;</mo><mi>z</mi></mrow></mfrac><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo>
+--R<msub><mi>y</mi><mrow><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi><mi>x</mi></mi><mo>,</mo><mi><mi>z</mi></mi><mo>)</mo>
 --R</math>
 --R
 --R                                                     Type: Expression Integer
 --E 14
+
 )clear all
 
 --S 15 of 21
@@ -234,12 +236,13 @@ y:=operator 'y
 
 --S 16 of 21
 D(y x,x,2)
+--R 
 --R
 --R         ,,
 --R   (2)  y  (x)
 --R
 --R<math xmlns="http://www.w3.org/1998/Math/MathML" mathsize="big" display="block">
---R<mfrac><mrow><msup><mo>&#x02146;</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>&#x02146;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>&#x02061;</mo><mo>(</mo><mi>x</mi><mo>)</mo>
+--R<msup><mi>y</mi><mrow><mo>&#x02032;</mo><mo>&#x02032;</mo></mrow></msup><mo>&#x02061;</mo><mo>(</mo><mi>x</mi><mo>)</mo>
 --R</math>
 --R
 --R                                                     Type: Expression Integer
diff --git a/src/input/ndftip.input.pamphlet b/src/input/ndftip.input.pamphlet
index 7b838a5..bbed72e 100644
--- a/src/input/ndftip.input.pamphlet
+++ b/src/input/ndftip.input.pamphlet
@@ -298,6 +298,7 @@ hdftA := nagHermitianDFT seqA;
 --R   Cannot convert from type Polynomial Integer to Vector Complex Float 
 --R      for value
 --R   hdftA
+--R
 --E 19
 --       +         2.48361         +
 --       |                         |
@@ -361,6 +362,7 @@ dftC :: Vector Float
 --RDaly Bug
 --R   Cannot convert from type Variable dftC to Vector Float for value
 --R   dftC
+--R
 --E 22
 -- [1.82616,1.86862,- 0.017503,0.502001,- 0.598725,- 0.0314404,- 2.62557]
 
@@ -824,7 +826,7 @@ nagDFT badSeqs
 --RDaly Bug
 --R   Cannot find a definition or applicable library operation named 
 --R      nagDFT with argument type(s) 
---R                                 List Symbol
+--R                              Variable badSeqs
 --R      
 --R      Perhaps you should use "@" to indicate the required return type, 
 --R      or "$" to specify which version of the function you need.
diff --git a/src/input/pmint.input.pamphlet b/src/input/pmint.input.pamphlet
index 280ab6e..ede0da7 100644
--- a/src/input/pmint.input.pamphlet
+++ b/src/input/pmint.input.pamphlet
@@ -22,7 +22,7 @@
 @
 \section{Rational Functions}
 <<*>>=
---S 1 of 28
+--S 1 of 29
 f:=(x^7-24*x^4-4*x^2+8*x-8)/(x^8+6*x^6+12*x^4+8*x^2)
 --R 
 --R
@@ -34,7 +34,7 @@ f:=(x^7-24*x^4-4*x^2+8*x-8)/(x^8+6*x^6+12*x^4+8*x^2)
 --R                                            Type: Fraction Polynomial Integer
 --E 1
 
---S 2 of 28
+--S 2 of 29
 g:=integrate(f,x)
 --R 
 --R
@@ -46,7 +46,7 @@ g:=integrate(f,x)
 --R                                          Type: Union(Expression Integer,...)
 --E 2
 
---S 3 of 28
+--S 3 of 29
 differentiate(g,x)
 --R 
 --R
@@ -63,7 +63,7 @@ differentiate(g,x)
 @
 \section{Trigonometric Functions}
 <<*>>=
---S 4 of 28
+--S 4 of 29
 f:=(x-tan(x))/tan(x)^2 + tan(x)
 --R 
 --R
@@ -75,7 +75,7 @@ f:=(x-tan(x))/tan(x)^2 + tan(x)
 --R                                                     Type: Expression Integer
 --E 4
 
---S 5 of 28
+--S 5 of 29
 g:=integrate(f,x)
 --R 
 --R
@@ -86,7 +86,7 @@ g:=integrate(f,x)
 --R                                          Type: Union(Expression Integer,...)
 --E 5
 
---S 6 of 28
+--S 6 of 29
 differentiate(g,x)
 --R 
 --R
@@ -102,8 +102,8 @@ differentiate(g,x)
 
 @
 \section{Log-Exp Functions}
-
---S 7 of 28
+<<*>>=
+--S 7 of 29
 f:=(1+x+x*exp(x))*(x+log(x)+exp(x)-1)/(x+log(x)+exp(x))^2/x
 --R 
 --R
@@ -115,7 +115,7 @@ f:=(1+x+x*exp(x))*(x+log(x)+exp(x)-1)/(x+log(x)+exp(x))^2/x
 --R                                                     Type: Expression Integer
 --E 7
 
---S 8 of 28
+--S 8 of 29
 g:=integrate(f,x)
 --R 
 --R
@@ -127,7 +127,7 @@ g:=integrate(f,x)
 --R                                          Type: Union(Expression Integer,...)
 --E 8
 
---S 9 of 28
+--S 9 of 29
 differentiate(g,x)
 --R 
 --R
@@ -144,7 +144,7 @@ differentiate(g,x)
 @
 \section{Liouvillian special functions}
 <<*>>=
---S 10 of 28
+--S 10 of 29
 f:=exp(-x^2)+erf(x)/(erf(x)^3-erf(x)^2-erf(x)+1)
 --R 
 --R
@@ -157,82 +157,154 @@ f:=exp(-x^2)+erf(x)/(erf(x)^3-erf(x)^2-erf(x)+1)
 --R                                                     Type: Expression Integer
 --E 10
 
---S 11 of 28
+--S 11 of 29 
+g:=integrate(f,x)
+--R
+--R                                                      2
+--I           x         3          2                 - %G
+--I         ++  (erf(%G)  - erf(%G)  - erf(%G) + 1)%e      + erf(%G)
+--I   (2)   |   ---------------------------------------------------- d%G
+--R        ++                    3          2
+--I                       erf(%G)  - erf(%G)  - erf(%G) + 1
+--R                                          Type: Union(Expression Integer,...)
+--E 11
+
+--S 12 of 29
+differentiate(g,x)
+--R
+--R                                             2
+--R               3         2                - x
+--R        (erf(x)  - erf(x)  - erf(x) + 1)%e     + erf(x)
+--R   (3)  -----------------------------------------------
+--R                       3         2
+--R                 erf(x)  - erf(x)  - erf(x) + 1
+--R                                                     Type: Expression Integer
+--E 12
+
+)clear all
+
+--S 13 of 29
 f:=(exp(-x^2)+erf(x))/(erf(x)^3-erf(x)^2-erf(x)+1)
 --R 
 --R
 --R                     2
 --R                  - x
 --R                %e     + erf(x)
---R   (2)  ------------------------------
+--R   (1)  ------------------------------
 --R              3         2
 --R        erf(x)  - erf(x)  - erf(x) + 1
 --R                                                     Type: Expression Integer
---E 11
+--E 13
 
---S 12 of 28 used to work!
-integrate(f,x)
+--S 14 of 29 used to work!
+g:=integrate(f,x)
 --R 
 --R
 --R                           2
---R           x           - %G
---R         ++          %e      + erf(%G)
---R   (3)   |   --------------------------------- d%G
+--I           x           - %G
+--I         ++          %e      + erf(%G)
+--I   (3)   |   --------------------------------- d%G
 --R        ++          3          2
---R             erf(%G)  - erf(%G)  - erf(%G) + 1
+--I             erf(%G)  - erf(%G)  - erf(%G) + 1
 --R                                          Type: Union(Expression Integer,...)
---E 12
+--E 14
 -- should be:
 --    1   sqrt(%pi)     1                           1
 -- -  - ------------  - - sqrt(%pi) log(erf(x)+1) + - sqrt(%pi) log(erf(x)-1)
 --    4  erf(x) - 1     8                           8
 
+--S 15 of 29
+differentiate(g,x)
+--R
+--R                     2
+--R                  - x
+--R                %e     + erf(x)
+--R   (3)  ------------------------------
+--R              3         2
+--R        erf(x)  - erf(x)  - erf(x) + 1
+--R                                                     Type: Expression Integer
+--E 15
+
 )clear all
  
 @
 \section{Airy Functions}
 <<*>>=
 -- Axiom does not have a 2 argument form of the airyAi function
---S 13 of 28 will certainly fail
-f:=(x-airyAi(x)*airyAi(1,x))/(x^2-airyAi(x)^2)
---R
---E 13
-
---S 14 of 28 will certainly fail
-integrate(f,x)
+--  f:=(x-airyAi(x)*airyAi(1,x))/(x^2-airyAi(x)^2)
+--it has the integral
 --R
 --R  1                    1
 --R  - log(x+airyAi(x)) + - log(x-airyAi(x))
 --R  2                    2
---E 14
 
---S 15 of 28
+
+--S 16 of 29 will certainly fail
+f:=(x-airyAi(x))/(x^2-airyAi(x)^2)
+--R
+--R              1
+--R   (1)  -------------
+--R        airyAi(x) + x
+--R                                                     Type: Expression Integer
+--E 16
+
+--S 17 of 29 will certainly fail
+g:=integrate(f,x)
+--R
+--R           x
+--R         ++         1
+--R   (2)   |   --------------- d%G
+--R        ++   airyAi(%G) + %G
+--R                                          Type: Union(Expression Integer,...)
+--E 17
+
+--S 18 of 29
+differentiate(g,x)
+--R
+--R              1
+--R   (3)  -------------
+--R        airyAi(x) + x
+--R                                                     Type: Expression Integer
+--E 18
+
+)clear all
+
+--S 19 of 29
 f:=x^2*airyAi(x)
 --R 
 --R
 --R         2
 --R   (1)  x airyAi(x)
 --R                                                     Type: Expression Integer
---E 15
+--E 19
 
---S 16 of 28 used to work
+--S 20 of 29 used to work
 g:=integrate(f,x)
 --R 
 --R
 --R           x
 --R         ++    2
---R   (2)   |   %G airyAi(%G)d%G
+--I   (2)   |   %G airyAi(%G)d%G
 --R        ++
 --R                                          Type: Union(Expression Integer,...)
---E 16
+--E 20
 -- should be:
 --  -airyAi(x) + airyAi(1,x) x
 
+--S 21 of 29
+differentiate(g,x)
+--R
+--R         2
+--R   (3)  x airyAi(x)
+--R                                                     Type: Expression Integer
+--E 21
+
 )clear all
 
 @
 \section{Bessel functions}
---S 17 of 28
+<<*>>=
+--S 22 of 29
 f:=besselJ(y+1,x)/besselJ(y,x)
 --R 
 --R
@@ -240,49 +312,96 @@ f:=besselJ(y+1,x)/besselJ(y,x)
 --R   (1)  ----------------
 --R          besselJ(y,x)
 --R                                                     Type: Expression Integer
---E 17
+--E 22
 
---S 18 of 28 used to work
+--S 23 of 29 used to work
 g:=integrate(f,x)
 --R 
 --R
 --R           x
---R         ++  besselJ(y + 1,%G)
---R   (2)   |   ----------------- d%G
---R        ++     besselJ(y,%G)
+--I         ++  besselJ(y + 1,%G)
+--I   (2)   |   ----------------- d%G
+--I        ++     besselJ(y,%G)
 --R                                          Type: Union(Expression Integer,...)
---E 18
+--E 23
 -- should be:
 --  y log(x) - log(besselJ(y,x))
 
+--S 24 of 29
+differentiate(g,x)
+--R
+--R        besselJ(y + 1,x)
+--R   (3)  ----------------
+--R          besselJ(y,x)
+--R                                                     Type: Expression Integer
+--E 24
+
+)clear all
+
+
 -- Axiom does not have Maple's normal function
---S 19 of 28 used to work
-f:=normal(y*besselJ(y,x)/x = besselJ(y+1,x))
+--S 25 of 29 used to work
+--f:=normal(y*besselJ(y,x)/x - besselJ(y+1,x))
+f:=y*besselJ(y,x)/x - besselJ(y+1,x)
 --R
---E 19
--- should be:
---   besselJ(y+1,x) x - y besselJ(y,x)
--- - ---------------------------------
---                  x
+--R        - x besselJ(y + 1,x) + y besselJ(y,x)
+--R   (1)  -------------------------------------
+--R                          x
+--R                                                     Type: Expression Integer
+--E 25
+
+--S 26 of 29
+g:=integrate(f,x)
+--R
+--R           x
+--I         ++  - %G besselJ(y + 1,%G) + y besselJ(y,%G)
+--I   (2)   |   ---------------------------------------- d%G
+--I        ++                      %G
+--R                                          Type: Union(Expression Integer,...)
+--E 26
 
+--S 27 of 29
+differentiate(g,x)
+--R
+--R        - x besselJ(y + 1,x) + y besselJ(y,x)
+--R   (3)  -------------------------------------
+--R                          x
+--R                                                     Type: Expression Integer
+--E 27
 )clear all
 
 @
 \section{Whittaker functions}
---S 20 of 28 used to work
+<<*>>=
+--S 28 of 29 used to work
 f:=WhittakerW(u+1,n,x)/(WhittakerW(u,n,x)*x)
---R
---E 20
+--R 
+--R   There are no library operations named WhittakerW 
+--R      Use HyperDoc Browse or issue
+--R                             )what op WhittakerW
+--R      to learn if there is any operation containing " WhittakerW " in 
+--R      its name.
+--R 
+--RDaly Bug
+--R   Cannot find a definition or applicable library operation named 
+--R      WhittakerW with argument type(s) 
+--R                             Polynomial Integer
+--R                                 Variable n
+--R                                 Variable x
+--R      
+--R      Perhaps you should use "@" to indicate the required return type, 
+--R      or "$" to specify which version of the function you need.
+--E 28
+
 -- Axiom does not implement WhittakerW
 -- should be:
 --  Whittaker(u+1,n,x)
 --  ------------------
 --  Whittaker(u,n,x) x
 
---S 21 of 28 used to work
-integrate(f,x)
---R
---E 21
+-- of 29 used to work
+--integrate(f,x)
+-- 22
 -- should be:
 --  x
 --  -  - u log(x) - log(WhattakerW(u,n,x))
@@ -294,25 +413,36 @@ integrate(f,x)
 \section{The Lambert W function} 
 <<*>>=
 -- Axiom does not implement LambertW
---S 22 of 28 used to work
+--S 29 of 29 used to work
 f:=LambertW(x)
---R
---E 22
-
---S 23 of 28 used to work
-g:=integrate(f,x)
---R
---E 23
+--R 
+--R   There are no library operations named LambertW 
+--R      Use HyperDoc Browse or issue
+--R                              )what op LambertW
+--R      to learn if there is any operation containing " LambertW " in its
+--R      name.
+--R 
+--RDaly Bug
+--R   Cannot find a definition or applicable library operation named 
+--R      LambertW with argument type(s) 
+--R                                 Variable x
+--R      
+--R      Perhaps you should use "@" to indicate the required return type, 
+--R      or "$" to specify which version of the function you need.
+--E 29
+
+-- of 29 used to work
+-- g:=integrate(f,x)
+-- 24
 -- should be:
 --    2             2  2                2
 --   x + LambertW(x)  x  - LambertW(x) x
 --   ------------------------------------
 --          x LambertW(x)
 
---S 24 of 28 used to work
-integrate(sin(LambertW(x)),x)
---R
---E 24
+-- of 29 used to work
+-- integrate(sin(LambertW(x)),x)
+-- 25
 --should be:
 -- +-                                                  -+
 -- |                                     2              |
@@ -343,10 +473,9 @@ integrate(sin(LambertW(x)),x)
 --                  |         +-             -+   |
 --                  +-                           -+
 
---S 25 of 28 used to work
-f:=((x^2+2)*LambertW(x^2)^2+x^2*(2*LambertW(x^2)+1))/(x*(1+LambertW(x^2)^3))
---R
---E 25
+-- of 29 used to work
+--f:=((x^2+2)*LambertW(x^2)^2+x^2*(2*LambertW(x^2)+1))/(x*(1+LambertW(x^2)^3))
+-- 26
 --should be:
 --                       2
 --    2                2      2              2
@@ -356,10 +485,9 @@ f:=((x^2+2)*LambertW(x^2)^2+x^2*(2*LambertW(x^2)+1))/(x*(1+LambertW(x^2)^3))
 --                            2  
 --           x (1 + LambertW(x ))
 
---S 26 of 28 used to work
-integrate(f,x)
---R
---E 26
+-- of 29 used to work
+--integrate(f,x)
+-- 27
 --should be:
 --                 2                    4
 --1  4           2     4          2    x              2   2    2           2
@@ -374,10 +502,10 @@ integrate(f,x)
 --                     2
 --   log(1 + LambertW(x ))
 
---S 27 of 28 used to work
-f:=(2*LambertW(x^2)*cos(LambertW(x^2))*(a*x+LambertW(x^2))+a*x*(1+LambertW(x^2)) + 2*LambertW(x^2))/((1+LambertW(x^2))*(a*x+LambertW(x^2))*x)
---R
---E 27
+-- of 29 used to work
+--f:=(2*LambertW(x^2)*cos(LambertW(x^2))*(a*x+LambertW(x^2))+a*x*(1+LambertW(x^2)) + 2*LambertW(x^2))/((1+LambertW(x^2))*(a*x+LambertW(x^2))*x)
+--
+-- 28
 --+-                                                       -+
 --|                                                         |
 --|             2                2                    2     |
@@ -392,10 +520,10 @@ f:=(2*LambertW(x^2)*cos(LambertW(x^2))*(a*x+LambertW(x^2))+a*x*(1+LambertW(x^2))
 -- (1 + LambertW(x ))(a x+LambertW(x )) x
 --
 
---S 28 of 28 used to work
+-- 29 of 29 used to work
 integrate(f,x)
---R
---E 28
+--
+-- 29
 --   
 --        +-              -+
 --        | 1           2  |
diff --git a/src/input/r20bugs.input.pamphlet b/src/input/r20bugs.input.pamphlet
index b28c18b..d91d6ff 100644
--- a/src/input/r20bugs.input.pamphlet
+++ b/src/input/r20bugs.input.pamphlet
@@ -196,7 +196,8 @@ positiveRemainder(-1::SINT,-5::SINT)
 complexRoots([u**2-v+1,v**2-4],[u,v],0.01)
 --R 
 --R
---R   (1)  [[1.732421875 %i,- 2.0],[- 1.732421875 %i,- 2.0],[- 1.0,2.0],[1.0,2.0]]
+--R   (1)
+--R   [[1.732421875 %i,- 2.0],[- 1.732421875 %i,- 2.0],[- 1.0,2.0],[1.0,2.0]]
 --R                                                Type: List List Complex Float
 --E 17
 
diff --git a/src/input/summation.input.pamphlet b/src/input/summation.input.pamphlet
index 78b7aa7..a1e2c42 100644
--- a/src/input/summation.input.pamphlet
+++ b/src/input/summation.input.pamphlet
@@ -16,6 +16,7 @@ This is bug 7057: summation is ambiguous.
 <<*>>=
 )spool summation.output
 )set message test on
+)set output mathml on
 )set message auto off
 )clear all
  
diff --git a/src/scripts/tex/axiom.sty b/src/scripts/tex/axiom.sty
index fdd8e27..3581b97 100755
--- a/src/scripts/tex/axiom.sty
+++ b/src/scripts/tex/axiom.sty
@@ -943,6 +943,8 @@ I'll try to avoid such incompatible changes in the future.}%
 % We use providecommand so it doesn't clash with amsmath package
 \providecommand{\eth}[1]{{#1}-th}
 
+% add the binom function for combfunc
+\providecommand{\binom}[2]{\left(\begin{array}{c}#1\\#2\end{array}\right)}
 
 % spadcommands are the actual text that you type at the axiom prompt
 \providecommand{\spadcommand}[1]%



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This patch adds two files missing from prior updates. -- Tim

========================================================================
diff --git a/changelog b/changelog
index 9c87000..41ba008 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080107 tpd src/input/gamma.input missing file added
+20080107 tpd src/input/pfaffian.input missing file added
 20080107 tpd src/input/Makefile add free to test bug 7089/343
 20080107 tpd src/input/free.input added to test bug 7089/343
 20080107 mxr src/input/free.spad fix FreeAbelianGroup bug 7089/343
diff --git a/src/input/gamma.input.pamphlet b/src/input/gamma.input.pamphlet
new file mode 100644
index 0000000..608cf75
--- /dev/null
+++ b/src/input/gamma.input.pamphlet
@@ -0,0 +1,1596 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input gamma.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a set of tests for the Gamma and polygamma functions
+as well as a set of graphics that illustrate these curves.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of Gamma, ranging from 1.000 to 1.995. The
+second column is the reference value of Gamma from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+
+Notice that certain entries have reasonably large differences:
+\begin{verbatim}
+[1.215, 0.9143261399876792,  0.91432614997006778, 9.9823885824079639E-9]
+[1.455, 0.88562206999631599, 0.88562207999314335, 9.9968273659101214E-9]
+\end{verbatim}
+
+<<*>>=
+)spool gamma.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 3
+[[1.000,1.0000000000,Gamma(1.000),Gamma(1.000)-1.0000000000],_
+ [1.005,0.9971385354,Gamma(1.005),Gamma(1.005)-0.9971385354],_
+ [1.010,0.9943258512,Gamma(1.010),Gamma(1.010)-0.9943258512],_
+ [1.015,0.9915612888,Gamma(1.015),Gamma(1.015)-0.9915612888],_
+ [1.020,0.9888442033,Gamma(1.020),Gamma(1.020)-0.9888442033],_
+ [1.025,0.9861739633,Gamma(1.025),Gamma(1.025)-0.9861739633],_
+ [1.030,0.9835499506,Gamma(1.030),Gamma(1.030)-0.9835499506],_
+ [1.035,0.9809715606,Gamma(1.035),Gamma(1.035)-0.9809715606],_
+ [1.040,0.9784382009,Gamma(1.040),Gamma(1.040)-0.9784382009],_
+ [1.045,0.9759492919,Gamma(1.045),Gamma(1.045)-0.9759492919],_
+ [1.050,0.9735042656,Gamma(1.050),Gamma(1.050)-0.9735042656],_
+ [1.055,0.9711025663,Gamma(1.055),Gamma(1.055)-0.9711025663],_
+ [1.060,0.9687436495,Gamma(1.060),Gamma(1.060)-0.9687436495],_
+ [1.065,0.9664269823,Gamma(1.065),Gamma(1.065)-0.9664269823],_
+ [1.070,0.9641520425,Gamma(1.070),Gamma(1.070)-0.9641520425],_
+ [1.075,0.9619183189,Gamma(1.075),Gamma(1.075)-0.9619183189],_
+ [1.080,0.9597253107,Gamma(1.080),Gamma(1.080)-0.9597253107],_
+ [1.085,0.9575725273,Gamma(1.085),Gamma(1.085)-0.9575725273],_
+ [1.090,0.9554594882,Gamma(1.090),Gamma(1.090)-0.9554594882],_
+ [1.095,0.9533857227,Gamma(1.095),Gamma(1.095)-0.9533857227],_
+ [1.100,0.9513507699,Gamma(1.100),Gamma(1.100)-0.9513507699],_
+ [1.105,0.9493541778,Gamma(1.105),Gamma(1.105)-0.9493541778],_
+ [1.110,0.9473955040,Gamma(1.110),Gamma(1.110)-0.9473955040],_
+ [1.115,0.9454743149,Gamma(1.115),Gamma(1.115)-0.9454743149],_
+ [1.120,0.9435901856,Gamma(1.120),Gamma(1.120)-0.9435901856],_
+ [1.125,0.9417426997,Gamma(1.125),Gamma(1.125)-0.9417426997],_
+ [1.130,0.9399314497,Gamma(1.130),Gamma(1.130)-0.9399314497],_
+ [1.135,0.9381560356,Gamma(1.135),Gamma(1.135)-0.9381560356],_
+ [1.140,0.9364160657,Gamma(1.140),Gamma(1.140)-0.9364160657],_
+ [1.145,0.9347111562,Gamma(1.145),Gamma(1.145)-0.9347111562],_
+ [1.150,0.9330409311,Gamma(1.150),Gamma(1.150)-0.9330409311],_
+ [1.155,0.9314050217,Gamma(1.155),Gamma(1.155)-0.9314050217],_
+ [1.160,0.9298030666,Gamma(1.160),Gamma(1.160)-0.9298030666],_
+ [1.165,0.9282347120,Gamma(1.165),Gamma(1.165)-0.9282347120],_
+ [1.170,0.9266996106,Gamma(1.170),Gamma(1.170)-0.9266996106],_
+ [1.175,0.9251974225,Gamma(1.175),Gamma(1.175)-0.9251974225],_
+ [1.180,0.9237278143,Gamma(1.180),Gamma(1.180)-0.9237278143],_
+ [1.185,0.9222904591,Gamma(1.185),Gamma(1.185)-0.9222904591],_
+ [1.190,0.9208850371,Gamma(1.190),Gamma(1.190)-0.9208850371],_
+ [1.195,0.9195112341,Gamma(1.195),Gamma(1.195)-0.9195112341],_
+ [1.200,0.9181687424,Gamma(1.200),Gamma(1.200)-0.9181687424],_
+ [1.205,0.9168572606,Gamma(1.205),Gamma(1.205)-0.9168572606],_
+ [1.210,0.9155764930,Gamma(1.210),Gamma(1.210)-0.9155764930],_
+ [1.215,0.9143261400,Gamma(1.215),Gamma(1.215)-0.9143261400],_
+ [1.220,0.9131059475,Gamma(1.220),Gamma(1.220)-0.9131059475],_
+ [1.225,0.9119156071,Gamma(1.225),Gamma(1.225)-0.9119156071],_
+ [1.230,0.9107548564,Gamma(1.230),Gamma(1.230)-0.9107548564],_
+ [1.235,0.9096234274,Gamma(1.235),Gamma(1.235)-0.9096234274],_
+ [1.240,0.9085210583,Gamma(1.240),Gamma(1.240)-0.9085210583],_
+ [1.245,0.9074474922,Gamma(1.245),Gamma(1.245)-0.9074474922],_
+ [1.250,0.9064024771,Gamma(1.250),Gamma(1.250)-0.9064024771],_
+ [1.255,0.9053857663,Gamma(1.255),Gamma(1.255)-0.9053857663],_
+ [1.260,0.9043971178,Gamma(1.260),Gamma(1.260)-0.9043971178],_
+ [1.265,0.9034362946,Gamma(1.265),Gamma(1.265)-0.9034362946],_
+ [1.270,0.9025030645,Gamma(1.270),Gamma(1.270)-0.9025030645],_
+ [1.275,0.9015971994,Gamma(1.275),Gamma(1.275)-0.9015971994],_
+ [1.280,0.9007184765,Gamma(1.280),Gamma(1.280)-0.9007184765],_
+ [1.285,0.8998666769,Gamma(1.285),Gamma(1.285)-0.8998666769],_
+ [1.290,0.8990415863,Gamma(1.290),Gamma(1.290)-0.8990415863],_
+ [1.295,0.8982429947,Gamma(1.295),Gamma(1.295)-0.8982429947],_
+ [1.300,0.8974706963,Gamma(1.300),Gamma(1.300)-0.8974706963],_
+ [1.305,0.8967244895,Gamma(1.305),Gamma(1.305)-0.8967244895],_
+ [1.310,0.8960041767,Gamma(1.310),Gamma(1.310)-0.8960041767],_
+ [1.315,0.8953095644,Gamma(1.315),Gamma(1.315)-0.8953095644],_
+ [1.320,0.8946404630,Gamma(1.320),Gamma(1.320)-0.8946404630],_
+ [1.325,0.8939966866,Gamma(1.325),Gamma(1.325)-0.8939966866],_
+ [1.330,0.8933780535,Gamma(1.330),Gamma(1.330)-0.8933780535],_
+ [1.335,0.8927843850,Gamma(1.335),Gamma(1.335)-0.8927843850],_
+ [1.340,0.8922155072,Gamma(1.340),Gamma(1.340)-0.8922155072],_
+ [1.345,0.8916712485,Gamma(1.345),Gamma(1.345)-0.8916712485],_
+ [1.350,0.8911514420,Gamma(1.350),Gamma(1.350)-0.8911514420],_
+ [1.355,0.8906559235,Gamma(1.355),Gamma(1.355)-0.8906559235],_
+ [1.360,0.8901845324,Gamma(1.360),Gamma(1.360)-0.8901845324],_
+ [1.365,0.8897371116,Gamma(1.365),Gamma(1.365)-0.8897371116],_
+ [1.370,0.8893135074,Gamma(1.370),Gamma(1.370)-0.8893135074],_
+ [1.375,0.8889135692,Gamma(1.375),Gamma(1.375)-0.8889135692],_
+ [1.380,0.8885371494,Gamma(1.380),Gamma(1.380)-0.8885371494],_
+ [1.385,0.8881841041,Gamma(1.385),Gamma(1.385)-0.8881841041],_
+ [1.390,0.8878542918,Gamma(1.390),Gamma(1.390)-0.8878542918],_
+ [1.395,0.8875475748,Gamma(1.395),Gamma(1.395)-0.8875475748],_
+ [1.400,0.8872638175,Gamma(1.400),Gamma(1.400)-0.8872638175],_
+ [1.405,0.8870028884,Gamma(1.405),Gamma(1.405)-0.8870028884],_
+ [1.410,0.8867646576,Gamma(1.410),Gamma(1.410)-0.8867646576],_
+ [1.415,0.8865489993,Gamma(1.415),Gamma(1.415)-0.8865489993],_
+ [1.420,0.8863557896,Gamma(1.420),Gamma(1.420)-0.8863557896],_
+ [1.425,0.8861849081,Gamma(1.425),Gamma(1.425)-0.8861849081],_
+ [1.430,0.8860362361,Gamma(1.430),Gamma(1.430)-0.8860362361],_
+ [1.435,0.8859096587,Gamma(1.435),Gamma(1.435)-0.8859096587],_
+ [1.440,0.8858050635,Gamma(1.440),Gamma(1.440)-0.8858050635],_
+ [1.445,0.8857223397,Gamma(1.445),Gamma(1.445)-0.8857223397],_
+ [1.450,0.8856613803,Gamma(1.450),Gamma(1.450)-0.8856613803],_
+ [1.455,0.8856220700,Gamma(1.455),Gamma(1.455)-0.8856220700],_
+ [1.460,0.8856043364,Gamma(1.460),Gamma(1.460)-0.8856043364],_
+ [1.465,0.8856080495,Gamma(1.465),Gamma(1.465)-0.8856080495],_
+ [1.470,0.8856331217,Gamma(1.470),Gamma(1.470)-0.8856331217],_
+ [1.475,0.8856794575,Gamma(1.475),Gamma(1.475)-0.8856794575],_
+ [1.480,0.8857469646,Gamma(1.480),Gamma(1.480)-0.8857469646],_
+ [1.485,0.8858355520,Gamma(1.485),Gamma(1.485)-0.8858355520],_
+ [1.490,0.8859451316,Gamma(1.490),Gamma(1.490)-0.8859451316],_
+ [1.495,0.8860756174,Gamma(1.495),Gamma(1.495)-0.8860756174],_
+ [1.500,0.8862269255,Gamma(1.500),Gamma(1.500)-0.8862269255],_
+ [1.505,0.8863989744,Gamma(1.505),Gamma(1.505)-0.8863989744],_
+ [1.510,0.8865916850,Gamma(1.510),Gamma(1.510)-0.8865916850],_
+ [1.515,0.8868049797,Gamma(1.515),Gamma(1.515)-0.8868049797],_
+ [1.520,0.8870387833,Gamma(1.520),Gamma(1.520)-0.8870387833],_
+ [1.525,0.8872930231,Gamma(1.525),Gamma(1.525)-0.8872930231],_
+ [1.530,0.8875676278,Gamma(1.530),Gamma(1.530)-0.8875676278],_
+ [1.535,0.8878625287,Gamma(1.535),Gamma(1.535)-0.8878625287],_
+ [1.540,0.8881776586,Gamma(1.540),Gamma(1.540)-0.8881776586],_
+ [1.545,0.8885129527,Gamma(1.545),Gamma(1.545)-0.8885129527],_
+ [1.550,0.8888683478,Gamma(1.550),Gamma(1.550)-0.8888683478],_
+ [1.555,0.8892437830,Gamma(1.555),Gamma(1.555)-0.8892437830],_
+ [1.560,0.8896391990,Gamma(1.560),Gamma(1.560)-0.8896391990],_
+ [1.565,0.8900545387,Gamma(1.565),Gamma(1.565)-0.8900545387],_
+ [1.570,0.8904897463,Gamma(1.570),Gamma(1.570)-0.8904897463],_
+ [1.575,0.8909447686,Gamma(1.575),Gamma(1.575)-0.8909447686],_
+ [1.580,0.8914195537,Gamma(1.580),Gamma(1.580)-0.8914195537],_
+ [1.585,0.8919140515,Gamma(1.585),Gamma(1.585)-0.8919140515],_
+ [1.590,0.8924282141,Gamma(1.590),Gamma(1.590)-0.8924282141],_
+ [1.595,0.8929619949,Gamma(1.595),Gamma(1.595)-0.8929619949],_
+ [1.600,0.8935153493,Gamma(1.600),Gamma(1.600)-0.8935153493],_
+ [1.605,0.8940882342,Gamma(1.605),Gamma(1.605)-0.8940882342],_
+ [1.610,0.8946806085,Gamma(1.610),Gamma(1.610)-0.8946806085],_
+ [1.615,0.8952924327,Gamma(1.615),Gamma(1.615)-0.8952924327],_
+ [1.620,0.8959236685,Gamma(1.620),Gamma(1.620)-0.8959236685],_
+ [1.625,0.8965742800,Gamma(1.625),Gamma(1.625)-0.8965742800],_
+ [1.630,0.8972442326,Gamma(1.630),Gamma(1.630)-0.8972442326],_
+ [1.635,0.8979334930,Gamma(1.635),Gamma(1.635)-0.8979334930],_
+ [1.640,0.8986420302,Gamma(1.640),Gamma(1.640)-0.8986420302],_
+ [1.645,0.8993698138,Gamma(1.645),Gamma(1.645)-0.8993698138],_
+ [1.650,0.9001168163,Gamma(1.650),Gamma(1.650)-0.9001168163],_
+ [1.655,0.9008830104,Gamma(1.655),Gamma(1.655)-0.9008830104],_
+ [1.660,0.9016683712,Gamma(1.660),Gamma(1.660)-0.9016683712],_
+ [1.665,0.9024728748,Gamma(1.665),Gamma(1.665)-0.9024728748],_
+ [1.670,0.9032964995,Gamma(1.670),Gamma(1.670)-0.9032964995],_
+ [1.675,0.9041392243,Gamma(1.675),Gamma(1.675)-0.9041392243],_
+ [1.680,0.9050010302,Gamma(1.680),Gamma(1.680)-0.9050010302],_
+ [1.685,0.9058818996,Gamma(1.685),Gamma(1.685)-0.9058818996],_
+ [1.690,0.9067818160,Gamma(1.690),Gamma(1.690)-0.9067818160],_
+ [1.695,0.9077007650,Gamma(1.695),Gamma(1.695)-0.9077007650],_
+ [1.700,0.9086387329,Gamma(1.700),Gamma(1.700)-0.9086387329],_
+ [1.705,0.9095957079,Gamma(1.705),Gamma(1.705)-0.9095957079],_
+ [1.710,0.9105716796,Gamma(1.710),Gamma(1.710)-0.9105716796],_
+ [1.715,0.9115666390,Gamma(1.715),Gamma(1.715)-0.9115666390],_
+ [1.720,0.9125805779,Gamma(1.720),Gamma(1.720)-0.9125805779],_
+ [1.725,0.9136134904,Gamma(1.725),Gamma(1.725)-0.9136134904],_
+ [1.730,0.9146653712,Gamma(1.730),Gamma(1.730)-0.9146653712],_
+ [1.735,0.9157362171,Gamma(1.735),Gamma(1.735)-0.9157362171],_
+ [1.740,0.9168260252,Gamma(1.740),Gamma(1.740)-0.9168260252],_
+ [1.745,0.9179347950,Gamma(1.745),Gamma(1.745)-0.9179347950],_
+ [1.750,0.9190625268,Gamma(1.750),Gamma(1.750)-0.9190625268],_
+ [1.755,0.9202092224,Gamma(1.755),Gamma(1.755)-0.9202092224],_
+ [1.760,0.9213748846,Gamma(1.760),Gamma(1.760)-0.9213748846],_
+ [1.765,0.9225595178,Gamma(1.765),Gamma(1.765)-0.9225595178],_
+ [1.770,0.9237631277,Gamma(1.770),Gamma(1.770)-0.9237631277],_
+ [1.775,0.9249857211,Gamma(1.775),Gamma(1.775)-0.9249857211],_
+ [1.780,0.9262273062,Gamma(1.780),Gamma(1.780)-0.9262273062],_
+ [1.785,0.9274878926,Gamma(1.785),Gamma(1.785)-0.9274878926],_
+ [1.790,0.9287674904,Gamma(1.790),Gamma(1.790)-0.9287674904],_
+ [1.795,0.9300661123,Gamma(1.795),Gamma(1.795)-0.9300661123],_
+ [1.800,0.9313837710,Gamma(1.800),Gamma(1.800)-0.9313837710],_
+ [1.805,0.9327204811,Gamma(1.805),Gamma(1.805)-0.9327204811],_
+ [1.810,0.9340762585,Gamma(1.810),Gamma(1.810)-0.9340762585],_
+ [1.815,0.9354511198,Gamma(1.815),Gamma(1.815)-0.9354511198],_
+ [1.820,0.9368450832,Gamma(1.820),Gamma(1.820)-0.9368450832],_
+ [1.825,0.9382581682,Gamma(1.825),Gamma(1.825)-0.9382581682],_
+ [1.830,0.9396903951,Gamma(1.830),Gamma(1.830)-0.9396903951],_
+ [1.835,0.9411417859,Gamma(1.835),Gamma(1.835)-0.9411417859],_
+ [1.840,0.9426123634,Gamma(1.840),Gamma(1.840)-0.9426123634],_
+ [1.845,0.9441021519,Gamma(1.845),Gamma(1.845)-0.9441021519],_
+ [1.850,0.9456111764,Gamma(1.850),Gamma(1.850)-0.9456111764],_
+ [1.855,0.9471394637,Gamma(1.855),Gamma(1.855)-0.9471394637],_
+ [1.860,0.9486870417,Gamma(1.860),Gamma(1.860)-0.9486870417],_
+ [1.865,0.9502539389,Gamma(1.865),Gamma(1.865)-0.9502539389],_
+ [1.870,0.9518401855,Gamma(1.870),Gamma(1.870)-0.9518401855],_
+ [1.875,0.9534458127,Gamma(1.875),Gamma(1.875)-0.9534458127],_
+ [1.880,0.9550708530,Gamma(1.880),Gamma(1.880)-0.9550708530],_
+ [1.885,0.9567153398,Gamma(1.885),Gamma(1.885)-0.9567153398],_
+ [1.890,0.9583793077,Gamma(1.890),Gamma(1.890)-0.9583793077],_
+ [1.895,0.9600627927,Gamma(1.895),Gamma(1.895)-0.9600627927],_
+ [1.900,0.9617658319,Gamma(1.900),Gamma(1.900)-0.9617658319],_
+ [1.905,0.9634884632,Gamma(1.905),Gamma(1.905)-0.9634884632],_
+ [1.910,0.9652307261,Gamma(1.910),Gamma(1.910)-0.9652307261],_
+ [1.915,0.9669926608,Gamma(1.915),Gamma(1.915)-0.9669926608],_
+ [1.920,0.9687743090,Gamma(1.920),Gamma(1.920)-0.9687743090],_
+ [1.925,0.9705757134,Gamma(1.925),Gamma(1.925)-0.9705757134],_
+ [1.930,0.9723969178,Gamma(1.930),Gamma(1.930)-0.9723969178],_
+ [1.935,0.9742379672,Gamma(1.935),Gamma(1.935)-0.9742379672],_
+ [1.940,0.9760989075,Gamma(1.940),Gamma(1.940)-0.9760989075],_
+ [1.945,0.9779797861,Gamma(1.945),Gamma(1.945)-0.9779797861],_
+ [1.950,0.9798806513,Gamma(1.950),Gamma(1.950)-0.9798806513],_
+ [1.955,0.9818015524,Gamma(1.955),Gamma(1.955)-0.9818015524],_
+ [1.960,0.9837425404,Gamma(1.960),Gamma(1.960)-0.9837425404],_
+ [1.965,0.9857036664,Gamma(1.965),Gamma(1.965)-0.9857036664],_
+ [1.970,0.9876849838,Gamma(1.970),Gamma(1.970)-0.9876849838],_
+ [1.975,0.9896865462,Gamma(1.975),Gamma(1.975)-0.9896865462],_
+ [1.980,0.9917084087,Gamma(1.980),Gamma(1.980)-0.9917084087],_
+ [1.985,0.9937506274,Gamma(1.985),Gamma(1.985)-0.9937506274],_
+ [1.990,0.9958132598,Gamma(1.990),Gamma(1.990)-0.9958132598],_
+ [1.995,0.9978963643,Gamma(1.995),Gamma(1.995)-0.9978963643]]
+--R 
+--R
+--R   (1)
+--R   [[1.,1.,1.,0.],
+--R
+--R     [1.0049999999999999, 0.99713853539999997, 0.99713853525101781,
+--R      - 1.4898215994207931E-10]
+--R     ,
+--R    [1.01,0.99432585120000005,0.99432585119150585,- 8.4942053391046102E-12],
+--R
+--R     [1.0149999999999999, 0.99156128880000005, 0.99156128884897066,
+--R      4.8970605348586105E-11]
+--R     ,
+--R    [1.02,0.9888442033,0.98884420326391309,- 3.6086911237021013E-11],
+--R
+--R     [1.0249999999999999, 0.98617396329999996, 0.98617396314825367,
+--R      - 1.5174628220648856E-10]
+--R     ,
+--R    [1.03,0.98354995059999994,0.98354995055382399,- 4.6175951950999661E-11],
+--R
+--R     [1.0349999999999999, 0.98097156060000001, 0.98097156055058576,
+--R      - 4.9414250469226317E-11]
+--R     ,
+--R    [1.04,0.9784382009,0.97843820091424472,1.4244716517453071E-11],
+--R
+--R     [1.0449999999999999, 0.97594929190000002, 0.97594929182295154,
+--R      - 7.7048478708263701E-11]
+--R     ,
+--R    [1.05,0.97350426560000003,0.97350426556277558,- 3.7224445748051949E-11],
+--R
+--R     [1.0549999999999999, 0.97110256630000003, 0.97110256624166991,
+--R      - 5.8330118513083562E-11]
+--R     ,
+--R
+--R     [1.0600000000000001, 0.96874364950000003, 0.9687436495116386,
+--R      1.1638578989447979E-11]
+--R     ,
+--R
+--R     [1.0649999999999999, 0.96642698230000001, 0.96642698229884005,
+--R      - 1.1599610161283636E-12]
+--R     ,
+--R
+--R     [1.0700000000000001, 0.96415204249999997, 0.96415204254136644,
+--R      4.1366465808323483E-11]
+--R     ,
+--R    [1.075,0.96191831890000001,0.96191831893444502,3.4445002405902869E-11],
+--R
+--R     [1.0800000000000001, 0.9597253107, 0.95972531068282241,
+--R      - 1.7177592681605347E-11]
+--R     ,
+--R    [1.085,0.95757252729999998,0.957572527260101,- 3.9898973014373951E-11],
+--R
+--R     [1.0900000000000001, 0.95545948820000004, 0.95545948817480109,
+--R      - 2.5198954034522103E-11]
+--R     ,
+--R    [1.095,0.95338572269999999,0.95338572274293298,4.2932990496069579E-11],
+--R
+--R     [1.1000000000000001, 0.9513507699, 0.95135076986687306,
+--R      - 3.3126945631067883E-11]
+--R     ,
+--R    [1.105,0.94935417779999998,0.94935417782034381,2.0343837725533831E-11],
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+--R      - 2.7262525570392881E-11]
+--R     ,
+--R    [1.845,0.94410215190000002,0.94410215176943868,- 1.3056133951749871E-10],
+--R
+--R     [1.8500000000000001, 0.94561117640000003, 0.94561117640619552,
+--R      6.1954885666182236E-12]
+--R     ,
+--R    [1.855,0.94713946370000002,0.94713946380763558,1.0763556712589661E-10],
+--R
+--R     [1.8600000000000001, 0.94868704169999996, 0.94868704167794815,
+--R      - 2.2051804826617172E-11]
+--R     ,
+--R    [1.865,0.95025393889999998,0.95025393889642473,- 3.5752512062003916E-12],
+--R
+--R     [1.8700000000000001, 0.95184018550000005, 0.95184018551317828,
+--R      1.3178236279998146E-11]
+--R     ,
+--R    [1.875,0.95344581269999995,0.95344581274503493,4.5034975748592387E-11],
+--R
+--R     [1.8799999999999999, 0.95507085300000005, 0.95507085297159322,
+--R      - 2.840683244187403E-11]
+--R     ,
+--R    [1.885,0.95671533980000001,0.95671533973145539,- 6.8544614428844852E-11],
+--R
+--R     [1.8899999999999999, 0.95837930770000002, 0.95837930771862279,
+--R      1.8622769992759913E-11]
+--R     ,
+--R    [1.895,0.96006279269999995,0.96006279277905426,7.9054318646853972E-11],
+--R
+--R     [1.8999999999999999, 0.96176583189999998, 0.96176583190738751,
+--R      7.3875350281582541E-12]
+--R     ,
+--R    [1.905,0.96348846320000003,0.96348846324382054,4.3820502781954929E-11],
+--R
+--R     [1.9099999999999999, 0.96523072610000005, 0.96523072607114846,
+--R      - 2.8851587785538868E-11]
+--R     ,
+--R    [1.915,0.96699266080000001,0.96699266081195756,1.1957546064422786E-11],
+--R
+--R     [1.9199999999999999, 0.96877430899999994, 0.96877430902597383,
+--R      2.5973889705710462E-11]
+--R     ,
+--R    [1.925,0.97057571340000004,0.97057571340755988,7.5598416415800784E-12],
+--R
+--R     [1.9299999999999999, 0.97239691780000004, 0.97239691778336623,
+--R      - 1.6633805444143945E-11]
+--R     ,
+--R
+--R     [1.9350000000000001, 0.9742379672, 0.97423796711012756,
+--R      - 8.9872442821103959E-11]
+--R     ,
+--R
+--R     [1.9399999999999999, 0.97609890749999995, 0.97609890747260597,
+--R      - 2.73939759765085E-11]
+--R     ,
+--R
+--R     [1.9450000000000001, 0.97797978610000003, 0.97797978608168112,
+--R      - 1.8318901950920008E-11]
+--R     ,
+--R    [1.95,0.97988065130000002,0.97988065127258051,- 2.7419511106074879E-11],
+--R    [1.9550000000000001,0.9818015524,0.98180155250325418,1.032541829815159E-10],
+--R    [1.96,0.98374254039999998,0.98374254035288666,- 4.711331325069068E-11],
+--R
+--R     [1.9650000000000001, 0.98570366639999996, 0.98570366652054964,
+--R      1.2054968134833643E-10]
+--R     ,
+--R    [1.97,0.98768498380000003,0.98768498382399139,2.399136445063732E-11],
+--R
+--R     [1.9750000000000001, 0.98968654619999996, 0.98968654619855956,
+--R      - 1.4404033521486781E-12]
+--R     ,
+--R    [1.98,0.99170840869999999,0.99170840869626087,- 3.7391201246350647E-12],
+--R
+--R     [1.9850000000000001, 0.99375062739999998, 0.99375062748495313,
+--R      8.4953155621292353E-11]
+--R     ,
+--R    [1.99,0.99581325980000002,0.995813259847667,4.7666981473071246E-11],
+--R
+--R     [1.9950000000000001, 0.99789636429999995, 0.99789636418206007,
+--R      - 1.1793988008435008E-10]
+--R     ]
+--R                                                  Type: List List DoubleFloat
+--E 1
+
+--S 2 of 3
+Psi(x:DFLOAT):DFLOAT==polygamma(0,x)
+--R 
+--R   Function declaration Psi : DoubleFloat -> DoubleFloat has been added
+--R      to workspace.
+--R                                                                   Type: Void
+--E 2
+@
+In the following table there are 4 columns. The first column
+is the argument of Psi, ranging from 1.000 to 1.995. The
+second column is the reference value of Psi from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+
+<<*>>=
+
+--S 3 of 3
+[[1.000, -0.5772156649, Psi(1.000), Psi(1.000)- -0.5772156649],_
+ [1.005, -0.5690209113, Psi(1.005), Psi(1.005)- -0.5690209113],_
+ [1.010, -0.5608854579, Psi(1.010), Psi(1.010)- -0.5608854579],_
+ [1.015, -0.5528085156, Psi(1.015), Psi(1.015)- -0.5528085156],_
+ [1.020, -0.5447893105, Psi(1.020), Psi(1.020)- -0.5447893105],_
+ [1.025, -0.5368270828, Psi(1.025), Psi(1.025)- -0.5368270828],_
+ [1.030, -0.5289210873, Psi(1.030), Psi(1.030)- -0.5289210873],_
+ [1.035, -0.5210705921, Psi(1.035), Psi(1.035)- -0.5210705921],_
+ [1.040, -0.5132748789, Psi(1.040), Psi(1.040)- -0.5132748789],_
+ [1.045, -0.5055332428, Psi(1.045), Psi(1.045)- -0.5055332428],_
+ [1.050, -0.4978449913, Psi(1.050), Psi(1.050)- -0.4978449913],_
+ [1.055, -0.4902094448, Psi(1.055), Psi(1.055)- -0.4902094448],_
+ [1.060, -0.4826259358, Psi(1.060), Psi(1.060)- -0.4826259358],_
+ [1.065, -0.4750938088, Psi(1.065), Psi(1.065)- -0.4750938088],_
+ [1.070, -0.4676124199, Psi(1.070), Psi(1.070)- -0.4676124199],_
+ [1.075, -0.4601811367, Psi(1.075), Psi(1.075)- -0.4601811367],_
+ [1.080, -0.4527993380, Psi(1.080), Psi(1.080)- -0.4527993380],_
+ [1.085, -0.4454664135, Psi(1.085), Psi(1.085)- -0.4454664135],_
+ [1.090, -0.4381817635, Psi(1.090), Psi(1.090)- -0.4381817635],_
+ [1.095, -0.4309447988, Psi(1.095), Psi(1.095)- -0.4309447988],_
+ [1.100, -0.4237549404, Psi(1.100), Psi(1.100)- -0.4237549404],_
+ [1.105, -0.4166116193, Psi(1.105), Psi(1.105)- -0.4166116193],_
+ [1.110, -0.4095142761, Psi(1.110), Psi(1.110)- -0.4095142761],_
+ [1.115, -0.4024623611, Psi(1.115), Psi(1.115)- -0.4024623611],_
+ [1.120, -0.3954553339, Psi(1.120), Psi(1.120)- -0.3954553339],_
+ [1.125, -0.3884926633, Psi(1.125), Psi(1.125)- -0.3884926633],_
+ [1.130, -0.3815738268, Psi(1.130), Psi(1.130)- -0.3815738268],_
+ [1.135, -0.3746983110, Psi(1.135), Psi(1.135)- -0.3746983110],_
+ [1.140, -0.3678656106, Psi(1.140), Psi(1.140)- -0.3678656106],_
+ [1.145, -0.3610752291, Psi(1.145), Psi(1.145)- -0.3610752291],_
+ [1.150, -0.3543266780, Psi(1.150), Psi(1.150)- -0.3543266780],_
+ [1.155, -0.3476194768, Psi(1.155), Psi(1.155)- -0.3476194768],_
+ [1.160, -0.3409531528, Psi(1.160), Psi(1.160)- -0.3409531528],_
+ [1.165, -0.3343272413, Psi(1.165), Psi(1.165)- -0.3343272413],_
+ [1.170, -0.3277412847, Psi(1.170), Psi(1.170)- -0.3277412847],_
+ [1.175, -0.3211948332, Psi(1.175), Psi(1.175)- -0.3211948332],_
+ [1.180, -0.3146874438, Psi(1.180), Psi(1.180)- -0.3146874438],_
+ [1.185, -0.3082186809, Psi(1.185), Psi(1.185)- -0.3082186809],_
+ [1.190, -0.3017881156, Psi(1.190), Psi(1.190)- -0.3017881156],_
+ [1.195, -0.2953953259, Psi(1.195), Psi(1.195)- -0.2953953259],_
+ [1.200, -0.2890398966, Psi(1.200), Psi(1.200)- -0.2890398966],_
+ [1.205, -0.2827214187, Psi(1.205), Psi(1.205)- -0.2827214187],_
+ [1.210, -0.2764394897, Psi(1.210), Psi(1.210)- -0.2764394897],_
+ [1.215, -0.2701937135, Psi(1.215), Psi(1.215)- -0.2701937135],_
+ [1.220, -0.2639837000, Psi(1.220), Psi(1.220)- -0.2639837000],_
+ [1.225, -0.2578090652, Psi(1.225), Psi(1.225)- -0.2578090652],_
+ [1.230, -0.2516694307, Psi(1.230), Psi(1.230)- -0.2516694307],_
+ [1.235, -0.2455644243, Psi(1.235), Psi(1.235)- -0.2455644243],_
+ [1.240, -0.2394936791, Psi(1.240), Psi(1.240)- -0.2394936791],_
+ [1.245, -0.2334568341, Psi(1.245), Psi(1.245)- -0.2334568341],_
+ [1.250, -0.2274535334, Psi(1.250), Psi(1.250)- -0.2274535334],_
+ [1.255, -0.2214834266, Psi(1.255), Psi(1.255)- -0.2214834266],_
+ [1.260, -0.2155461686, Psi(1.260), Psi(1.260)- -0.2155461686],_
+ [1.265, -0.2096414193, Psi(1.265), Psi(1.265)- -0.2096414193],_
+ [1.270, -0.2037688437, Psi(1.270), Psi(1.270)- -0.2037688437],_
+ [1.275, -0.1979281118, Psi(1.275), Psi(1.275)- -0.1979281118],_
+ [1.280, -0.1921188983, Psi(1.280), Psi(1.280)- -0.1921188983],_
+ [1.285, -0.1863408828, Psi(1.285), Psi(1.285)- -0.1863408828],_
+ [1.290, -0.1805937494, Psi(1.290), Psi(1.290)- -0.1805937494],_
+ [1.295, -0.1748771870, Psi(1.295), Psi(1.295)- -0.1748771870],_
+ [1.300, -0.1691908889, Psi(1.300), Psi(1.300)- -0.1691908889],_
+ [1.305, -0.1635345526, Psi(1.305), Psi(1.305)- -0.1635345526],_
+ [1.310, -0.1579078803, Psi(1.310), Psi(1.310)- -0.1579078803],_
+ [1.315, -0.1523105782, Psi(1.315), Psi(1.315)- -0.1523105782],_
+ [1.320, -0.1467423568, Psi(1.320), Psi(1.320)- -0.1467423568],_
+ [1.325, -0.1412029305, Psi(1.325), Psi(1.325)- -0.1412029305],_
+ [1.330, -0.1356920180, Psi(1.330), Psi(1.330)- -0.1356920180],_
+ [1.335, -0.1302093416, Psi(1.335), Psi(1.335)- -0.1302093416],_
+ [1.340, -0.1247546279, Psi(1.340), Psi(1.340)- -0.1247546279],_
+ [1.345, -0.1193276069, Psi(1.345), Psi(1.345)- -0.1193276069],_
+ [1.350, -0.1139280127, Psi(1.350), Psi(1.350)- -0.1139280127],_
+ [1.355, -0.1085555827, Psi(1.355), Psi(1.355)- -0.1085555827],_
+ [1.360, -0.1032100582, Psi(1.360), Psi(1.360)- -0.1032100582],_
+ [1.365, -0.0978911840, Psi(1.365), Psi(1.365)- -0.0978911840],_
+ [1.370, -0.0925987082, Psi(1.370), Psi(1.370)- -0.0925987082],_
+ [1.375, -0.0873323825, Psi(1.375), Psi(1.375)- -0.0873323825],_
+ [1.380, -0.0820919619, Psi(1.380), Psi(1.380)- -0.0820919619],_
+ [1.385, -0.0768772046, Psi(1.385), Psi(1.385)- -0.0768772046],_
+ [1.390, -0.0716878723, Psi(1.390), Psi(1.390)- -0.0716878723],_
+ [1.395, -0.0665237297, Psi(1.395), Psi(1.395)- -0.0665237297],_
+ [1.400, -0.0613845446, Psi(1.400), Psi(1.400)- -0.0613845446],_
+ [1.405, -0.0562700879, Psi(1.405), Psi(1.405)- -0.0562700879],_
+ [1.410, -0.0511801337, Psi(1.410), Psi(1.410)- -0.0511801337],_
+ [1.415, -0.0461144589, Psi(1.415), Psi(1.415)- -0.0461144589],_
+ [1.420, -0.0410728433, Psi(1.420), Psi(1.420)- -0.0410728433],_
+ [1.425, -0.0360550697, Psi(1.425), Psi(1.425)- -0.0360550697],_
+ [1.430, -0.0310609237, Psi(1.430), Psi(1.430)- -0.0310609237],_
+ [1.435, -0.0260901935, Psi(1.435), Psi(1.435)- -0.0260901935],_
+ [1.440, -0.0211426703, Psi(1.440), Psi(1.440)- -0.0211426703],_
+ [1.445, -0.0162181479, Psi(1.445), Psi(1.445)- -0.0162181479],_
+ [1.450, -0.0113164226, Psi(1.450), Psi(1.450)- -0.0113164226],_
+ [1.455, -0.0064372934, Psi(1.455), Psi(1.455)- -0.0064372934],_
+ [1.460, -0.0015805620, Psi(1.460), Psi(1.460)- -0.0015805620],_
+ [1.465,  0.0032539677, Psi(1.465), Psi(1.465)-  0.0032539677],_
+ [1.470,  0.0080664890, Psi(1.470), Psi(1.470)-  0.0080664890],_
+ [1.475,  0.0128571930, Psi(1.475), Psi(1.475)-  0.0128571930],_
+ [1.480,  0.0176262684, Psi(1.480), Psi(1.480)-  0.0176262684],_
+ [1.485,  0.0223739013, Psi(1.485), Psi(1.485)-  0.0223739013],_
+ [1.490,  0.0271002758, Psi(1.490), Psi(1.490)-  0.0271002758],_
+ [1.495,  0.0318055736, Psi(1.495), Psi(1.495)-  0.0318055736],_
+ [1.500,  0.0364899740, Psi(1.500), Psi(1.500)-  0.0364899740],_
+ [1.505,  0.0411536543, Psi(1.505), Psi(1.505)-  0.0411536543],_
+ [1.510,  0.0457967896, Psi(1.510), Psi(1.510)-  0.0457967896],_
+ [1.515,  0.0504195527, Psi(1.515), Psi(1.515)-  0.0504195527],_
+ [1.520,  0.0550221146, Psi(1.520), Psi(1.520)-  0.0550221146],_
+ [1.525,  0.0596046439, Psi(1.525), Psi(1.525)-  0.0596046439],_
+ [1.530,  0.0641673074, Psi(1.530), Psi(1.530)-  0.0641673074],_
+ [1.535,  0.0687102697, Psi(1.535), Psi(1.535)-  0.0687102697],_
+ [1.540,  0.0732336936, Psi(1.540), Psi(1.540)-  0.0732336936],_
+ [1.545,  0.0777377300, Psi(1.545), Psi(1.545)-  0.0777377300],_
+ [1.550,  0.0822225675, Psi(1.550), Psi(1.550)-  0.0822225675],_
+ [1.555,  0.0866883334, Psi(1.555), Psi(1.555)-  0.0866883334],_
+ [1.560,  0.0911351925, Psi(1.560), Psi(1.560)-  0.0911351925],_
+ [1.565,  0.0955632984, Psi(1.565), Psi(1.565)-  0.0955632984],_
+ [1.570,  0.0999728024, Psi(1.570), Psi(1.570)-  0.0999728024],_
+ [1.575,  0.1043638544, Psi(1.575), Psi(1.575)-  0.1043638544],_
+ [1.580,  0.1087366023, Psi(1.580), Psi(1.580)-  0.1087366023],_
+ [1.585,  0.1130911923, Psi(1.585), Psi(1.585)-  0.1130911923],_
+ [1.590,  0.1174277690, Psi(1.590), Psi(1.590)-  0.1174277690],_
+ [1.595,  0.1217464754, Psi(1.595), Psi(1.595)-  0.1217464754],_
+ [1.600,  0.1260474528, Psi(1.600), Psi(1.600)-  0.1260474528],_
+ [1.605,  0.1303308407, Psi(1.605), Psi(1.605)-  0.1303308407],_
+ [1.610,  0.1345967772, Psi(1.610), Psi(1.610)-  0.1345967772],_
+ [1.615,  0.1388453988, Psi(1.615), Psi(1.615)-  0.1388453988],_
+ [1.620,  0.1430768404, Psi(1.620), Psi(1.620)-  0.1430768404],_
+ [1.625,  0.1472912354, Psi(1.625), Psi(1.625)-  0.1472912354],_
+ [1.630,  0.1514887158, Psi(1.630), Psi(1.630)-  0.1514887158],_
+ [1.635,  0.1556694120, Psi(1.635), Psi(1.635)-  0.1556694120],_
+ [1.640,  0.1598334529, Psi(1.640), Psi(1.640)-  0.1598334529],_
+ [1.645,  0.1639809660, Psi(1.645), Psi(1.645)-  0.1639809660],_
+ [1.650,  0.1681120776, Psi(1.650), Psi(1.650)-  0.1681120776],_
+ [1.655,  0.1722269122, Psi(1.655), Psi(1.655)-  0.1722269122],_
+ [1.660,  0.1763255933, Psi(1.660), Psi(1.660)-  0.1763255933],_
+ [1.665,  0.1804082427, Psi(1.665), Psi(1.665)-  0.1804082427],_
+ [1.670,  0.1844749813, Psi(1.670), Psi(1.670)-  0.1844749813],_
+ [1.675,  0.1885259282, Psi(1.675), Psi(1.675)-  0.1885259282],_
+ [1.680,  0.1925612015, Psi(1.680), Psi(1.680)-  0.1925612015],_
+ [1.685,  0.1965809180, Psi(1.685), Psi(1.685)-  0.1965809180],_
+ [1.690,  0.2005851931, Psi(1.690), Psi(1.690)-  0.2005851931],_
+ [1.695,  0.2045741410, Psi(1.695), Psi(1.695)-  0.2045741410],_
+ [1.700,  0.2085478749, Psi(1.700), Psi(1.700)-  0.2085478749],_
+ [1.705,  0.2125065064, Psi(1.705), Psi(1.705)-  0.2125065064],_
+ [1.710,  0.2164501462, Psi(1.710), Psi(1.710)-  0.2164501462],_
+ [1.715,  0.2203789037, Psi(1.715), Psi(1.715)-  0.2203789037],_
+ [1.720,  0.2242928871, Psi(1.720), Psi(1.720)-  0.2242928871],_
+ [1.725,  0.2281922037, Psi(1.725), Psi(1.725)-  0.2281922037],_
+ [1.730,  0.2320769593, Psi(1.730), Psi(1.730)-  0.2320769593],_
+ [1.735,  0.2359472589, Psi(1.735), Psi(1.735)-  0.2359472589],_
+ [1.740,  0.2398032061, Psi(1.740), Psi(1.740)-  0.2398032061],_
+ [1.745,  0.2436449038, Psi(1.745), Psi(1.745)-  0.2436449038],_
+ [1.750,  0.2474724535, Psi(1.750), Psi(1.750)-  0.2474724535],_
+ [1.755,  0.2512859559, Psi(1.755), Psi(1.755)-  0.2512859559],_
+ [1.760,  0.2550855103, Psi(1.760), Psi(1.760)-  0.2550855103],_
+ [1.765,  0.2588712154, Psi(1.765), Psi(1.765)-  0.2588712154],_
+ [1.770,  0.2626431686, Psi(1.770), Psi(1.770)-  0.2626431686],_
+ [1.775,  0.2664014664, Psi(1.775), Psi(1.775)-  0.2664014664],_
+ [1.780,  0.2701462043, Psi(1.780), Psi(1.780)-  0.2701462043],_
+ [1.785,  0.2738774769, Psi(1.785), Psi(1.785)-  0.2738774769],_
+ [1.790,  0.2775953776, Psi(1.790), Psi(1.790)-  0.2775953776],_
+ [1.795,  0.2812999992, Psi(1.795), Psi(1.795)-  0.2812999992],_
+ [1.800,  0.2849914333, Psi(1.800), Psi(1.800)-  0.2849914333],_
+ [1.805,  0.2886697707, Psi(1.805), Psi(1.805)-  0.2886697707],_
+ [1.810,  0.2923351012, Psi(1.810), Psi(1.810)-  0.2923351012],_
+ [1.815,  0.2959875138, Psi(1.815), Psi(1.815)-  0.2959875138],_
+ [1.820,  0.2996270966, Psi(1.820), Psi(1.820)-  0.2996270966],_
+ [1.825,  0.3032539367, Psi(1.825), Psi(1.825)-  0.3032539367],_
+ [1.830,  0.3068681205, Psi(1.830), Psi(1.830)-  0.3068681205],_
+ [1.835,  0.3104697335, Psi(1.835), Psi(1.835)-  0.3104697335],_
+ [1.840,  0.3140588602, Psi(1.840), Psi(1.840)-  0.3140588602],_
+ [1.845,  0.3176355846, Psi(1.845), Psi(1.845)-  0.3176355846],_
+ [1.850,  0.3211999895, Psi(1.850), Psi(1.850)-  0.3211999895],_
+ [1.855,  0.3247521572, Psi(1.855), Psi(1.855)-  0.3247521572],_
+ [1.860,  0.3282921691, Psi(1.860), Psi(1.860)-  0.3282921691],_
+ [1.865,  0.3318201056, Psi(1.865), Psi(1.865)-  0.3318201056],_
+ [1.870,  0.3353360467, Psi(1.870), Psi(1.870)-  0.3353360467],_
+ [1.875,  0.3388400713, Psi(1.875), Psi(1.875)-  0.3388400713],_
+ [1.880,  0.3423322577, Psi(1.880), Psi(1.880)-  0.3423322577],_
+ [1.885,  0.3458126835, Psi(1.885), Psi(1.885)-  0.3458126835],_
+ [1.890,  0.3492814255, Psi(1.890), Psi(1.890)-  0.3492814255],_
+ [1.895,  0.3527385596, Psi(1.895), Psi(1.895)-  0.3527385596],_
+ [1.900,  0.3561841612, Psi(1.900), Psi(1.900)-  0.3561841612],_
+ [1.905,  0.3596183049, Psi(1.905), Psi(1.905)-  0.3596183049],_
+ [1.910,  0.3630410646, Psi(1.910), Psi(1.910)-  0.3630410646],_
+ [1.915,  0.3664525136, Psi(1.915), Psi(1.915)-  0.3664525136],_
+ [1.920,  0.3698527244, Psi(1.920), Psi(1.920)-  0.3698527244],_
+ [1.925,  0.3732417688, Psi(1.925), Psi(1.925)-  0.3732417688],_
+ [1.930,  0.3766197179, Psi(1.930), Psi(1.930)-  0.3766197179],_
+ [1.935,  0.3799866424, Psi(1.935), Psi(1.935)-  0.3799866424],_
+ [1.940,  0.3833426119, Psi(1.940), Psi(1.940)-  0.3833426119],_
+ [1.945,  0.3866876959, Psi(1.945), Psi(1.945)-  0.3866876959],_
+ [1.950,  0.3900219627, Psi(1.950), Psi(1.950)-  0.3900219627],_
+ [1.955,  0.3933454805, Psi(1.955), Psi(1.955)-  0.3933454805],_
+ [1.960,  0.3966583163, Psi(1.960), Psi(1.960)-  0.3966583163],_
+ [1.965,  0.3999605371, Psi(1.965), Psi(1.965)-  0.3999605371],_
+ [1.970,  0.4032522088, Psi(1.970), Psi(1.970)-  0.4032522088],_
+ [1.975,  0.4065333970, Psi(1.975), Psi(1.975)-  0.4065333970],_
+ [1.980,  0.4098041664, Psi(1.980), Psi(1.980)-  0.4098041664],_
+ [1.985,  0.4130645816, Psi(1.985), Psi(1.985)-  0.4130645816],_
+ [1.990,  0.4163147060, Psi(1.990), Psi(1.990)-  0.4163147060],_
+ [1.995,  0.4195546030, Psi(1.995), Psi(1.995)-  0.4195546030],_
+ [2.000,  0.4227843351, Psi(2.000), Psi(2.000)-  0.4227843351]]
+--R 
+--R   Compiling function Psi with type DoubleFloat -> DoubleFloat 
+--R
+--R   (3)
+--R   [[1.,- 0.57721566489999998,- 0.57721566490153275,- 1.5327739077974911E-12],
+--R
+--R     [1.0049999999999999, - 0.56902091129999999, - 0.56902091134438304,
+--R      - 4.4383052788532495E-11]
+--R     ,
+--R    [1.01,- 0.5608854579,- 0.56088545786867472,3.1325275706706179E-11],
+--R
+--R     [1.0149999999999999, - 0.55280851559999999, - 0.55280851559434629,
+--R      5.6536997306011472E-12]
+--R     ,
+--R    [1.02,- 0.54478931050000001,- 0.54478931045617984,4.3820169715047541E-11],
+--R
+--R     [1.0249999999999999, - 0.53682708280000002, - 0.53682708284938863,
+--R      - 4.9388604317357476E-11]
+--R     ,
+--R    [1.03,- 0.52892108729999998,- 0.5289210872854303,1.4569678796760854E-11],
+--R
+--R     [1.0349999999999999, - 0.52107059209999995, - 0.52107059205771,
+--R      4.2289949320206688E-11]
+--R     ,
+--R    [1.04,- 0.51327487890000001,- 0.51327487891683021,- 1.6830203897200136E-11],
+--R
+--R     [1.0449999999999999, - 0.5055332428, - 0.50553324275508449,
+--R      4.4915515751142721E-11]
+--R     ,
+--R    [1.05,- 0.49784499129999998,- 0.49784499129987031,1.2967404927621828E-13],
+--R
+--R     [1.0549999999999999, - 0.4902094448, - 0.49020944481574569,
+--R      - 1.5745682535595051E-11]
+--R     ,
+--R
+--R     [1.0600000000000001, - 0.48262593580000002, - 0.48262593581482538,
+--R      - 1.4825363159332028E-11]
+--R     ,
+--R
+--R     [1.0649999999999999, - 0.47509380880000002, - 0.47509380877526647,
+--R      2.4733548542599237E-11]
+--R     ,
+--R
+--R     [1.0700000000000001, - 0.46761241990000002, - 0.46761241986755342,
+--R      3.2446600961577587E-11]
+--R     ,
+--R    [1.075,- 0.46018113669999999,- 0.4601811366883593,1.1640688413194766E-11],
+--R
+--R     [1.0800000000000001, - 0.452799338, - 0.45279933800171246,
+--R      - 1.7124635043330727E-12]
+--R     ,
+--R    [1.085,- 0.44546641349999999,- 0.44546641348725191,1.2748080369107129E-11],
+--R
+--R     [1.0900000000000001, - 0.43818176349999999, - 0.43818176349533489,
+--R      4.6651016383236765E-12]
+--R     ,
+--R    [1.095,- 0.43094479879999997,- 0.43094479880878706,- 8.7870821730007265E-12]
+--R     ,
+--R
+--R     [1.1000000000000001, - 0.4237549404, - 0.42375494041107653,
+--R      - 1.1076528583231493E-11]
+--R     ,
+--R    [1.105,- 0.41661161930000001,- 0.41661161926071655,3.9283465369521764E-11],
+--R
+--R     [1.1100000000000001, - 0.40951427610000002, - 0.40951427607169383,
+--R      2.830619072469176E-11]
+--R     ,
+--R    [1.115,- 0.4024623611,- 0.40246236109974648,2.535194276731545E-13],
+--R
+--R     [1.1200000000000001, - 0.39545533389999998, - 0.39545533393429283,
+--R      - 3.4292846340377992E-11]
+--R     ,
+--R    [1.125,- 0.38849266329999999,- 0.38849266329585463,4.1453507293454095E-12],
+--R
+--R     [1.1299999999999999, - 0.38157382680000002, - 0.38157382683879215,
+--R      - 3.8792136169973901E-11]
+--R     ,
+--R    [1.135,- 0.37469831100000001,- 0.37469831095919082,4.0809189361112885E-11],
+--R
+--R     [1.1399999999999999, - 0.3678656106, - 0.36786561060774969,
+--R      - 7.7496897787909802E-12]
+--R     ,
+--R    [1.145,- 0.36107522910000001,- 0.361075229107509,- 7.5089934270522463E-12],
+--R
+--R     [1.1499999999999999, - 0.35432667800000001, - 0.35432667797627904,
+--R      2.3720969632989863E-11]
+--R     ,
+--R    [1.155,- 0.34761947679999999,- 0.34761947675362337,4.6376624762700658E-11],
+--R
+--R     [1.1599999999999999, - 0.34095315279999999, - 0.34095315283226135,
+--R      - 3.226136024991888E-11]
+--R     ,
+--R    [1.165,- 0.33432724130000002,- 0.3343272412937619,6.2381211307638296E-12],
+--R
+--R     [1.1699999999999999, - 0.32774128470000002, - 0.3277412847483927,
+--R      - 4.839267875311748E-11]
+--R     ,
+--R    [1.175,- 0.32119483319999997,- 0.3211948331790081,2.0991874905007535E-11],
+--R
+--R     [1.1799999999999999, - 0.31468744380000002, - 0.31468744378886082,
+--R      1.1139200672971583E-11]
+--R     ,
+--R
+--R     [1.1850000000000001, - 0.30821868089999999, - 0.30821868085320625,
+--R      4.6793735553052329E-11]
+--R     ,
+--R
+--R     [1.1899999999999999, - 0.30178811560000002, - 0.30178811557461016,
+--R      2.5389856883606399E-11]
+--R     ,
+--R
+--R     [1.1950000000000001, - 0.29539532590000001, - 0.2953953259418296,
+--R      - 4.1829595343045867E-11]
+--R     ,
+--R    [1.2,- 0.28903989660000001,- 0.28903989659218843,7.8115847124138327E-12],
+--R
+--R     [1.2050000000000001, - 0.2827214187, - 0.28272141867731704,
+--R      2.2682966616116573E-11]
+--R     ,
+--R    [1.21,- 0.27643948969999999,- 0.2764394897321919,- 3.2191915799728577E-11],
+--R
+--R     [1.2150000000000001, - 0.27019371349999999, - 0.27019371354735244,
+--R      - 4.7352455290194939E-11]
+--R     ,
+--R    [1.22,- 0.26398369999999999,- 0.26398370004422023,- 4.4220238581971216E-11],
+--R
+--R     [1.2250000000000001, - 0.25780906520000002, - 0.25780906515343338,
+--R      4.6566639433365253E-11]
+--R     ,
+--R    [1.23,- 0.25166943069999997,- 0.25166943069609982,3.9001579743569437E-12],
+--R
+--R     [1.2350000000000001, - 0.2455644243, - 0.24556442426789726,
+--R      3.2102737135275561E-11]
+--R     ,
+--R    [1.24,- 0.2394936791,- 0.23949367912593666,- 2.5936669478809904E-11],
+--R
+--R     [1.2450000000000001, - 0.23345683410000001, - 0.23345683407831253,
+--R      2.1687485141086427E-11]
+--R     ,
+--R    [1.25,- 0.22745353339999999,- 0.22745353337626528,2.37347086429196E-11],
+--R
+--R     [1.2549999999999999, - 0.22148342660000001, - 0.22148342660888165,
+--R      - 8.8816454191231742E-12]
+--R     ,
+--R    [1.26,- 0.2155461686,- 0.21554616860026521,- 2.6520452500733427E-13],
+--R
+--R     [1.2649999999999999, - 0.20964141929999999, - 0.20964141930911384,
+--R      - 9.1138485647235257E-12]
+--R     ,
+--R    [1.27,- 0.20376884370000001,- 0.20376884373062343,- 3.0623420466113771E-11],
+--R
+--R     [1.2749999999999999, - 0.1979281118, - 0.19792811180067393,
+--R      - 6.7393313152308565E-13]
+--R     ,
+--R    [1.28,- 0.19211889830000001,- 0.19211889830222173,- 2.221722805728632E-12],
+--R
+--R     [1.2849999999999999, - 0.1863408828, - 0.18634088277384209,
+--R      2.6157909172042082E-11]
+--R     ,
+--R    [1.29,- 0.1805937494,- 0.1805937494203691,- 2.0369095299344053E-11],
+--R
+--R     [1.2949999999999999, - 0.17487718699999999, - 0.17487718702556942,
+--R      - 2.5569435457839518E-11]
+--R     ,
+--R    [1.3,- 0.1691908889,- 0.16919088886679934,3.3200664439902994E-11],
+--R
+--R     [1.3049999999999999, - 0.16353455259999999, - 0.163534552631597,
+--R      - 3.1597002791983186E-11]
+--R     ,
+--R
+--R     [1.3100000000000001, - 0.1579078803, - 0.15790788033614178,
+--R      - 3.6141784010013112E-11]
+--R     ,
+--R
+--R     [1.3149999999999999, - 0.15231057819999999, - 0.15231057824555994,
+--R      - 4.5559944705786393E-11]
+--R     ,
+--R
+--R     [1.3200000000000001, - 0.1467423568, - 0.1467423567959959,
+--R      4.0041026050374739E-12]
+--R     ,
+--R    [1.325,- 0.14120293049999999,- 0.14120293051842803,- 1.8428036874240661E-11]
+--R     ,
+--R
+--R     [1.3300000000000001, - 0.135692018, - 0.13569201796416941,
+--R      3.583058849621068E-11]
+--R     ,
+--R    [1.335,- 0.1302093416,- 0.13020934163201769,- 3.2017694051589274E-11],
+--R
+--R     [1.3400000000000001, - 0.1247546279, - 0.12475462789700376,
+--R      2.9962421432827568E-12]
+--R     ,
+--R    [1.345,- 0.11932760689999999,- 0.11932760694070754,- 4.0707548443208452E-11]
+--R     ,
+--R
+--R     [1.3500000000000001, - 0.1139280127, - 0.11392801268308839,
+--R      1.6911611000480775E-11]
+--R     ,
+--R    [1.355,- 0.1085555827,- 0.10855558271580501,- 1.5805010078473458E-11],
+--R
+--R     [1.3600000000000001, - 0.1032100582, - 0.10321005823697738,
+--R      - 3.6977379491709428E-11]
+--R     ,
+--R
+--R     [1.365, - 9.7891184000000006E-2, - 9.7891183987354968E-2,
+--R      1.2645037794634106E-11]
+--R     ,
+--R
+--R     [1.3700000000000001, - 9.2598708200000004E-2, - 9.2598708187860979E-2,
+--R      1.2139025895585576E-11]
+--R     ,
+--R    [1.375,- 8.73323825E-2,- 8.7332382478473081E-2,2.1526919136150013E-11],
+--R
+--R     [1.3799999999999999, - 8.2091961899999996E-2, - 8.2091961858406615E-2,
+--R      4.1593381516769057E-11]
+--R     ,
+--R
+--R     [1.385, - 7.6877204599999999E-2, - 7.6877204627574525E-2,
+--R      - 2.7574525995888166E-11]
+--R     ,
+--R
+--R     [1.3899999999999999, - 7.1687872299999997E-2, - 7.1687872329281643E-2,
+--R      - 2.928164655191523E-11]
+--R     ,
+--R
+--R     [1.395, - 6.6523729700000006E-2, - 6.6523729694132228E-2,
+--R      5.867778485324493E-12]
+--R     ,
+--R
+--R     [1.3999999999999999, - 6.13845446E-2, - 6.1384544585116108E-2,
+--R      1.4883892729411485E-11]
+--R     ,
+--R
+--R     [1.405, - 5.6270087900000001E-2, - 5.6270087943841696E-2,
+--R      - 4.3841694163937461E-11]
+--R     ,
+--R
+--R     [1.4099999999999999, - 5.1180133699999998E-2, - 5.1180133737897426E-2,
+--R      - 3.7897428251110199E-11]
+--R     ,
+--R
+--R     [1.415, - 4.6114458900000002E-2, - 4.6114458909301992E-2,
+--R      - 9.3019897340340663E-12]
+--R     ,
+--R
+--R     [1.4199999999999999, - 4.1072843300000002E-2, - 4.1072843324024277E-2,
+--R      - 2.4024275624423552E-11]
+--R     ,
+--R
+--R     [1.425, - 3.6055069699999998E-2, - 3.6055069722547906E-2,
+--R      - 2.2547907985170923E-11]
+--R     ,
+--R
+--R     [1.4299999999999999, - 3.10609237E-2, - 3.1060923671447194E-2,
+--R      2.8552805952930527E-11]
+--R     ,
+--R
+--R     [1.4350000000000001, - 2.6090193500000001E-2, - 2.609019351596098E-2,
+--R      - 1.5960979066198533E-11]
+--R     ,
+--R
+--R     [1.4399999999999999, - 2.1142670299999999E-2, - 2.1142670333530678E-2,
+--R      - 3.3530678233972822E-11]
+--R     ,
+--R
+--R     [1.4450000000000001, - 1.62181479E-2, - 1.6218147888283685E-2,
+--R      1.1716315417853451E-11]
+--R     ,
+--R
+--R     [1.45, - 1.1316422600000001E-2, - 1.1316422586445718E-2,
+--R      1.3554282696226494E-11]
+--R     ,
+--R
+--R     [1.4550000000000001, - 6.4372934000000003E-3, - 6.4372934326406561E-3,
+--R      - 3.2640655803217733E-11]
+--R     ,
+--R    [1.46,- 1.580562E-3,- 1.5805619870833398E-3,1.2916660145981251E-11],
+--R
+--R     [1.4650000000000001, 3.2539677000000002E-3, 3.2539676763745362E-3,
+--R      - 2.3625463998339091E-11]
+--R     ,
+--R    [1.47,8.0664889999999996E-3,8.0664890113649745E-3,1.1364974933369965E-11],
+--R
+--R     [1.4750000000000001, 1.2857192999999999E-2, 1.2857193039295334E-2,
+--R      3.9295334347544397E-11]
+--R     ,
+--R    [1.48,1.7626268399999999E-2,1.7626268388849287E-2,- 1.1150712297958165E-11],
+--R
+--R     [1.4850000000000001, 2.2373901299999999E-2, 2.2373901334705404E-2,
+--R      3.47054052163287E-11]
+--R     ,
+--R    [1.49,2.71002758E-2,2.7100275835486465E-2,3.5486464461387257E-11],
+--R
+--R     [1.4950000000000001, 3.1805573599999998E-2, 3.1805573570971468E-2,
+--R      - 2.9028529580088502E-11]
+--R     ,
+--R    [1.5,3.6489974000000001E-2,3.6489973978576673E-2,- 2.1423328389058582E-11],
+--R
+--R     [1.5049999999999999, 4.1153654300000002E-2, 4.1153654289123542E-2,
+--R      - 1.0876459455300136E-11]
+--R     ,
+--R    [1.51,4.5796789599999999E-2,4.5796789561914686E-2,- 3.8085312681346295E-11],
+--R
+--R     [1.5149999999999999, 5.0419552700000002E-2, 5.0419552719128236E-2,
+--R      1.9128233719190035E-11]
+--R     ,
+--R    [1.52,5.5022114599999998E-2,5.5022114579551307E-2,- 2.0448691351315773E-11],
+--R
+--R     [1.5249999999999999, 5.9604643899999997E-2, 5.960464389166209E-2,
+--R      - 8.3379067539190999E-12]
+--R     ,
+--R    [1.53,6.4167307399999998E-2,6.4167307366077231E-2,- 3.3922767372907003E-11],
+--R
+--R     [1.5349999999999999, 6.8710269700000007E-2, 6.8710269707385141E-2,
+--R      7.3851341708675022E-12]
+--R     ,
+--R    [1.54,7.3233693599999997E-2,7.3233693645366138E-2,4.5366141399050264E-11],
+--R
+--R     [1.5449999999999999, 7.7737730000000005E-2, 7.7737739965624497E-2,
+--R      9.9656244922918802E-9]
+--R     ,
+--R    [1.55,8.2222567499999996E-2,8.2222567539644631E-2,3.9644634797220135E-11],
+--R
+--R     [1.5549999999999999, 8.6688333399999998E-2, 8.6688333354268288E-2,
+--R      - 4.5731710085483712E-11]
+--R     ,
+--R
+--R     [1.5600000000000001, 9.1135192500000003E-2, 9.1135192540635401E-2,
+--R      4.0635397824395625E-11]
+--R     ,
+--R
+--R     [1.5649999999999999, 9.5563298399999996E-2, 9.5563298402570163E-2,
+--R      2.5701663020072374E-12]
+--R     ,
+--R
+--R     [1.5700000000000001, 9.9972802400000005E-2, 9.9972802444444731E-2,
+--R      4.4444725677550423E-11]
+--R     ,
+--R    [1.575,0.1043638544,0.10436385439851947,- 1.4805240367010697E-12],
+--R
+--R     [1.5800000000000001, 0.1087366023, 0.10873660225178161,
+--R      - 4.8218387616039138E-11]
+--R     ,
+--R    [1.585,0.1130911923,0.11309119227228603,- 2.7713970007781086E-11],
+--R    [1.5900000000000001,0.117427769,0.11742776903501095,3.5010952470493351E-11],
+--R    [1.595,0.1217464754,0.12174647544723916,4.7239157030531942E-11],
+--R
+--R     [1.6000000000000001, 0.12604745279999999, 0.12604745277347584,
+--R      - 2.6524143992290306E-11]
+--R     ,
+--R    [1.605,0.13033084070000001,0.13033084065991318,- 4.0086822750140527E-11],
+--R
+--R     [1.6100000000000001, 0.13459677719999999, 0.13459677715844587,
+--R      - 4.155412125506075E-11]
+--R     ,
+--R    [1.615,0.13884539879999999,0.13884539875025736,- 4.9742626684334823E-11],
+--R
+--R     [1.6200000000000001, 0.14307684039999999, 0.14307684036898005,
+--R      - 3.1019936619358646E-11]
+--R     ,
+--R    [1.625,0.1472912354,0.14729123542343325,2.3433255336158254E-11],
+--R
+--R     [1.6299999999999999, 0.15148871580000001, 0.15148871581995815,
+--R      1.9958146246779052E-11]
+--R     ,
+--R    [1.635,0.15566941200000001,0.15566941198435302,- 1.5646983708705875E-11],
+--R
+--R     [1.6399999999999999, 0.15983345290000001, 0.15983345288341522,
+--R      - 1.6584789097606745E-11]
+--R     ,
+--R    [1.645,0.16398096600000001,0.16398096604610457,4.6104564610516263E-11],
+--R
+--R     [1.6499999999999999, 0.16811207759999999, 0.16811207758432767,
+--R      - 1.5672324549242944E-11]
+--R     ,
+--R    [1.655,0.1722269122,0.17222691221335784,1.335784260980688E-11],
+--R
+--R     [1.6599999999999999, 0.1763255933, 0.17632559327189457,
+--R      - 2.8105434646263916E-11]
+--R     ,
+--R    [1.665,0.1804082427,0.18040824274177392,4.1773917658360915E-11],
+--R
+--R     [1.6699999999999999, 0.18447498130000001, 0.1844749812673292,
+--R      - 3.2670810501400638E-11]
+--R     ,
+--R    [1.675,0.1885259282,0.18852592817442249,- 2.5577512330343666E-11],
+--R
+--R     [1.6799999999999999, 0.1925612015, 0.19256120148913258,
+--R      - 1.0867418076543345E-11]
+--R     ,
+--R
+--R     [1.6850000000000001, 0.19658091799999999, 0.19658091795613342,
+--R      - 4.3866577037476873E-11]
+--R     ,
+--R
+--R     [1.6899999999999999, 0.2005851931, 0.20058519305674649,
+--R      - 4.3253511883278861E-11]
+--R     ,
+--R
+--R     [1.6950000000000001, 0.20457414099999999, 0.20457414102668603,
+--R      2.6686042264856269E-11]
+--R     ,
+--R    [1.7,0.20854787490000001,0.20854787487349435,- 2.6505658778930297E-11],
+--R
+--R     [1.7050000000000001, 0.21250650639999999, 0.21250650639368796,
+--R      - 6.3120342286282494E-12]
+--R     ,
+--R    [1.71,0.2164501462,0.21645014618960501,- 1.0394990423989725E-11],
+--R
+--R     [1.7150000000000001, 0.22037890369999999, 0.2203789036859658,
+--R      - 1.4034190476408526E-11]
+--R     ,
+--R    [1.72,0.2242928871,0.22429288714615725,4.6157244693034727E-11],
+--R
+--R     [1.7250000000000001, 0.22819220370000001, 0.22819220368823745,
+--R      - 1.1762563145722993E-11]
+--R     ,
+--R    [1.73,0.2320769593,0.23207695930067274,6.7273964177161361E-13],
+--R
+--R     [1.7350000000000001, 0.23594725890000001, 0.23594725885781176,
+--R      - 4.2188252891151024E-11]
+--R     ,
+--R    [1.74,0.2398032061,0.23980320613509676,3.5096758832509067E-11],
+--R
+--R     [1.7450000000000001, 0.24364490380000001, 0.24364490382402559,
+--R      2.4025587075371391E-11]
+--R     ,
+--R    [1.75,0.24747245349999999,0.2474724535468612,4.6861209357373923E-11],
+--R
+--R     [1.7549999999999999, 0.25128595590000002, 0.25128595587109781,
+--R      - 2.8902213955461775E-11]
+--R     ,
+--R    [1.76,0.25508551029999998,0.25508551032368809,2.3688107031460959E-11],
+--R
+--R     [1.7649999999999999, 0.25887121540000002, 0.25887121540503744,
+--R      5.0374149296317228E-12]
+--R     ,
+--R    [1.77,0.26264316859999998,0.26264316860276249,2.7625124410235458E-12],
+--R
+--R     [1.7749999999999999, 0.26640146640000001, 0.2664014664052331,
+--R      5.2330917377219066E-12]
+--R     ,
+--R    [1.78,0.27014620430000003,0.27014620431488368,1.4883649868124849E-11],
+--R
+--R     [1.7849999999999999, 0.27387747690000003, 0.27387747686131236,
+--R      - 3.8687664183356674E-11]
+--R     ,
+--R    [1.79,0.27759537760000003,0.27759537761416786,1.4167833572997779E-11],
+--R
+--R     [1.7949999999999999, 0.28129999919999998, 0.2812999991958266,
+--R      - 4.1733838607171947E-12]
+--R     ,
+--R    [1.8,0.28499143329999999,0.2849914332938619,- 6.138090036245103E-12],
+--R
+--R     [1.8049999999999999, 0.28866977069999999, 0.28866977067331689,
+--R      - 2.6683100173841012E-11]
+--R     ,
+--R
+--R     [1.8100000000000001, 0.2923351012, 0.29233510118877948,
+--R      - 1.1220524509525376E-11]
+--R     ,
+--R
+--R     [1.8149999999999999, 0.29598751379999999, 0.29598751379626109,
+--R      - 3.7388980800301397E-12]
+--R     ,
+--R
+--R     [1.8200000000000001, 0.29962709659999998, 0.29962709656488773,
+--R      - 3.5112246443702588E-11]
+--R     ,
+--R    [1.825,0.3032539367,0.30325393668840539,- 1.1594614157672822E-11],
+--R
+--R     [1.8300000000000001, 0.30686812050000001, 0.30686812049650136,
+--R      - 3.4986458175012558E-12]
+--R     ,
+--R    [1.835,0.31046973350000001,0.31046973346594764,- 3.4052372033244183E-11],
+--R
+--R     [1.8400000000000001, 0.31405886020000001, 0.31405886023156859,
+--R      3.1568581082552782E-11]
+--R     ,
+--R    [1.845,0.31763558460000002,0.31763558459703256,- 2.9674596113693497E-12],
+--R    [1.8500000000000001,0.3211999895,0.32119998954547946,4.5479453536501069E-11]
+--R     ,
+--R    [1.855,0.3247521572,0.32475215724997797,4.9977966209979741E-11],
+--R
+--R     [1.8600000000000001, 0.32829216909999998, 0.32829216908382075,
+--R      - 1.6179224626711175E-11]
+--R     ,
+--R    [1.865,0.33182010560000003,0.33182010563065989,3.0659863536897092E-11],
+--R
+--R     [1.8700000000000001, 0.3353360467, 0.33533604669448569,
+--R      - 5.5143112298594588E-12]
+--R     ,
+--R    [1.875,0.33884007129999999,0.33884007130944738,9.4473873168965383E-12],
+--R
+--R     [1.8799999999999999, 0.34233225769999998, 0.34233225774952925,
+--R      4.9529269574577484E-11]
+--R     ,
+--R    [1.885,0.3458126835,0.34581268353806771,3.8067715646405986E-11],
+--R
+--R     [1.8899999999999999, 0.34928142550000002, 0.34928142545713492,
+--R      - 4.286510035811375E-11]
+--R     ,
+--R    [1.895,0.3527385596,0.35273855955676792,- 4.3232084578903596E-11],
+--R
+--R     [1.8999999999999999, 0.35618416120000002, 0.35618416116406026,
+--R      - 3.5939751175106949E-11]
+--R     ,
+--R    [1.905,0.35961830490000002,0.35961830489211799,- 7.8820283633262989E-12],
+--R
+--R     [1.9099999999999999, 0.36304106460000002, 0.36304106464888108,
+--R      4.8881065861650086E-11]
+--R     ,
+--R    [1.915,0.36645251359999997,0.36645251364580167,4.580169576939852E-11],
+--R    [1.9199999999999999,0.3698527244,0.36985272440640171,6.4017124934423464E-12]
+--R     ,
+--R    [1.925,0.37324176879999998,0.37324176877469795,- 2.5302038242358549E-11],
+--R
+--R     [1.9299999999999999, 0.37661971789999998, 0.37661971792349891,
+--R      2.3498925028064832E-11]
+--R     ,
+--R
+--R     [1.9350000000000001, 0.37998664240000002, 0.37998664236258128,
+--R      - 3.7418734777361351E-11]
+--R     ,
+--R
+--R     [1.9399999999999999, 0.38334261190000002, 0.38334261194674013,
+--R      4.6740111780962934E-11]
+--R     ,
+--R
+--R     [1.9450000000000001, 0.38668769590000002, 0.38668769588372298,
+--R      - 1.6277035275180651E-11]
+--R     ,
+--R    [1.95,0.3900219627,0.39002196274204304,4.2043035719530053E-11],
+--R
+--R     [1.9550000000000001, 0.39334548050000001, 0.39334548045868012,
+--R      - 4.1319891952440457E-11]
+--R     ,
+--R    [1.96,0.39665831629999998,0.39665831634666171,4.6661730035424398E-11],
+--R
+--R     [1.9650000000000001, 0.39996053710000001, 0.39996053710254509,
+--R      2.5450752616507089E-12]
+--R     ,
+--R    [1.97,0.40325220880000001,0.40325220881377177,1.3771761508962754E-11],
+--R
+--R     [1.9750000000000001, 0.40653339700000002, 0.40653339696592627,
+--R      - 3.4073743826468217E-11]
+--R     ,
+--R    [1.98,0.4098041664,0.40980416644989071,4.9890702680244203E-11],
+--R
+--R     [1.9850000000000001, 0.41306458159999998, 0.41306458156888626,
+--R      - 3.1113722709363856E-11]
+--R     ,
+--R    [1.99,0.41631470599999998,0.41631470604541487,4.5414894067619116E-11],
+--R
+--R     [1.9950000000000001, 0.41955460300000003, 0.41955460302810832,
+--R      2.8108293470552326E-11]
+--R     ,
+--R    [2.,0.42278433510000002,0.42278433509846725,- 1.5327739077974911E-12]]
+--R                                                  Type: List List DoubleFloat
+--E 3
+
+)spool 
+)lisp (bye)
+ 
+@
+
+\subsection{Gamma}
+Because we use this file for regression testing and regression tests
+do not yet allow graphics the rest of this file is commented out.
+
+This is the graph of the gamma and 1/gamma functions.
+We are going to combine two graphs in a single viewport so we enable
+color differences for the two graphs.
+\begin{verbatim}
+)clear all
+lineColorDefault(green())
+viewport1:=draw(Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+lineColorDefault(blue())
+viewport2:=draw(1/Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+putGraph(viewport2,getGraph(viewport1,1),2)
+points(viewport2,1,"off")
+points(viewport2,2,"off")
+makeViewport2D(viewport2)
+close(viewport1)
+close(viewport2)
+\end{verbatim}
+
+This is a graph of the log(gamma) function.
+\begin{verbatim}
+)clear all
+viewport:=draw(log Gamma(i), i=0..8, adaptive==true, unit==[1.0,1.0])
+points(viewport1,1,"off")
+close(viewport)
+\end{verbatim}
+
+This is a graph of the Psi function
+\begin{verbatim}
+)clear all
+Psi(x)==polygamma(0,x)
+viewport:=draw(Psi(y),y=-3.5..4,adaptive==true)
+scale(viewport,1,0.9,22.5)
+translate(viewport,1,0,-0.02)
+points(viewport,1,"off")
+close(viewport)
+\end{verbatim}
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf Digital Library of Mathematical Functions},
+http://dlmf.nist.gov
+\bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp267-270
+\end{thebibliography}
+\end{document}
diff --git a/src/input/pfaffian.input.pamphlet b/src/input/pfaffian.input.pamphlet
new file mode 100644
index 0000000..0e2f33b
--- /dev/null
+++ b/src/input/pfaffian.input.pamphlet
@@ -0,0 +1,524 @@
+\documentclass{article}
+\usepackage{amsfonts}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input pfaffian}
+\author{Timothy Daly, Gunter Rote, Martin Rubey}
+\maketitle
+\begin{abstract}
+The Pfaffian is a function defined on skew-symmetric $2n\times 2n$ matrices.
+It is the square root of the determinant.\cite{1}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{Examples}
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cc}
+0 & a\\
+-a & 0
+\end{array}
+\right] = a
+$$
+
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cccc}
+0 & a & b & c\\
+-a & 0 & d & e\\
+-b & -d & 0 & f\\
+-c & -e & -f & 0
+\end{array}
+\right] = af -be + dc
+$$
+
+$$
+{\rm Pfaffian}\left[
+\begin{array}{cccc}
+\begin{array}{cc}
+0 & \lambda_1\\
+-\lambda_1 & 0
+\end{array} & 0 & \cdots & 0\\
+0 & 
+\begin{array}{cc}
+0 & \lambda_2\\
+-\lambda_2 & 0
+\end{array} & & 0\\
+\vdots & & \ddots & \vdots\\
+0 & 0 & \cdots &
+\begin{array}{cc}
+0 & \lambda_n\\
+-\lambda_n & 0
+\end{array}
+\end{array}
+\right] = \lambda_1\lambda_2\cdots\lambda_n
+$$
+\section{Formal definition}
+
+A matrix is skew-symmetric if $A^T = -A$
+
+Let $A = \{a_{i,j}\}$ be a $2n \times 2n$ skew-symmetric matrix. 
+The Pfaffian of A is defined by the equation
+
+$$Pf(A) = \frac{1}{s^n n!}\sum_{\sigma\in{}S_{2n}}sgn(\sigma)
+\prod_{i=1}^n{a_{\sigma(2i-1),\sigma(2i)}}$$
+
+where $S_{2n}$ is the symmetric group and $sgn(\sigma)$ 
+is the signature of $\sigma$.
+
+The Pfaffian of a $n\times n$ skew-symmetric matrix for n odd is
+defined to be zero.
+
+\section{Axiom code}
+<<*>>=
+)spool pfaffian.output
+)set message test on
+)set message auto off
+)clear all
+
+@
+The function B0 has an input range of PositiveInteger. It computes
+a square matrix.  
+<<*>>=
+--S 1 of 26
+B0 n == matrix [[(if i=j+1 and odd? j then -1 else _
+                   if i=j-1 and odd? i then 1 else 0) _
+                     for j in 1..n] for i in 1..n]
+--R 
+--R                                                                   Type: Void
+--E 1
+@
+
+The first 8 values computed are:
+<<*>>=
+--S 2 of 26
+B0 1
+--R 
+--R   Compiling function B0 with type PositiveInteger -> Matrix Integer 
+--R
+--R   (2)  [0]
+--R                                                         Type: Matrix Integer
+--E 2
+
+--S 3 of 26
+B0 2
+--R 
+--R
+--R        + 0   1+
+--R   (3)  |      |
+--R        +- 1  0+
+--R                                                         Type: Matrix Integer
+--E 3
+
+--S 4 of 26
+B0 3
+--R 
+--R
+--R        + 0   1  0+
+--R        |         |
+--R   (4)  |- 1  0  0|
+--R        |         |
+--R        + 0   0  0+
+--R                                                         Type: Matrix Integer
+--E 4
+
+--S 5 of 26
+B0 4
+--R 
+--R
+--R        + 0   1   0   0+
+--R        |              |
+--R        |- 1  0   0   0|
+--R   (5)  |              |
+--R        | 0   0   0   1|
+--R        |              |
+--R        + 0   0  - 1  0+
+--R                                                         Type: Matrix Integer
+--E 5
+
+--S 6 of 26
+B0 5
+--R 
+--R
+--R        + 0   1   0   0  0+
+--R        |                 |
+--R        |- 1  0   0   0  0|
+--R        |                 |
+--R   (6)  | 0   0   0   1  0|
+--R        |                 |
+--R        | 0   0  - 1  0  0|
+--R        |                 |
+--R        + 0   0   0   0  0+
+--R                                                         Type: Matrix Integer
+--E 6
+
+--S 7 of 26
+B0 6
+--R 
+--R
+--R        + 0   1   0   0   0   0+
+--R        |                      |
+--R        |- 1  0   0   0   0   0|
+--R        |                      |
+--R        | 0   0   0   1   0   0|
+--R   (7)  |                      |
+--R        | 0   0  - 1  0   0   0|
+--R        |                      |
+--R        | 0   0   0   0   0   1|
+--R        |                      |
+--R        + 0   0   0   0  - 1  0+
+--R                                                         Type: Matrix Integer
+--E 7
+
+--S 8 of 26
+B0 7
+--R 
+--R
+--R        + 0   1   0   0   0   0  0+
+--R        |                         |
+--R        |- 1  0   0   0   0   0  0|
+--R        |                         |
+--R        | 0   0   0   1   0   0  0|
+--R        |                         |
+--R   (8)  | 0   0  - 1  0   0   0  0|
+--R        |                         |
+--R        | 0   0   0   0   0   1  0|
+--R        |                         |
+--R        | 0   0   0   0  - 1  0  0|
+--R        |                         |
+--R        + 0   0   0   0   0   0  0+
+--R                                                         Type: Matrix Integer
+--E 8
+
+--S 9 of 26
+B0 8
+--R 
+--R
+--R        + 0   1   0   0   0   0   0   0+
+--R        |                              |
+--R        |- 1  0   0   0   0   0   0   0|
+--R        |                              |
+--R        | 0   0   0   1   0   0   0   0|
+--R        |                              |
+--R        | 0   0  - 1  0   0   0   0   0|
+--R   (9)  |                              |
+--R        | 0   0   0   0   0   1   0   0|
+--R        |                              |
+--R        | 0   0   0   0  - 1  0   0   0|
+--R        |                              |
+--R        | 0   0   0   0   0   0   0   1|
+--R        |                              |
+--R        + 0   0   0   0   0   0  - 1  0+
+--R                                                         Type: Matrix Integer
+--E 9
+
+--S 10 of 26
+PfChar(lambda, A) ==
+    n := nrows A
+    odd? n => 0
+    (n = 2) => lambda^2 + A.(1,2)
+    M := subMatrix(A, 3, n, 3, n)
+    r := subMatrix(A, 1, 1, 3, n)
+    s := subMatrix(A, 3, n, 2, 2)
+
+    p := PfChar(lambda, M)
+    d := degree(p, lambda)
+
+    B := B0(n-2)
+    C := r*B
+    g := [(C*s).(1,1), A.(1,2), 1]
+    if d >= 4 then 
+        B := M*B
+        for i in 4..d by 2 repeat
+            C := C*B
+            g := cons((C*s).(1,1), g)
+    g := reverse! g
+
+    res := 0
+    for i in 0..d by 2 for j in 2..d+2 repeat
+        c := coefficient(p, lambda, i)
+        for e in first(g, j) for k in 2..-d by -2 repeat
+            res := res +  c * e * lambda^(k+i)
+
+    res
+--R 
+--R                                                                   Type: Void
+--E 10
+
+--S 11 of 26
+pfaffian A == eval(PfChar(l, A), l=0)
+--R 
+--R                                                                   Type: Void
+--E 11
+
+--S 12 of 26
+m:Matrix(Integer):=[[0,15],[-15,0]]
+--R 
+--R
+--R         + 0    15+
+--R   (12)  |        |
+--R         +- 15  0 +
+--R                                                         Type: Matrix Integer
+--E 12
+
+--S 13 of 26
+pfaffian m
+--R 
+--R   Compiling function B0 with type Integer -> Matrix Integer 
+--R   The type of the local variable res has changed in the computation.
+--R   We will attempt to interpret the code.
+--R   Cannot compile map: PfChar 
+--R   We will attempt to interpret the code.
+--R
+--R   (13)  15
+--R                                                     Type: Polynomial Integer
+--E 13
+
+--S 14 of 26
+m1:Matrix(Polynomial(Integer)):=[[0,a,b,c],[-a,0,d,e],[-b,-d,0,f],[-c,-e,-f,0]]
+--R 
+--R
+--R         + 0    a    b   c+
+--R         |                |
+--R         |- a   0    d   e|
+--R   (14)  |                |
+--R         |- b  - d   0   f|
+--R         |                |
+--R         +- c  - e  - f  0+
+--R                                              Type: Matrix Polynomial Integer
+--E 14
+
+--S 15 of 26
+pfaffian m1
+--R 
+--R
+--R   (15)  a f - b e + c d
+--R                                                     Type: Polynomial Integer
+--E 15
+
+--S 16 of 26
+(a,b,c,d,e,f):=(3,5,7,11,13,17)
+--R 
+--R
+--R   (16)  17
+--R                                                        Type: PositiveInteger
+--E 16
+
+--S 17 of 26
+m1
+--R 
+--R
+--R         + 0    a    b   c+
+--R         |                |
+--R         |- a   0    d   e|
+--R   (17)  |                |
+--R         |- b  - d   0   f|
+--R         |                |
+--R         +- c  - e  - f  0+
+--R                                              Type: Matrix Polynomial Integer
+--E 17
+
+--S 18 of 26
+a*f-b*e+d*c
+--R 
+--R
+--R   (18)  63
+--R                                                        Type: PositiveInteger
+--E 18
+
+--S 19 of 26
+n:=pfaffian m1
+--R 
+--R
+--R   (19)  a f - b e + c d
+--R                                                     Type: Polynomial Integer
+--E 19
+
+--S 20 of 26
+eval(n,['a,'b,'c,'d,'e,'f]::List(Symbol),[a,b,c,d,e,f])
+--R 
+--R
+--R   (20)  63
+--R                                                     Type: Polynomial Integer
+--E 20
+
+@
+Here we demonstrate the pfaffian of a block matrix which is
+anti-symmetric at each level. The proper result is to return
+a single matrix in the ground field. Maybe. It's not clear.
+<<*>>=
+
+)clear properties z
+)clear properties d
+)clear properties v
+
+--S 21 of 26
+z:SQMATRIX(2,INT):=[[0,0],[0,0]]
+--R 
+--R
+--R         +0  0+
+--R   (21)  |    |
+--R         +0  0+
+--R                                                Type: SquareMatrix(2,Integer)
+--E 21
+
+--S 22 of 26
+d:SQMATRIX(2,INT):=[[0,1],[-1,0]]
+--R 
+--R
+--R         + 0   1+
+--R   (22)  |      |
+--R         +- 1  0+
+--R                                                Type: SquareMatrix(2,Integer)
+--E 22
+
+--S 23 of 26
+v:SQMATRIX(4,SQMATRIX(2,INT)):=[[z,d,d,d],[-d,z,d,d],[-d,-d,z,d],[-d,-d,-d,z]]
+--R 
+--R
+--R         + +0  0+   + 0   1+  + 0   1+  + 0   1++
+--R         | |    |   |      |  |      |  |      ||
+--R         | +0  0+   +- 1  0+  +- 1  0+  +- 1  0+|
+--R         |                                      |
+--R         |+0  - 1+   +0  0+   + 0   1+  + 0   1+|
+--R         ||      |   |    |   |      |  |      ||
+--R         |+1   0 +   +0  0+   +- 1  0+  +- 1  0+|
+--R   (23)  |                                      |
+--R         |+0  - 1+  +0  - 1+   +0  0+   + 0   1+|
+--R         ||      |  |      |   |    |   |      ||
+--R         |+1   0 +  +1   0 +   +0  0+   +- 1  0+|
+--R         |                                      |
+--R         |+0  - 1+  +0  - 1+  +0  - 1+   +0  0+ |
+--R         ||      |  |      |  |      |   |    | |
+--R         ++1   0 +  +1   0 +  +1   0 +   +0  0+ +
+--R                                Type: SquareMatrix(4,SquareMatrix(2,Integer))
+--E 23
+
+--S 24 of 26
+pfaffian v
+--R 
+--R   There are 1 exposed and 0 unexposed library operations named 
+--R      subMatrix having 5 argument(s) but none was determined to be 
+--R      applicable. Use HyperDoc Browse, or issue
+--R                            )display op subMatrix
+--R      to learn more about the available operations. Perhaps 
+--R      package-calling the operation or using coercions on the arguments
+--R      will allow you to apply the operation.
+--R 
+--RDaly Bug
+--R   Cannot find a definition or applicable library operation named 
+--R      subMatrix with argument type(s) 
+--R                   SquareMatrix(4,SquareMatrix(2,Integer))
+--R                               PositiveInteger
+--R                               PositiveInteger
+--R                               PositiveInteger
+--R                               PositiveInteger
+--R      
+--R      Perhaps you should use "@" to indicate the required return type, 
+--R      or "$" to specify which version of the function you need.
+--E 24
+
+--S 25 of 26
+mypf(m) ==
+ nr:= nrows m
+ odd? nr => 0
+-- not zero? (nr mod 2)
+ not square? m => 0
+ not antisymmetric? m => 0
+ nr = 2 => m.(1,2)
+ nr = 4 => m.(1,2)*m.(3,4)-m.(1,3)*m.(2,4)+m.(2,3)*m.(1,4)
+ 0
+--R 
+--R                                                                   Type: Void
+--E 25
+
+@
+We can construct an anti-symmetric matrix from a sequence by taking
+the first $n$ elements of the sequence:
+<<*>>=
+--S 26 of 26
+antisymmetric(seq,n) == 
+  m:= matrix [[(if i<j then (seq.(j-i)) _
+                 else if i>j then -(seq.(i-j))    
+                  else 0) for j in 1..n] for i in 1..n]
+--R 
+--R                                                                   Type: Void
+--E 26
+
+@
+According to Austin\cite{1} we can construct a map, called the
+``Pfaffian transformation'' of a sequence which maps the sequence
+$(a_0, a_1, a_2, \ldots)$ to $(Pf(A_0), Pf(A_1), Pf(A_2), \ldots)$.
+Here we construct and test the examples given in the paper:
+\begin{verbatim}
+--
+seq:=[i for i in 1..]
+--R 
+--R
+--R   (26)  [1,2,3,4,5,6,7,8,9,10,...]
+--R                                                 Type: Stream PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[2 for i in 1..]
+--R 
+--R
+--R   (27)  [2,2,2,2,2,2,2,2,2,2,...]
+--R                                                 Type: Stream PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[2^i for i in 0..]
+--R 
+--R
+--R   (28)  [1,2,4,8,16,32,64,128,256,512,...]
+--R                                                 Type: Stream PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+
+--
+seq:=[1,1,2,3,5,8,13,21,34]
+--R 
+--R
+--R   (29)  [1,1,2,3,5,8,13,21,34]
+--R                                                   Type: List PositiveInteger
+--
+
+--
+[pfaffian makepfaff(seq,m) for m in 2..10 by 2]
+--
+\end{verbatim}
+
+@
+<<*>>=
+)spool 
+)lisp (bye)
+
+@
+The Pfaffian is the square root of the determinant. The determinant
+can be calculated by
+\begin{verbatim}
+|a b c|
+|d e f| = a|e f| - b|d f| + c|d e|
+|g h i|    |h i|    |g i|    |g h|
+\end{verbatim}
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Austin, T., Bantilan, H., Jonas, I., Kory, P., 
+``The Pfaffian Transformation'', March 9, 2007
+{\bf ``apps.carleton.edu/curricular/math/assets/pfaffian\_comps\_06\_07.pdf''}
+\bibitem{2} {\bf ``http://en.wikipedia.org/wiki/Pfaffian''}
+\end{thebibliography}
+\end{document}



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Date: Sat, 12 Jan 2008 07:29:54 -0800
From: Arthur Ralfs <acralfs@shaw.ca>
Subject: Re: [Axiom-developer] mathml sums and products patch
In-reply-to: <9qy7bccdog.fsf@aquin.mat.univie.ac.at>
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Martin Rubey wrote:
> 3) it seems to me that the domains FORMULA, TEX and MATHML share a *lot* of
>    code.  Woldn't it make sense to factor that out?  Won't the precedence order
>    be the same for all output formats?  I do not know whether anybody still
>    uses SCRIPT, but that's really beside the point... 
>
> 4) Finally, in line with 3), what do you think about moving at least some of
>    the code from i-output.boot to a domain "ASCII"?  I imagine that this might
>    be easier than fixing i-output.boot...  I tried to look into it again,
>    because I really want these terrible display bugs fixed, but I fail to see
>    how it works.  For example, I do not see any precedence order set...
>
> Many thanks again,
>
> Martin
>
>
>   
I spent some time looking closely at the code in i-output.boot.  It
looks awful
to me, probably not something I want to spend my volunteer hours on.  To get
an idea of what's involved I put a write-line statement at the start of
each function
and then issued the command "1", this is what I got back:

(19) -> 1

output
mathprintWithNumber
outputTran
maprin
maprin0
maprinChk
maPrin
charybdis
keyp
charyTop
charyTopWidth
putWidth
keyp
getBindingPowerOf
getOpBindingPower
getBindingPowerOf
getOpBindingPower
putWidth
putWidth
infixArgNeedsParens
getBindingPowerOf
WIDTH
infixArgNeedsParens
getBindingPowerOf
WIDTH
opWidth
keyp
WIDTH
putWidth
APP
appInfix
keyp
getOpBindingPower
getOpBindingPower
WIDTH
appInfixArg
infixArgNeedsParens
getBindingPowerOf
appparu
subspan
superspan
applpar
APP
atom
appChar
appChar
APP
atom
stringer
appChar
WIDTH
apprpar
APP
atom
appChar
WIDTH
appconc
appChar
appInfixArg
infixArgNeedsParens
getBindingPowerOf
APP
atom
stringer
appChar
WIDTH
appconc
superspan
superspan
superspan
superspan
superspan
superspan
superspan
superspan
subspan
subspan
subspan
subspan
subspan
subspan
subspan
subspan
scylla
   (19)  1

That's 84 function calls to print out "1".  The code has almost no
documentation.
I seriously ask the questions: does anybody want to deal with this? is
it worth
spending time on?  Is this type of output anachronistic? What about
fixing up
OutputForm and using that as the default text output?

Arthur



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Arthur Ralfs <acralfs@shaw.ca> writes:

> I spent some time looking closely at the code in i-output.boot.  It
> looks awful
> to me, 

it looks awful.  Not only to you :-)

> probably not something I want to spend my volunteer hours on.  To get an idea
> of what's involved I put a write-line statement at the start of each function
> and then issued the command "1", this is what I got back:

[...]

> I seriously ask the questions: does anybody want to deal with this? is it
> worth spending time on?  

I don't think so.  Rather, I believe we should have a few commands in boot
(something like print a string, get output width, etc.), the rest should move
to spad.

> Is this type of output anachronistic? What about fixing up OutputForm and
> using that as the default text output?

No, OutputForm is (or rather, will be) the domain abstracting output.  I guess,
if we get OutputForm to be good enough for content mathml, TeX and ASCII, it
will be really good enough for most what is to come.

What we need is a domain ASCII, that converts OutputForm to ASCII art.  I don't
think that this can be too difficult:  you could start with the mathml domain
and gradually replace mathml markup by ascii art.

Martin




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Hi,
Just checking if this is the right email address for you.  Will try to send you a pic later.  hope the day went as well as can be expected considering...love you and miss you so very much.
J.


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<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:10pt"><DIV>Hi,</DIV>
<DIV>Just checking if this is the right email address for you.&nbsp; Will try to send you a pic later.&nbsp; hope the day went as well as can be expected considering...love you and miss you so very much.</DIV>
<DIV>J.</DIV></div><br>
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Subject: Re: [fricas-devel] Re: [Axiom-developer] mathml sums and
	products	patch
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On 12 Jan 2008 19:06:09 +0100
  Martin Rubey <martin.rubey@univie.ac.at> wrote:
>What we need is a domain ASCII, that converts OutputForm 
>to ASCII art.  I don't
>think that this can be too difficult:  you could start 
>with the mathml domain
>and gradually replace mathml markup by ascii art.
>
>Martin

I think we should aim for a UNICODE domain, which would 
include ASCII art as part of unicode art and at the same 
time allows all types of fonts for future expansion, like 
math symbols and CJK (that's Chinese, Japanese and 
Korean).  

William



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These files compare Axiom output to Abramowitz and Stegun, 
Handbook of Mathematical Functions (1965). So far it has
uncovered two bugs. It is expected that these numeric results
may be unstable in the trailing digits on different architectures 
(e.g. 64bit machines).

=======================================================================
diff --git a/changelog b/changelog
index 693db6a..45e1388 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,15 @@
+20080112 tpd src/input/besselk.input complex gamma A&S reference regression
+20080110 tpd src/input/Makefile add new regression test files
+20080110 tpd src/input/asinhatanh.input create A&S reference regression
+20080110 tpd src/input/sinhcosh.input create A&S reference regression
+20080110 tpd src/input/tanhcoth.input create A&S reference regression
+20080110 tpd src/input/asinatan.input create A&S reference regression
+20080110 tpd src/input/seccsc.input create A&S reference regression
+20080110 tpd src/input/tancot.input create A&S reference regression
+20080109 tpd src/input/log.input create A&S reference regression
+20080109 tpd src/input/sincos.input create A&S reference regression
+20080108 tpd src/input/exp.input create A&S reference regression
+20080108 tpd src/input/besselk.input update bibliography
 20080107 tpd src/input/Makefile besselk.input regression test added
 20080107 tpd src/input/besselk.input regression test added
 20080107 tpd src/input/gamma.input missing file added
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index d727a41..07fca93 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -284,7 +284,8 @@ OUTS= ffrac.output     \
 REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     allfact.regress   antoine.regress \
     arith.regress     array1.regress   array2.regress \
-    arrows.regress    assign.regress   atansqrt.regress \
+    arrows.regress    asinatan.regress asinhatanh.regress \
+    assign.regress    atansqrt.regress \
     asec.regress      bags.regress     bbtree.regress   besselk.regress \
     binary.regress    bop.regress      bstree.regress   bouquet.regress \
     bug100.regress    bug101.regress \
@@ -306,6 +307,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     elt.regress       eq.regress       eqtbl.regress    equation2.regress \
     equation.regress  evalex.regress   eval.regress     exdiff.regress \
     exint.regress     exit.regress     exlap.regress    exlimit.regress \
+    exp.regress \
     expexpan.regress  explim.regress   expr1.regress    exprode.regress \
     expr.regress      exseries.regress exsum.regress    exprpoly.regress \
     farray.regress    ffdemo.regress   fferr.regress    ffx72.regress \
@@ -329,6 +331,7 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     laplace.regress   leg.regress      limit.regress    lindep.regress \
     list.regress      lode.regress     lodesys.regress  lodo1.regress \
     lodo2.regress     lodo3.regress    lodof.regress    lodo.regress \
+    log.regress \
     lpoly.regress     lupfact.regress  lword.regress    macbug.regress \
     macros.regress    magma.regress    mapleok.regress  mappkg1.regress \
     matbug.regress    mathml.regress   \
@@ -350,14 +353,16 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     realclos.regress  reclos.regress   repa6.regress    robidoux.regress \
     roman.regress     roots.regress    ruleset.regress  rules.regress \
     schaum1.regress \
-    scherk.regress    scope.regress    segbind.regress  seg.regress \
+    scherk.regress    scope.regress    seccsc.regress \
+    segbind.regress   seg.regress \
     series2.regress   series.regress   sersolve.regress set.regress \
+    sincos.regress    sinhcosh.regress \
     sincosex.regress  sint.regress     skew.regress     slowint.regress \
     solvetra.regress  space3.regress   sqmatrix.regress sregset.regress \
     stbl.regress      stream2.regress  stream.regress   streams.regress \
     string.regress    strtbl.regress   summation.regress \
-    symbol.regress    t111293.regress \
-    table.regress     tanatan.regress  textfile.regress torus.regress \
+    symbol.regress    t111293.regress  table.regress    tancot.regress \
+    tanhcoth.regress  tanatan.regress  textfile.regress torus.regress \
     triglim.regress   tsetcatvermeer.regress            tutchap1.regress \
     void.regress      uniseg.regress
 
@@ -502,6 +507,7 @@ NAGLIB=${OUT}/c02aff.input   ${OUT}/c02agf.input     ${OUT}/c05adf.input \
 FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/alist.input    ${OUT}/allfact.input    ${OUT}/antoine.input  \
        ${OUT}/array1.input   ${OUT}/array2.input     ${OUT}/arrows.input \
+       ${OUT}/asinatan.input ${OUT}/asinhatanh.input \
        ${OUT}/assign.input   ${OUT}/atansqrt.input   ${OUT}/axiom.input \
        ${OUT}/asec.input \
        ${OUT}/bags.input     ${OUT}/bbtree.input     ${OUT}/besselk.input \
@@ -539,7 +545,8 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/elt.input \
        ${OUT}/eq.input       ${OUT}/eqtbl.input      ${OUT}/equation.input \
        ${OUT}/errortrap.input \
-       ${OUT}/eval.input     ${OUT}/exit.input       ${OUT}/expexpan.input \
+       ${OUT}/eval.input     ${OUT}/exit.input       ${OUT}/exp.input \
+       ${OUT}/expexpan.input \
        ${OUT}/explim.input   ${OUT}/explot2d.input   ${OUT}/explot3d.input \
        ${OUT}/expr1.input    ${OUT}/exprpoly.input \
        ${OUT}/farray.input   ${OUT}/ffdemo.input \
@@ -575,8 +582,8 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/lindep.input   ${OUT}/liss1.input      ${OUT}/liss2.input \
        ${OUT}/list.input     ${OUT}/lode.input       ${OUT}/lodesys.input \
        ${OUT}/lodo1.input    ${OUT}/lodo2.input      ${OUT}/lodof.input \
-       ${OUT}/lodo.input     ${OUT}/lodo3.input      ${OUT}/lpoly.input \
-       ${OUT}/lump.input \
+       ${OUT}/lodo.input     ${OUT}/lodo3.input      ${OUT}/log.input \
+       ${OUT}/lpoly.input    ${OUT}/lump.input \
        ${OUT}/lupfact.input  ${OUT}/lword.input      ${OUT}/macbug.input \
        ${OUT}/macros.input   ${OUT}/marcbench.input  ${OUT}/magma.input \
        ${OUT}/mapleok.input  ${OUT}/mappkg1.input    ${OUT}/matbug.input \
@@ -616,9 +623,10 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/robidoux.input ${OUT}/roman.input      ${OUT}/roots.input \
        ${OUT}/ruleset.input  ${OUT}/rules.input      ${OUT}/schaum1.input \
        ${OUT}/saddle.input \
-       ${OUT}/scherk.input   ${OUT}/scope.input \
+       ${OUT}/scherk.input   ${OUT}/scope.input      ${OUT}/seccsc.input \
        ${OUT}/segbind.input  ${OUT}/seg.input        ${OUT}/series2.input \
        ${OUT}/series.input   ${OUT}/sersolve.input   ${OUT}/set.input \
+       ${OUT}/sincos.input   ${OUT}/sinhcosh.input \
        ${OUT}/sincosex.input ${OUT}/sininv.input     ${OUT}/sinsin.input \
        ${OUT}/sinsin2.input  ${OUT}/sint.input       ${OUT}/skew.input \
        ${OUT}/slowint.input  ${OUT}/solvetra.input   ${OUT}/space3.input \
@@ -628,7 +636,8 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/string.input   ${OUT}/strtbl.input     ${OUT}/summation.input \
        ${OUT}/symbol.input \
        ${OUT}/synonym.input  ${OUT}/t111293.input \
-       ${OUT}/table.input    ${OUT}/tanatan.input    ${OUT}/test.input \
+       ${OUT}/table.input    ${OUT}/tancot.input     ${OUT}/tanhcoth.input \
+       ${OUT}/tanatan.input  ${OUT}/test.input \
        ${OUT}/tetra.input    ${OUT}/textfile.input   ${OUT}/tknot.input \
        ${OUT}/tree.input \
        ${OUT}/triglim.input  ${OUT}/tschirn.input ${OUT}/tsetcatbutcher.input \
@@ -684,7 +693,8 @@ DOCFILES= \
   ${DOC}/as-eg3.input.dvi      ${DOC}/as-eg4.input.dvi     \
   ${DOC}/as-eg5.input.dvi      ${DOC}/aseg6.as.dvi         \
   ${DOC}/as-eg6.input.dvi      ${DOC}/aseg7.as.dvi         \
-  ${DOC}/asec.input.dvi \
+  ${DOC}/asec.input.dvi        ${DOC}/asinatan.input.dvi   \
+  ${DOC}/asinhatanh.input.dvi \
   ${DOC}/assign.input.dvi      ${DOC}/atansqrt.input.dvi   \
   ${DOC}/axiom.input.dvi       ${DOC}/bags.input.dvi       \
   ${DOC}/bbtree.input.dvi      ${DOC}/besselk.input.dvi    \
@@ -773,6 +783,7 @@ DOCFILES= \
   ${DOC}/eval.input.dvi        ${DOC}/exdiff.input.dvi     \
   ${DOC}/exint.input.dvi       ${DOC}/exit.input.dvi       \
   ${DOC}/exlap.input.dvi       ${DOC}/exlimit.input.dvi    \
+  ${DOC}/exp.input.dvi         \
   ${DOC}/expexpan.input.dvi    ${DOC}/explim.input.dvi     \
   ${DOC}/explot2d.input.dvi    ${DOC}/explot3d.input.dvi   \
   ${DOC}/expr1.input.dvi       ${DOC}/expr.input.dvi       \
@@ -854,6 +865,7 @@ DOCFILES= \
   ${DOC}/lodesys.input.dvi     ${DOC}/lodo1.input.dvi      \
   ${DOC}/lodo2.input.dvi       ${DOC}/lodo3.input.dvi      \
   ${DOC}/lodof.input.dvi       ${DOC}/lodo.input.dvi       \
+  ${DOC}/log.input.dvi \
   ${DOC}/loop.input.dvi        ${DOC}/lpoly.input.dvi      \
   ${DOC}/lump.input.dvi        ${DOC}/lupfact.input.dvi    \
   ${DOC}/lword.input.dvi       ${DOC}/macbug.input.dvi     \
@@ -926,10 +938,12 @@ DOCFILES= \
   ${DOC}/s21baf.input.dvi      ${DOC}/s21bbf.input.dvi     \
   ${DOC}/s21bcf.input.dvi      ${DOC}/s21bdf.input.dvi     \
   ${DOC}/saddle.input.dvi      ${DOC}/scherk.input.dvi     \
-  ${DOC}/scope.input.dvi       ${DOC}/segbind.input.dvi    \
+  ${DOC}/scope.input.dvi       ${DOC}/seccsc.input.dvi     \
+  ${DOC}/segbind.input.dvi    \
   ${DOC}/seg.input.dvi         ${DOC}/series2.input.dvi    \
   ${DOC}/series.input.dvi      ${DOC}/sersolve.input.dvi   \
-  ${DOC}/set.input.dvi         ${DOC}/sincosex.input.dvi   \
+  ${DOC}/set.input.dvi         ${DOC}/sincos.input.dvi     \
+  ${DOC}/sinhcosh.input.dvi    ${DOC}/sincosex.input.dvi   \
   ${DOC}/sininv.input.dvi      ${DOC}/sinsin2.input.dvi    \
   ${DOC}/sinsin.input.dvi      ${DOC}/sint.input.dvi       \
   ${DOC}/skew.input.dvi        ${DOC}/slowint.input.dvi    \
@@ -943,6 +957,7 @@ DOCFILES= \
   ${DOC}/symbol.input.dvi      ${DOC}/synonym2.input.dvi   \
   ${DOC}/synonym.input.dvi     ${DOC}/t111293.input.dvi    \
   ${DOC}/table.input.dvi       ${DOC}/tanatan.input.dvi    \
+  ${DOC}/tancot.input.dvi      ${DOC}/tanhcoth.input.dvi   \
   ${DOC}/test.input.dvi        ${DOC}/tetra.input.dvi      \
   ${DOC}/textfile.input.dvi    ${DOC}/tknot.input.dvi      \
   ${DOC}/torus.input.dvi       ${DOC}/tree.input.dvi       \
diff --git a/src/input/asinatan.input.pamphlet b/src/input/asinatan.input.pamphlet
new file mode 100644
index 0000000..a468c88
--- /dev/null
+++ b/src/input/asinatan.input.pamphlet
@@ -0,0 +1,456 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input asinatan.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's asin and atan functions with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of asin(x), ranging from 0.01 to 1.00. The
+second column is the reference value of tan from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp203-212
+<<*>>=
+)spool asinatan.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.01,0.010000166674,asin(0.01),asin(0.01)-0.010000166674],_
+[0.02,0.020001333573,asin(0.02),asin(0.02)-0.020001333573],_
+[0.03,0.030004501823,asin(0.03),asin(0.03)-0.030004501823],_
+[0.04,0.040010674354,asin(0.04),asin(0.04)-0.040010674354],_
+[0.05,0.050020856806,asin(0.05),asin(0.05)-0.050020856806],_
+[0.06,0.060036058445,asin(0.06),asin(0.06)-0.060036058445],_
+[0.07,0.070057293088,asin(0.07),asin(0.07)-0.070057293088],_
+[0.08,0.080085580034,asin(0.08),asin(0.08)-0.080085580034],_
+[0.09,0.090121945015,asin(0.09),asin(0.09)-0.090121945015],_
+[0.10,0.100167421162,asin(0.10),asin(0.10)-0.100167421162],_
+[0.11,0.110223049988,asin(0.11),asin(0.11)-0.110223049988],_
+[0.12,0.120289882395,asin(0.12),asin(0.12)-0.120289882395],_
+[0.13,0.130368979703,asin(0.13),asin(0.13)-0.130368979703],_
+[0.14,0.140461414710,asin(0.14),asin(0.14)-0.140461414710],_
+[0.15,0.150568272777,asin(0.15),asin(0.15)-0.150568272777],_
+[0.16,0.160690652952,asin(0.16),asin(0.16)-0.160690652952],_
+[0.17,0.170829669129,asin(0.17),asin(0.17)-0.170829669129],_
+[0.18,0.180986451247,asin(0.18),asin(0.18)-0.180986451247],_
+[0.19,0.191162146531,asin(0.19),asin(0.19)-0.191162146531],_
+[0.20,0.201357920790,asin(0.20),asin(0.20)-0.201357920790],_
+[0.21,0.211574959758,asin(0.21),asin(0.21)-0.211574959758],_
+[0.22,0.221814470497,asin(0.22),asin(0.22)-0.221814470497],_
+[0.23,0.232077682863,asin(0.23),asin(0.23)-0.232077682863],_
+[0.24,0.242365851039,asin(0.24),asin(0.24)-0.242365851039],_
+[0.25,0.252680255142,asin(0.25),asin(0.25)-0.252680255142],_
+[0.26,0.263022202908,asin(0.26),asin(0.26)-0.263022202908],_
+[0.27,0.273393031467,asin(0.27),asin(0.27)-0.273393031467],_
+[0.28,0.283794109208,asin(0.28),asin(0.28)-0.283794109208],_
+[0.29,0.294226837749,asin(0.29),asin(0.29)-0.294226837749],_
+[0.30,0.304692654015,asin(0.30),asin(0.30)-0.304692654015],_
+[0.31,0.315193032441,asin(0.31),asin(0.31)-0.315193032441],_
+[0.32,0.325729487295,asin(0.32),asin(0.32)-0.325729487295],_
+[0.33,0.336303575154,asin(0.33),asin(0.33)-0.336303575154],_
+[0.34,0.346916897527,asin(0.34),asin(0.34)-0.346916897527],_
+[0.35,0.357571103646,asin(0.35),asin(0.35)-0.357571103646],_
+[0.36,0.368267893437,asin(0.36),asin(0.36)-0.368267893437],_
+[0.37,0.379009020696,asin(0.37),asin(0.37)-0.379009020696],_
+[0.38,0.389796296474,asin(0.38),asin(0.38)-0.389796296474],_
+[0.39,0.400631592701,asin(0.39),asin(0.39)-0.400631592701],_
+[0.40,0.411516846067,asin(0.40),asin(0.40)-0.411516846067],_
+[0.41,0.422454062187,asin(0.41),asin(0.41)-0.422454062187],_
+[0.42,0.433445320070,asin(0.42),asin(0.42)-0.433445320070],_
+[0.43,0.444492776936,asin(0.43),asin(0.43)-0.444492776936],_
+[0.44,0.455598673396,asin(0.44),asin(0.44)-0.455598673396],_
+[0.45,0.466765339047,asin(0.45),asin(0.45)-0.466765339047],_
+[0.46,0.477995198519,asin(0.46),asin(0.46)-0.477995198519],_
+[0.47,0.489290778014,asin(0.47),asin(0.47)-0.489290778014],_
+[0.48,0.500654712405,asin(0.48),asin(0.48)-0.500654712405],_
+[0.49,0.512089752934,asin(0.49),asin(0.49)-0.512089752934],_
+[0.50,0.523598775598,asin(0.50),asin(0.50)-0.523598775598],_
+[0.51,0.535184790276,asin(0.51),asin(0.51)-0.535184790276],_
+[0.52,0.546850950696,asin(0.52),asin(0.52)-0.546850950696],_
+[0.53,0.558600565343,asin(0.53),asin(0.53)-0.558600565343],_
+[0.54,0.570437109400,asin(0.54),asin(0.54)-0.570437109400],_
+[0.55,0.582364237869,asin(0.55),asin(0.55)-0.582364237869],_
+[0.56,0.594385800001,asin(0.56),asin(0.56)-0.594385800001],_
+[0.57,0.606505855213,asin(0.57),asin(0.57)-0.606505855213],_
+[0.58,0.618728690672,asin(0.58),asin(0.58)-0.618728690672],_
+[0.59,0.631058840778,asin(0.59),asin(0.59)-0.631058840778],_
+[0.60,0.643501108793,asin(0.60),asin(0.60)-0.643501108793],_
+[0.61,0.656060590925,asin(0.61),asin(0.61)-0.656060590925],_
+[0.62,0.668742703202,asin(0.62),asin(0.62)-0.668742703202],_
+[0.63,0.681553211563,asin(0.63),asin(0.63)-0.681553211563],_
+[0.64,0.694498265627,asin(0.64),asin(0.64)-0.694498265627],_
+[0.65,0.707584436725,asin(0.65),asin(0.65)-0.707584436725],_
+[0.66,0.720818760870,asin(0.66),asin(0.66)-0.720818760870],_
+[0.67,0.734208787453,asin(0.67),asin(0.67)-0.734208787453],_
+[0.68,0.747762634660,asin(0.68),asin(0.68)-0.747762634660],_
+[0.69,0.761489052748,asin(0.69),asin(0.69)-0.761489052748],_
+[0.70,0.775397496611,asin(0.70),asin(0.70)-0.775397496611],_
+[0.71,0.789498209346,asin(0.71),asin(0.71)-0.789498209346],_
+[0.72,0.803802318933,asin(0.72),asin(0.72)-0.803802318933],_
+[0.73,0.818321950632,asin(0.73),asin(0.73)-0.818321950632],_
+[0.74,0.833070358342,asin(0.74),asin(0.74)-0.833070358342],_
+[0.75,0.848062078981,asin(0.75),asin(0.75)-0.848062078981],_
+[0.76,0.863313115016,asin(0.76),asin(0.76)-0.863313115016],_
+[0.77,0.878841151669,asin(0.77),asin(0.77)-0.878841151669],_
+[0.78,0.894665817234,asin(0.78),asin(0.78)-0.894665817234],_
+[0.79,0.910808997407,asin(0.79),asin(0.79)-0.910808997407],_
+[0.80,0.927295218002,asin(0.80),asin(0.80)-0.927295218002],_
+[0.81,0.944152115154,asin(0.81),asin(0.81)-0.944152115154],_
+[0.82,0.961411018764,asin(0.82),asin(0.82)-0.961411018764],_
+[0.83,0.979107684368,asin(0.83),asin(0.83)-0.979107684368],_
+[0.84,0.997283222372,asin(0.84),asin(0.84)-0.997283222372],_
+[0.85,1.015985293815,asin(0.85),asin(0.85)-1.015985293815],_
+[0.86,1.035269672481,asin(0.86),asin(0.86)-1.035269672481],_
+[0.87,1.055202320549,asin(0.87),asin(0.87)-1.055202320549],_
+[0.88,1.075862200454,asin(0.88),asin(0.88)-1.075862200454],_
+[0.89,1.097345169523,asin(0.89),asin(0.89)-1.097345169523],_
+[0.90,1.119769514999,asin(0.90),asin(0.90)-1.119769514999],_
+[0.91,1.143284061850,asin(0.91),asin(0.91)-1.143284061850],_
+[0.92,1.168080485214,asin(0.92),asin(0.92)-1.168080485214],_
+[0.93,1.194412844477,asin(0.93),asin(0.93)-1.194412844477],_
+[0.94,1.222630305522,asin(0.94),asin(0.94)-1.222630305522],_
+[0.95,1.253235897503,asin(0.95),asin(0.95)-1.253235897503],_
+[0.96,1.287002217587,asin(0.96),asin(0.96)-1.287002217587],_
+[0.97,1.325230809280,asin(0.97),asin(0.97)-1.325230809280],_
+[0.98,1.370461484472,asin(0.98),asin(0.98)-1.370461484472],_
+[0.99,1.429256853470,asin(0.99),asin(0.99)-1.429256853470],_
+[1.00,1.570796326795,asin(1.00),asin(1.00)-1.570796326795]]
+--R 
+--R
+--R   (1)
+--R   [[0.01,0.0100001666 74,0.0100001666 7416711312 6,0.167113126 E -12],
+--R    [0.02,0.0200013335 73,0.0200013335 7339049175 1,0.390491751 E -12],
+--R    [0.03,0.0300045018 23,0.0300045018 2347693769,0.47693769 E -12],
+--R    [0.04,0.0400106743 54,0.0400106743 5398892622 1,- 0.1107378 E -13],
+--R    [0.05,0.0500208568 06,0.0500208568 0577001466 3,- 0.229985337 E -12],
+--R    [0.06,0.0600360584 45,0.0600360584 452784225,0.2784225 E -12],
+--R    [0.07,0.0700572930 88,0.0700572930 8805025329 9,0.502533 E -13],
+--R    [0.08,0.0800855800 34,0.0800855800 3365901374 8,- 0.34098625 E -12],
+--R    [0.09,0.0901219450 15,0.0901219450 1459525581 5,- 0.40474419 E -12],
+--R    [0.1,0.1001674211 62,0.1001674211 6155979635,- 0.44020365 E -12],
+--R    [0.11,0.1102230499 88,0.1102230499 8774663318,- 0.25336682 E -12],
+--R    [0.12,0.1202898823 95,0.1202898823 9478807132,- 0.21192868 E -12],
+--R    [0.13,0.1303689797 03,0.1303689797 0314551294,0.14551294 E -12],
+--R    [0.14,0.1404614147 1,0.1404614147 0985580027,- 0.14419973 E -12],
+--R    [0.15,0.1505682727 77,0.1505682727 7668602642,- 0.31397358 E -12],
+--R    [0.16,0.1606906529 52,0.1606906529 5191060036,- 0.8939964 E -13],
+--R    [0.17,0.1708296691 29,0.1708296691 2910450072,0.1045007 E -12],
+--R    [0.18,0.1809864512 47,0.1809864512 4654770725,- 0.45229275 E -12],
+--R    [0.19,0.1911621465 31,0.1911621465 3105960346,0.5960346 E -13],
+--R    [0.2,0.2013579207 9,0.2013579207 9033079146,0.33079146 E -12],
+--R    [0.21,0.2115749597 58,0.2115749597 5809561381,0.9561381 E -13],
+--R    [0.22,0.2218144704 97,0.2218144704 9679441094,- 0.20558906 E -12],
+--R    [0.23,0.2320776828 63,0.2320776828 6271317742,- 0.28682258 E -12],
+--R    [0.24,0.2423658510 39,0.2423658510 3896323277,- 0.3676723 E -13],
+--R    [0.25,0.2526802551 42,0.2526802551 4207865349,0.7865349 E -13],
+--R    [0.26,0.2630222029 08,0.2630222029 0846889582,0.46889582 E -12],
+--R    [0.27,0.2733930314 67,0.2733930314 6747322258,0.47322258 E -12],
+--R    [0.28,0.2837941092 08,0.2837941092 0832784562,0.32784563 E -12],
+--R    [0.29,0.2942268377 49,0.2942268377 4898247936,- 0.175206 E -13],
+--R    [0.3,0.3046926540 15,0.3046926540 1539750797,0.39750797 E -12],
+--R    [0.31,0.3151930324 41,0.3151930324 4072444134,- 0.27555866 E -12],
+--R    [0.32,0.3257294872 95,0.3257294872 9463015931,- 0.36984069 E -12],
+--R    [0.33,0.3363035751 54,0.3363035751 5398034192,- 0.196581 E -13],
+--R    [0.34,0.3469168975 27,0.3469168975 2716170922,0.1617092 E -12],
+--R    [0.35,0.3575711036 46,0.3575711036 4551028671,- 0.48971329 E -12],
+--R    [0.36,0.3682678934 37,0.3682678934 3663998093,- 0.36001907 E -12],
+--R    [0.37,0.3790090206 96,0.3790090206 9595081407,- 0.4918592 E -13],
+--R    [0.38,0.3897962964 74,0.3897962964 7426056359,0.26056359 E -12],
+--R    [0.39,0.4006315927 01,0.4006315927 0137191121,0.37191121 E -12],
+--R    [0.4,0.4115168460 67,0.4115168460 6748801938,0.48801938 E -12],
+--R    [0.41,0.4224540621 87,0.4224540621 8675575657,- 0.24424343 E -12],
+--R    [0.42,0.4334453200 7,0.4334453200 6988597641,- 0.1140236 E -12],
+--R    [0.43,0.4444927769 36,0.4444927769 3581903875,- 0.1809612 E -12],
+--R    [0.44,0.4555986733 96,0.4555986733 9582336298,- 0.176637 E -12],
+--R    [0.45,0.4667653390 47,0.4667653390 4729636185,0.29636185 E -12],
+--R    [0.46,0.4779951985 19,0.4779951985 1895237921,- 0.4762079 E -13],
+--R    [0.47,0.4892907780 14,0.4892907780 1411571422,0.1157142 E -12],
+--R    [0.48,0.5006547124 05,0.5006547124 0458816211,- 0.4118379 E -12],
+--R    [0.49,0.5120897529 34,0.5120897529 3414777137,0.1477714 E -12],
+--R    [0.5,0.5235987755 98,0.5235987755 9829887308,0.2988731 E -12],
+--R    [0.51,0.5351847902 76,0.5351847902 7559984754,- 0.4001525 E -12],
+--R    [0.52,0.5468509506 96,0.5468509506 9594410879,- 0.558912 E -13],
+--R    [0.53,0.5586005653 43,0.5586005653 4280071945,- 0.1992805 E -12],
+--R    [0.54,0.5704371094,0.5704371093 9992190735,- 0.7809265 E -13],
+--R    [0.55,0.5823642378 69,0.5823642378 6874344183,- 0.2565582 E -12],
+--R    [0.56,0.5943858000 01,0.5943858000 0106220388,0.6220388 E -13],
+--R    [0.57,0.6065058552 13,0.6065058552 1308694718,0.8694718 E -13],
+--R    [0.58,0.6187286906 72,0.6187286906 7225109051,0.2510905 E -12],
+--R    [0.59,0.6310588407 78,0.6310588407 7802138008,0.213801 E -13],
+--R    [0.6,0.6435011087 93,0.6435011087 932843868,0.2843868 E -12],
+--R    [0.61,0.6560605909 25,0.6560605909 2492264451,- 0.7735549 E -13],
+--R    [0.62,0.6687427032 02,0.6687427032 0237175414,0.3717541 E -12],
+--R    [0.63,0.6815532115 63,0.6815532115 6311692392,0.1169239 E -12],
+--R    [0.64,0.6944982656 27,0.6944982656 2655598784,- 0.4440122 E -12],
+--R    [0.65,0.7075844367 25,0.7075844367 2535557545,0.3555755 E -12],
+--R    [0.66,0.7208187608 7,0.7208187608 7008963003,0.8963003 E -13],
+--R    [0.67,0.7342087874 53,0.7342087874 533589103,0.3589103 E -12],
+--R    [0.68,0.7477626346 6,0.7477626346 5992058046,- 0.7941954 E -13],
+--R    [0.69,0.7614890527 48,0.7614890527 4763322831,- 0.3667717 E -12],
+--R    [0.7,0.7753974966 11,0.7753974966 1075306374,- 0.2469363 E -12],
+--R    [0.71,0.7894982093 46,0.7894982093 4617197301,0.171973 E -12],
+--R    [0.72,0.8038023189 33,0.8038023189 3303003221,0.300322 E -13],
+--R    [0.73,0.8183219506 32,0.8183219506 3155981721,- 0.4401828 E -12],
+--R    [0.74,0.8330703583 42,0.8330703583 4164781149,- 0.3521885 E -12],
+--R    [0.75,0.8480620789 81,0.8480620789 8148100805,0.48100805 E -12],
+--R    [0.76,0.8633131150 16,0.8633131150 1555364761,- 0.4463524 E -12],
+--R    [0.77,0.8788411516 69,0.8788411516 6857970602,- 0.420294 E -12],
+--R    [0.78,0.8946658172 34,0.8946658172 3423520948,0.2352095 E -12],
+--R    [0.79,0.9108089974 07,0.9108089974 0739818925,0.3981893 E -12],
+--R    [0.8,0.9272952180 02,0.9272952180 0161223243,- 0.3877676 E -12],
+--R    [0.81,0.9441521151 54,0.9441521151 5415595048,0.1559505 E -12],
+--R    [0.82,0.9614110187 64,0.9614110187 6410160329,0.1016033 E -12],
+--R    [0.83,0.9791076843 68,0.9791076843 6835265479,0.3526548 E -12],
+--R    [0.84,0.9972832223 72,0.9972832223 7179986605,- 0.2001339 E -12],
+--R    [0.85,1.0159852938 15,1.0159852938 148251312,- 0.1748688 E -12],
+--R    [0.86,1.0352696724 81,1.0352696724 805087973,- 0.4912027 E -12],
+--R    [0.87,1.0552023205 49,1.0552023205 488061213,- 0.1938787 E -12],
+--R    [0.88,1.0758622004 54,1.0758622004 540010964,0.10964 E -14],
+--R    [0.89,1.0973451695 23,1.0973451695 228303977,- 0.1696023 E -12],
+--R    [0.9,1.1197695149 99,1.1197695149 986341867,- 0.3658133 E -12],
+--R    [0.91,1.1432840618 5,1.1432840618 500270323,0.270323 E -13],
+--R    [0.92,1.1680804852 14,1.1680804852 142350363,0.2350363 E -12],
+--R    [0.93,1.1944128444 77,1.1944128444 771683741,0.1683741 E -12],
+--R    [0.94,1.2226303055 22,1.2226303055 21935738,- 0.64262 E -13],
+--R    [0.95,1.2532358975 03,1.2532358975 033752587,0.3752587 E -12],
+--R    [0.96,1.2870022175 87,1.2870022175 865687736,- 0.4312264 E -12],
+--R    [0.97,1.3252308092 8,1.3252308092 796046112,- 0.3953888 E -12],
+--R    [0.98,1.3704614844 72,1.3704614844 717770265,- 0.2229735 E -12],
+--R    [0.99,1.4292568534 7,1.4292568534 704694005,0.4694005 E -12],
+--R    [1.0,1.5707963267 95,1.5707963267 948966192,- 0.103381 E -12]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of atan(x), ranging from 0.01 to 1.00. The
+second column is the reference value of tan from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp203-212
+<<*>>=
+
+--S 2 of 2
+[[0.01,0.009999666687,atan(0.01),atan(0.01)-0.009999666687],_
+[0.02,0.019997333973,atan(0.02),atan(0.02)-0.019997333973],_
+[0.03,0.029991004857,atan(0.03),atan(0.03)-0.029991004857],_
+[0.04,0.039978687123,atan(0.04),atan(0.04)-0.039978687123],_
+[0.05,0.049958395722,atan(0.05),atan(0.05)-0.049958395722],_
+[0.06,0.059928155121,atan(0.06),atan(0.06)-0.059928155121],_
+[0.07,0.069886001635,atan(0.07),atan(0.07)-0.069886001635],_
+[0.08,0.079829985712,atan(0.08),atan(0.08)-0.079829985712],_
+[0.09,0.089758174190,atan(0.09),atan(0.09)-0.089758174190],_
+[0.10,0.099668652491,atan(0.10),atan(0.10)-0.099668652491],_
+[0.11,0.109559526774,atan(0.11),atan(0.11)-0.109559526774],_
+[0.12,0.119428926018,atan(0.12),atan(0.12)-0.119428926018],_
+[0.13,0.129275004048,atan(0.13),atan(0.13)-0.129275004048],_
+[0.14,0.139095941482,atan(0.14),atan(0.14)-0.139095941482],_
+[0.15,0.148889947609,atan(0.15),atan(0.15)-0.148889947609],_
+[0.16,0.158655262186,atan(0.16),atan(0.16)-0.158655262186],_
+[0.17,0.168390157148,atan(0.17),atan(0.17)-0.168390157148],_
+[0.18,0.178092938231,atan(0.18),atan(0.18)-0.178092938231],_
+[0.19,0.187761946514,atan(0.19),atan(0.19)-0.187761946514],_
+[0.20,0.197395559850,atan(0.20),atan(0.20)-0.197395559850],_
+[0.21,0.206992194220,atan(0.21),atan(0.21)-0.206992194220],_
+[0.22,0.216550304976,atan(0.22),atan(0.22)-0.216550304976],_
+[0.23,0.226068387994,atan(0.23),atan(0.23)-0.226068387994],_
+[0.24,0.235544980721,atan(0.24),atan(0.24)-0.235544980721],_
+[0.25,0.244978663127,atan(0.25),atan(0.25)-0.244978663127],_
+[0.26,0.254368058553,atan(0.26),atan(0.26)-0.254368058553],_
+[0.27,0.263711834462,atan(0.27),atan(0.27)-0.263711834462],_
+[0.28,0.273008703087,atan(0.28),atan(0.28)-0.273008703087],_
+[0.29,0.282257421981,atan(0.29),atan(0.29)-0.282257421981],_
+[0.30,0.291456794478,atan(0.30),atan(0.30)-0.291456794478],_
+[0.31,0.300605670042,atan(0.31),atan(0.31)-0.300605670042],_
+[0.32,0.309702944542,atan(0.32),atan(0.32)-0.309702944542],_
+[0.33,0.318747560421,atan(0.33),atan(0.33)-0.318747560421],_
+[0.34,0.327738506781,atan(0.34),atan(0.34)-0.327738506781],_
+[0.35,0.336674819387,atan(0.35),atan(0.35)-0.336674819387],_
+[0.36,0.345555580582,atan(0.36),atan(0.36)-0.345555580582],_
+[0.37,0.354379919123,atan(0.37),atan(0.37)-0.354379919123],_
+[0.38,0.363147009946,atan(0.38),atan(0.38)-0.363147009946],_
+[0.39,0.371856073849,atan(0.39),atan(0.39)-0.371856073849],_
+[0.40,0.380506377112,atan(0.40),atan(0.40)-0.380506377112],_
+[0.41,0.389097231055,atan(0.41),atan(0.41)-0.389097231055],_
+[0.42,0.397627991522,atan(0.42),atan(0.42)-0.397627991522],_
+[0.43,0.406098058318,atan(0.43),atan(0.43)-0.406098058318],_
+[0.44,0.414506874585,atan(0.44),atan(0.44)-0.414506874585],_
+[0.45,0.422853926133,atan(0.45),atan(0.45)-0.422853926133],_
+[0.46,0.431138740719,atan(0.46),atan(0.46)-0.431138740719],_
+[0.47,0.439360887285,atan(0.47),atan(0.47)-0.439360887285],_
+[0.48,0.447519975157,atan(0.48),atan(0.48)-0.447519975157],_
+[0.49,0.455615653211,atan(0.49),atan(0.49)-0.455615653211],_
+[0.50,0.463647609001,atan(0.50),atan(0.50)-0.463647609001],_
+[0.51,0.471615567862,atan(0.51),atan(0.51)-0.471615567862],_
+[0.52,0.479519291993,atan(0.52),atan(0.52)-0.479519291993],_
+[0.53,0.487358579505,atan(0.53),atan(0.53)-0.487358579505],_
+[0.54,0.495133263468,atan(0.54),atan(0.54)-0.495133263468],_
+[0.55,0.502843210928,atan(0.55),atan(0.55)-0.502843210928],_
+[0.56,0.510488321917,atan(0.56),atan(0.56)-0.510488321917],_
+[0.57,0.518068528457,atan(0.57),atan(0.57)-0.518068528457],_
+[0.58,0.525583793552,atan(0.58),atan(0.58)-0.525583793552],_
+[0.59,0.533034110177,atan(0.59),atan(0.59)-0.533034110177],_
+[0.60,0.540419500271,atan(0.60),atan(0.60)-0.540419500271],_
+[0.61,0.547740013716,atan(0.61),atan(0.61)-0.547740013716],_
+[0.62,0.554995727339,atan(0.62),atan(0.62)-0.554995727339],_
+[0.63,0.562186743900,atan(0.63),atan(0.63)-0.562186743900],_
+[0.64,0.569313191101,atan(0.64),atan(0.64)-0.569313191101],_
+[0.65,0.576375220591,atan(0.65),atan(0.65)-0.576375220591],_
+[0.66,0.583373006994,atan(0.66),atan(0.66)-0.583373006994],_
+[0.67,0.590306746935,atan(0.67),atan(0.67)-0.590306746935],_
+[0.68,0.597176658093,atan(0.68),atan(0.68)-0.597176658093],_
+[0.69,0.603982978253,atan(0.69),atan(0.69)-0.603982978253],_
+[0.70,0.610725964389,atan(0.70),atan(0.70)-0.610725964389],_
+[0.71,0.617405891752,atan(0.71),atan(0.71)-0.617405891752],_
+[0.72,0.624023052977,atan(0.72),atan(0.72)-0.624023052977],_
+[0.73,0.630577757215,atan(0.73),atan(0.73)-0.630577757215],_
+[0.74,0.637070329276,atan(0.74),atan(0.74)-0.637070329276],_
+[0.75,0.643501108793,atan(0.75),atan(0.75)-0.643501108793],_
+[0.76,0.649870449412,atan(0.76),atan(0.76)-0.649870449412],_
+[0.77,0.656178717991,atan(0.77),atan(0.77)-0.656178717991],_
+[0.78,0.662426293833,atan(0.78),atan(0.78)-0.662426293833],_
+[0.79,0.668613567928,atan(0.79),atan(0.79)-0.668613567928],_
+[0.80,0.674740942224,atan(0.80),atan(0.80)-0.674740942224],_
+[0.81,0.680808828916,atan(0.81),atan(0.81)-0.680808828916],_
+[0.82,0.686817649759,atan(0.82),atan(0.82)-0.686817649759],_
+[0.83,0.692767835397,atan(0.83),atan(0.83)-0.692767835397],_
+[0.84,0.698659824721,atan(0.84),atan(0.84)-0.698659824721],_
+[0.85,0.704494064242,atan(0.85),atan(0.85)-0.704494064242],_
+[0.86,0.710271007487,atan(0.86),atan(0.86)-0.710271007487],_
+[0.87,0.715991114416,atan(0.87),atan(0.87)-0.715991114416],_
+[0.88,0.721654850865,atan(0.88),atan(0.88)-0.721654850865],_
+[0.89,0.727262687997,atan(0.89),atan(0.89)-0.727262687997],_
+[0.90,0.732815101787,atan(0.90),atan(0.90)-0.732815101787],_
+[0.91,0.738312572517,atan(0.91),atan(0.91)-0.738312572517],_
+[0.92,0.743755584299,atan(0.92),atan(0.92)-0.743755584299],_
+[0.93,0.749144624606,atan(0.93),atan(0.93)-0.749144624606],_
+[0.94,0.754480183834,atan(0.94),atan(0.94)-0.754480183834],_
+[0.95,0.759762754876,atan(0.95),atan(0.95)-0.759762754876],_
+[0.96,0.764992832711,atan(0.96),atan(0.96)-0.764992832711],_
+[0.97,0.770170914020,atan(0.97),atan(0.97)-0.770170914020],_
+[0.98,0.775297496812,atan(0.98),atan(0.98)-0.775297496812],_
+[0.99,0.780373080067,atan(0.99),atan(0.99)-0.780373080067],_
+[1.00,0.785398163397,atan(1.00),atan(1.00)-0.785398163397]]
+--R 
+--R
+--R   (2)
+--R   [[0.01,0.0099996666 87,0.0099996666 8666523820 63,- 0.334761794 E -12],
+--R    [0.02,0.0199973339 73,0.0199973339 7315053306 1,0.150533061 E -12],
+--R    [0.03,0.0299910048 57,0.0299910048 5687789967 7,- 0.122100323 E -12],
+--R    [0.04,0.0399786871 23,0.0399786871 2329004141 4,0.290041414 E -12],
+--R    [0.05,0.0499583957 22,0.0499583957 2194276141,- 0.5723859 E -13],
+--R    [0.06,0.0599281551 21,0.0599281551 2120788443 2,0.20788443 E -12],
+--R    [0.07,0.0698860016 35,0.0698860016 3464249929 5,- 0.35750071 E -12],
+--R    [0.08,0.0798299857 12,0.0798299857 1223731589 3,0.23731589 E -12],
+--R    [0.09,0.0897581741 9,0.0897581741 8995052315,- 0.4947685 E -13],
+--R    [0.1,0.0996686524 91,0.0996686524 9116202737 9,0.16202738 E -12],
+--R    [0.11,0.1095595267 74,0.1095595267 7394434487,- 0.5565513 E -13],
+--R    [0.12,0.1194289260 18,0.1194289260 1833845181,0.33845181 E -12],
+--R    [0.13,0.1292750040 48,0.1292750040 4814305472,0.14305472 E -12],
+--R    [0.14,0.1390959414 82,0.1390959414 820713243,0.713243 E -13],
+--R    [0.15,0.1488899476 09,0.1488899476 0949725059,0.49725059 E -12],
+--R    [0.16,0.1586552621 86,0.1586552621 8640140386,0.40140386 E -12],
+--R    [0.17,0.1683901571 48,0.1683901571 4752989727,- 0.47010272 E -12],
+--R    [0.18,0.1780929382 31,0.1780929382 3119754967,0.19754967 E -12],
+--R    [0.19,0.1877619465 14,0.1877619465 135934152,- 0.4065848 E -12],
+--R    [0.2,0.1973955598 5,0.1973955598 4988075837,- 0.11924163 E -12],
+--R    [0.21,0.2069921942 2,0.2069921942 198210249,- 0.1789751 E -12],
+--R    [0.22,0.2165503049 76,0.2165503049 7608927648,0.8927648 E -13],
+--R    [0.23,0.2260683879 94,0.2260683879 9388390584,- 0.11609416 E -12],
+--R    [0.24,0.2355449807 21,0.2355449807 2086334143,- 0.13665857 E -12],
+--R    [0.25,0.2449786631 27,0.2449786631 2686415417,- 0.13584583 E -12],
+--R    [0.26,0.2543680585 53,0.2543680585 5326593143,0.26593143 E -12],
+--R    [0.27,0.2637118344 62,0.2637118344 6226612016,0.26612016 E -12],
+--R    [0.28,0.2730087030 87,0.2730087030 8671060295,- 0.28939705 E -12],
+--R    [0.29,0.2822574219 81,0.2822574219 8149112127,0.49112127 E -12],
+--R    [0.3,0.2914567944 78,0.2914567944 77867092,- 0.132908 E -12],
+--R    [0.31,0.3006056700 42,0.3006056700 4239540423,0.39540423 E -12],
+--R    [0.32,0.3097029445 42,0.3097029445 4245619992,0.45619992 E -12],
+--R    [0.33,0.3187475604 21,0.3187475604 2064443712,- 0.35556288 E -12],
+--R    [0.34,0.3277385067 81,0.3277385067 80555446,- 0.444554 E -12],
+--R    [0.35,0.3366748193 87,0.3366748193 867271814,- 0.2728186 E -12],
+--R    [0.36,0.3455555805 82,0.3455555805 8171213686,- 0.28786314 E -12],
+--R    [0.37,0.3543799191 23,0.3543799191 2343780983,0.43780983 E -12],
+--R    [0.38,0.3631470099 46,0.3631470099 4617628972,0.1762897 E -12],
+--R    [0.39,0.3718560738 49,0.3718560738 485812575,- 0.4187425 E -12],
+--R    [0.4,0.3805063771 12,0.3805063771 123648863,0.3648863 E -12],
+--R    [0.41,0.3890972310 55,0.3890972310 5527841924,0.27841924 E -12],
+--R    [0.42,0.3976279915 22,0.3976279915 22129314,0.129314 E -12],
+--R    [0.43,0.4060980583 18,0.4060980583 1761564783,- 0.38435217 E -12],
+--R    [0.44,0.4145068745 85,0.4145068745 8478593834,- 0.2140617 E -12],
+--R    [0.45,0.4228539261 33,0.4228539261 3294071297,- 0.5928703 E -13],
+--R    [0.46,0.4311387407 19,0.4311387407 1878218339,- 0.2178166 E -12],
+--R    [0.47,0.4393608872 85,0.4393608872 8459143742,- 0.40856258 E -12],
+--R    [0.48,0.4475199751 57,0.4475199751 5716987972,0.1698797 E -12],
+--R    [0.49,0.4556156532 11,0.4556156532 1122449214,0.2244921 E -12],
+--R    [0.5,0.4636476090 01,0.4636476090 0080611621,- 0.1938838 E -12],
+--R    [0.51,0.4716155678 62,0.4716155678 6232766012,0.32766012 E -12],
+--R    [0.52,0.4795192919 93,0.4795192919 9259616542,- 0.40383458 E -12],
+--R    [0.53,0.4873585795 05,0.4873585795 0519028312,0.1902831 E -12],
+--R    [0.54,0.4951332634 68,0.4951332634 6840412185,0.40412185 E -12],
+--R    [0.55,0.5028432109 28,0.5028432109 2786082733,- 0.1391727 E -12],
+--R    [0.56,0.5104883219 17,0.5104883219 1677576997,- 0.22423 E -12],
+--R    [0.57,0.5180685284 57,0.5180685284 56720949,- 0.279051 E -12],
+--R    [0.58,0.5255837935 52,0.5255837935 5161020277,- 0.3897972 E -12],
+--R    [0.59,0.5330341101 77,0.5330341101 7749002604,0.49002604 E -12],
+--R    [0.6,0.5404195002 71,0.5404195002 7058415544,- 0.4158446 E -12],
+--R    [0.61,0.5477400137 16,0.5477400137 1590245052,- 0.9754948 E -13],
+--R    [0.62,0.5549957273 39,0.5549957273 3858676242,- 0.4132376 E -12],
+--R    [0.63,0.5621867439,0.5621867439 0002917485,0.291748 E -13],
+--R    [0.64,0.5693131911 01,0.5693131911 0066188631,- 0.3381137 E -12],
+--R    [0.65,0.5763752205 91,0.5763752205 9118368022,0.1836802 E -12],
+--R    [0.66,0.5833730069 94,0.5833730069 9385593947,- 0.1440605 E -12],
+--R    [0.67,0.5903067469 35,0.5903067469 3537198239,0.3719824 E -12],
+--R    [0.68,0.5971766580 93,0.5971766580 9267754844,- 0.3224516 E -12],
+--R    [0.69,0.6039829782 53,0.6039829782 5299790738,- 0.20926 E -14],
+--R    [0.7,0.6107259643 89,0.6107259643 8920861654,0.2086165 E -12],
+--R    [0.71,0.6174058917 52,0.6174058917 5157266652,- 0.4273335 E -12],
+--R    [0.72,0.6240230529 77,0.6240230529 7675684759,- 0.2431524 E -12],
+--R    [0.73,0.6305777572 15,0.6305777572 1493480666,- 0.6519333 E -13],
+--R    [0.74,0.6370703292 76,0.6370703292 7568357172,- 0.3164283 E -12],
+--R    [0.75,0.6435011087 93,0.6435011087 932843868,0.2843868 E -12],
+--R    [0.76,0.6498704494 12,0.6498704494 1194757749,- 0.524225 E -13],
+--R    [0.77,0.6561787179 91,0.6561787179 9139487538,0.3948754 E -12],
+--R    [0.78,0.6624262938 33,0.6624262938 3315116177,0.1511618 E -12],
+--R    [0.79,0.6686135679 28,0.6686135679 2782091069,- 0.1790893 E -12],
+--R    [0.8,0.6747409422 24,0.6747409422 2355266306,- 0.4473369 E -12],
+--R    [0.81,0.6808088289 16,0.6808088289 1582756649,- 0.1724335 E -12],
+--R    [0.82,0.6868176497 59,0.6868176497 5864527553,- 0.3547245 E -12],
+--R    [0.83,0.6927678353 97,0.6927678353 9712221066,0.1222107 E -12],
+--R    [0.84,0.6986598247 21,0.6986598247 214631978,0.4631978 E -12],
+--R    [0.85,0.7044940642 42,0.7044940642 4221771666,0.2177167 E -12],
+--R    [0.86,0.7102710074 87,0.7102710074 8668623033,- 0.3137697 E -12],
+--R    [0.87,0.7159911144 16,0.7159911144 1630019894,0.3001989 E -12],
+--R    [0.88,0.7216548508 65,0.7216548508 6476123707,- 0.2387629 E -12],
+--R    [0.89,0.7272626879 97,0.7272626879 9669029805,- 0.309702 E -12],
+--R    [0.9,0.7328151017 87,0.7328151017 8650659164,- 0.49340836 E -12],
+--R    [0.91,0.7383125725 17,0.7383125725 1722800021,0.2280002 E -12],
+--R    [0.92,0.7437555842 99,0.7437555842 9885988576,- 0.1401142 E -12],
+--R    [0.93,0.7491446246 06,0.7491446246 0601721032,0.172103 E -13],
+--R    [0.94,0.7544801838 34,0.7544801838 3440566231,0.4056623 E -12],
+--R    [0.95,0.7597627548 76,0.7597627548 7577082892,- 0.2291711 E -12],
+--R    [0.96,0.7649928327 11,0.7649928327 1091022317,- 0.8977683 E -13],
+--R    [0.97,0.7701709140 2,0.7701709140 2033100726,0.3310073 E -12],
+--R    [0.98,0.7752974968 12,0.7752974968 1212640304,0.126403 E -12],
+--R    [0.99,0.7803730800 67,0.7803730800 666358989,- 0.3641011 E -12],
+--R    [1.0,0.7853981633 97,0.7853981633 9744830961,0.4483096 E -12]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp203-212
+\end{thebibliography}
+\end{document}
+
diff --git a/src/input/asinhatanh.input.pamphlet b/src/input/asinhatanh.input.pamphlet
new file mode 100644
index 0000000..b0c7783
--- /dev/null
+++ b/src/input/asinhatanh.input.pamphlet
@@ -0,0 +1,460 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input asinhatanh.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a set of tests for the tanh and coth functions.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of asinh, ranging from 0.00 to 1.00. The
+second column is the reference value of asinh from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp221
+
+<<*>>=
+)spool asinhatanh.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.00,0.000000000,asinh(0.00),asinh(0.00)-0.000000000],_
+[0.01,0.009999833,asinh(0.01),asinh(0.01)-0.009999833],_
+[0.02,0.019998667,asinh(0.02),asinh(0.02)-0.019998667],_
+[0.03,0.029995502,asinh(0.03),asinh(0.03)-0.029995502],_
+[0.04,0.039989341,asinh(0.04),asinh(0.04)-0.039989341],_
+[0.05,0.049979190,asinh(0.05),asinh(0.05)-0.049979190],_
+[0.06,0.059964058,asinh(0.06),asinh(0.06)-0.059964058],_
+[0.07,0.069942959,asinh(0.07),asinh(0.07)-0.069942959],_
+[0.08,0.079914912,asinh(0.08),asinh(0.08)-0.079914912],_
+[0.09,0.089878941,asinh(0.09),asinh(0.09)-0.089878941],_
+[0.10,0.099834079,asinh(0.10),asinh(0.10)-0.099834079],_
+[0.11,0.109779366,asinh(0.11),asinh(0.11)-0.109779366],_
+[0.12,0.119713851,asinh(0.12),asinh(0.12)-0.119713851],_
+[0.13,0.129636590,asinh(0.13),asinh(0.13)-0.129636590],_
+[0.14,0.139546654,asinh(0.14),asinh(0.14)-0.139546654],_
+[0.15,0.149443120,asinh(0.15),asinh(0.15)-0.149443120],_
+[0.16,0.159325080,asinh(0.16),asinh(0.16)-0.159325080],_
+[0.17,0.169191636,asinh(0.17),asinh(0.17)-0.169191636],_
+[0.18,0.179041904,asinh(0.18),asinh(0.18)-0.179041904],_
+[0.19,0.188875015,asinh(0.19),asinh(0.19)-0.188875015],_
+[0.20,0.198690110,asinh(0.20),asinh(0.20)-0.198690110],_
+[0.21,0.208486350,asinh(0.21),asinh(0.21)-0.208486350],_
+[0.22,0.218262908,asinh(0.22),asinh(0.22)-0.218262908],_
+[0.23,0.228018972,asinh(0.23),asinh(0.23)-0.228018972],_
+[0.24,0.237753749,asinh(0.24),asinh(0.24)-0.237753749],_
+[0.25,0.247466462,asinh(0.25),asinh(0.25)-0.247466462],_
+[0.26,0.257156349,asinh(0.26),asinh(0.26)-0.257156349],_
+[0.27,0.266822667,asinh(0.27),asinh(0.27)-0.266822667],_
+[0.28,0.276464691,asinh(0.28),asinh(0.28)-0.276464691],_
+[0.29,0.286081715,asinh(0.29),asinh(0.29)-0.286081715],_
+[0.30,0.295673048,asinh(0.30),asinh(0.30)-0.295673048],_
+[0.31,0.305238020,asinh(0.31),asinh(0.31)-0.305238020],_
+[0.32,0.314775980,asinh(0.32),asinh(0.32)-0.314775980],_
+[0.33,0.324286295,asinh(0.33),asinh(0.33)-0.324286295],_
+[0.34,0.333768352,asinh(0.34),asinh(0.34)-0.333768352],_
+[0.35,0.343221555,asinh(0.35),asinh(0.35)-0.343221555],_
+[0.36,0.352645330,asinh(0.36),asinh(0.36)-0.352645330],_
+[0.37,0.362039121,asinh(0.37),asinh(0.37)-0.362039121],_
+[0.38,0.371402391,asinh(0.38),asinh(0.38)-0.371402391],_
+[0.39,0.380734624,asinh(0.39),asinh(0.39)-0.380734624],_
+[0.40,0.390035320,asinh(0.40),asinh(0.40)-0.390035320],_
+[0.41,0.399304001,asinh(0.41),asinh(0.41)-0.399304001],_
+[0.42,0.408540208,asinh(0.42),asinh(0.42)-0.408540208],_
+[0.43,0.417743500,asinh(0.43),asinh(0.43)-0.417743500],_
+[0.44,0.426913454,asinh(0.44),asinh(0.44)-0.426913454],_
+[0.45,0.436049669,asinh(0.45),asinh(0.45)-0.436049669],_
+[0.46,0.445151759,asinh(0.46),asinh(0.46)-0.445151759],_
+[0.47,0.454219359,asinh(0.47),asinh(0.47)-0.454219359],_
+[0.48,0.463252120,asinh(0.48),asinh(0.48)-0.463252120],_
+[0.49,0.472249713,asinh(0.49),asinh(0.49)-0.472249713],_
+[0.50,0.481211825,asinh(0.50),asinh(0.50)-0.481211825],_
+[0.51,0.490138161,asinh(0.51),asinh(0.51)-0.490138161],_
+[0.52,0.499028444,asinh(0.52),asinh(0.52)-0.499028444],_
+[0.53,0.507882413,asinh(0.53),asinh(0.53)-0.507882413],_
+[0.54,0.516699824,asinh(0.54),asinh(0.54)-0.516699824],_
+[0.55,0.525480448,asinh(0.55),asinh(0.55)-0.525480448],_
+[0.56,0.534224074,asinh(0.56),asinh(0.56)-0.534224074],_
+[0.57,0.542930505,asinh(0.57),asinh(0.57)-0.542930505],_
+[0.58,0.551599562,asinh(0.58),asinh(0.58)-0.551599562],_
+[0.59,0.560231077,asinh(0.59),asinh(0.59)-0.560231077],_
+[0.60,0.568824899,asinh(0.60),asinh(0.60)-0.568824899],_
+[0.61,0.577380892,asinh(0.61),asinh(0.61)-0.577380892],_
+[0.62,0.585898932,asinh(0.62),asinh(0.62)-0.585898932],_
+[0.63,0.594378911,asinh(0.63),asinh(0.63)-0.594378911],_
+[0.64,0.602820733,asinh(0.64),asinh(0.64)-0.602820733],_
+[0.65,0.611224314,asinh(0.65),asinh(0.65)-0.611224314],_
+[0.66,0.619589584,asinh(0.66),asinh(0.66)-0.619589584],_
+[0.67,0.627916485,asinh(0.67),asinh(0.67)-0.627916485],_
+[0.68,0.636204970,asinh(0.68),asinh(0.68)-0.636204970],_
+[0.69,0.644455005,asinh(0.69),asinh(0.69)-0.644455005],_
+[0.70,0.652666566,asinh(0.70),asinh(0.70)-0.652666566],_
+[0.71,0.660839641,asinh(0.71),asinh(0.71)-0.660839641],_
+[0.72,0.668974227,asinh(0.72),asinh(0.72)-0.668974227],_
+[0.73,0.677070332,asinh(0.73),asinh(0.73)-0.677070332],_
+[0.74,0.685127974,asinh(0.74),asinh(0.74)-0.685127974],_
+[0.75,0.693147181,asinh(0.75),asinh(0.75)-0.693147181],_
+[0.76,0.701127988,asinh(0.76),asinh(0.76)-0.701127988],_
+[0.77,0.709070441,asinh(0.77),asinh(0.77)-0.709070441],_
+[0.78,0.716974594,asinh(0.78),asinh(0.78)-0.716974594],_
+[0.79,0.724840509,asinh(0.79),asinh(0.79)-0.724840509],_
+[0.80,0.732668256,asinh(0.80),asinh(0.80)-0.732668256],_
+[0.81,0.740457912,asinh(0.81),asinh(0.81)-0.740457912],_
+[0.82,0.748209563,asinh(0.82),asinh(0.82)-0.748209563],_
+[0.83,0.755923300,asinh(0.83),asinh(0.83)-0.755923300],_
+[0.84,0.763599222,asinh(0.84),asinh(0.84)-0.763599222],_
+[0.85,0.771237433,asinh(0.85),asinh(0.85)-0.771237433],_
+[0.86,0.778838046,asinh(0.86),asinh(0.86)-0.778838046],_
+[0.87,0.786401177,asinh(0.87),asinh(0.87)-0.786401177],_
+[0.88,0.793926950,asinh(0.88),asinh(0.88)-0.793926950],_
+[0.89,0.801415491,asinh(0.89),asinh(0.89)-0.801415491],_
+[0.90,0.808866936,asinh(0.90),asinh(0.90)-0.808866936],_
+[0.91,0.816281421,asinh(0.91),asinh(0.91)-0.816281421],_
+[0.92,0.823659091,asinh(0.92),asinh(0.92)-0.823659091],_
+[0.93,0.831000091,asinh(0.93),asinh(0.93)-0.831000091],_
+[0.94,0.838304575,asinh(0.94),asinh(0.94)-0.838304575],_
+[0.95,0.845572697,asinh(0.95),asinh(0.95)-0.845572697],_
+[0.96,0.852804617,asinh(0.96),asinh(0.96)-0.852804617],_
+[0.97,0.860000498,asinh(0.97),asinh(0.97)-0.860000498],_
+[0.98,0.867160507,asinh(0.98),asinh(0.98)-0.867160507],_
+[0.99,0.874284812,asinh(0.99),asinh(0.99)-0.874284812],_
+[1.00,0.881373587,asinh(1.00),asinh(1.00)-0.881373587]]
+--R 
+--R
+--R   (1)
+--R   [[0.0,0.0,0.0,0.0],
+--R    [0.01,0.009999833,0.0099998333 4083288693 52,0.3408328869 35 E -9],
+--R    [0.02,0.019998667,0.0199986669 0660953936,- 0.9339046064 E -10],
+--R    [0.03,0.029995502,0.0299955018 2152425832 6,- 0.1784757416 74 E -9],
+--R    [0.04,0.039989341,0.0399893410 0602700269 1,0.6027002691 E -11],
+--R    [0.05,0.04997919,0.0499791900 6934866523 2,0.6934866523 2 E -10],
+--R    [0.06,0.059964058,0.0599640581 9533394203 3,0.1953339420 3 E -9],
+--R    [0.07,0.069942959,0.0699429590 1940182241 9,0.1940182242 E -10],
+--R    [0.08,0.079914912,0.0799149114 9449676816 5,- 0.5055032318 35 E -9],
+--R    [0.09,0.089878941,0.0898789407 4394680193 8,- 0.2560531980 6 E -9],
+--R    [0.1,0.099834079,0.0998340788 9920756332 7,- 0.1007924366 7 E -9],
+--R    [0.11,0.109779366,0.1097793659 2054342078,- 0.7945657921 7 E -10],
+--R    [0.12,0.119713851,0.1197138503 9877738636,- 0.6012226136 36 E -9],
+--R    [0.13,0.12963659,0.1296365903 3633024308,0.3363302430 8 E -9],
+--R    [0.14,0.139546654,0.1395466539 0586528422,- 0.9413471578 E -10],
+--R    [0.15,0.14944312,0.1494431201 8495765616,0.1849576561 6 E -9],
+--R    [0.16,0.15932508,0.1593250798 6531572882,- 0.1346842711 8 E -9],
+--R    [0.17,0.169191636,0.1691916359 3519539889,- 0.6480460111 E -10],
+--R    [0.18,0.179041904,0.1790419043 3376594692,0.3337659469 2 E -9],
+--R    [0.19,0.188875015,0.1888750145 7630719416,- 0.4236928058 4 E -9],
+--R    [0.2,0.19869011,0.1986901103 4924140647,0.3492414064 7 E -9],
+--R    [0.21,0.20848635,0.2084863500 7412884481,0.7412884481 E -10],
+--R    [0.22,0.218262908,0.2182629074 3988224632,- 0.5601177536 8 E -9],
+--R    [0.23,0.228018972,0.2280189719 0258204081,- 0.9741795919 E -10],
+--R    [0.24,0.237753749,0.2377537491 5239999115,0.1523999911 5 E -9],
+--R    [0.25,0.247466462,0.2474664615 4726345294,- 0.4527365470 6 E -9],
+--R    [0.26,0.257156349,0.2571563485 1301487614,- 0.4869851238 6 E -9],
+--R    [0.27,0.266822667,0.2668226669 0994085825,- 0.9005914175 E -10],
+--R    [0.28,0.276464691,0.2764646913 6566139339,0.3656613933 9 E -9],
+--R    [0.29,0.286081715,0.2860817145 7448237948,- 0.4255176205 2 E -9],
+--R    [0.3,0.295673048,0.2956730475 634224391,- 0.4365775609 E -9],
+--R    [0.31,0.30523802,0.3052380199 2522822388,- 0.7477177612 E -10],
+--R    [0.32,0.31477598,0.3147759800 1879020946,0.187902095 E -10],
+--R    [0.33,0.324286295,0.3242862951 3746321043,0.1374632104 E -9],
+--R    [0.34,0.333768352,0.3337683516 4588216777,- 0.3541178322 3 E -9],
+--R    [0.35,0.343221555,0.3432215550 8594396213,0.8594396213 E -10],
+--R    [0.36,0.35264533,0.3526453302 5269991354,0.2526999135 4 E -9],
+--R    [0.37,0.362039121,0.3620391212 4097112736,0.2409711273 6 E -9],
+--R    [0.38,0.371402391,0.3714023914 6355987334,0.4635598733 4 E -9],
+--R    [0.39,0.380734624,0.3807346236 4198472547,- 0.3580152745 3 E -9],
+--R    [0.4,0.39003532,0.3900353197 7071527608,- 0.2292847239 E -9],
+--R    [0.41,0.399304001,0.3993040010 5592394477,0.5592394477 E -10],
+--R    [0.42,0.408540208,0.4085402078 298078441,- 0.1701921559 E -9],
+--R    [0.43,0.4177435,0.4177434994 4156299241,- 0.5584370075 9 E -9],
+--R    [0.44,0.426913454,0.4269134541 2611656164,0.1261165616 E -9],
+--R    [0.45,0.436049669,0.4360496688 517405265,- 0.1482594735 E -9],
+--R    [0.46,0.445151759,0.4451517591 4768227909,0.1476822791 E -9],
+--R    [0.47,0.454219359,0.4542193589 1295474595,- 0.8704525405 E -10],
+--R    [0.48,0.46325212,0.4632521202 0743057053,0.2074305705 E -9],
+--R    [0.49,0.472249713,0.4722497130 2638229031,0.263822903 E -10],
+--R    [0.5,0.481211825,0.4812118250 596034475,0.596034475 E -10],
+--R    [0.51,0.490138161,0.4901381614 3623452983,0.4362345298 3 E -9],
+--R    [0.52,0.499028444,0.4990284444 564028591,0.4564028591 E -9],
+--R    [0.53,0.507882413,0.5078824133 1076734063,0.3107673406 E -9],
+--R    [0.54,0.516699824,0.5166998237 890376715,- 0.2109623285 E -9],
+--R    [0.55,0.525480448,0.5254804479 7851348895,- 0.214865111 E -10],
+--R    [0.56,0.534224074,0.5342240739 5366232528,- 0.463376747 E -10],
+--R    [0.57,0.542930505,0.5429305054 5772642154,0.4577264215 E -9],
+--R    [0.58,0.551599562,0.5515995615 7731772498,- 0.422682275 E -9],
+--R    [0.59,0.560231077,0.5602310764 1092803468,- 0.5890719653 2 E -9],
+--R    [0.6,0.568824899,0.5688248987 322475301,- 0.2677524699 E -9],
+--R    [0.61,0.577380892,0.5773808916 4915007492,- 0.3508499251 E -9],
+--R    [0.62,0.585898932,0.5858989322 5916797142,0.2591679714 E -9],
+--R    [0.63,0.594378911,0.5943789113 0224247799,0.302242478 E -9],
+--R    [0.64,0.602820733,0.6028207328 1149960517,- 0.1885003948 E -9],
+--R    [0.65,0.611224314,0.6112243137 62763672,- 0.237236328 E -9],
+--R    [0.66,0.619589584,0.6195895837 234840191,- 0.2765159809 E -9],
+--R    [0.67,0.627916485,0.6279164845 0171330467,- 0.4982866953 3 E -9],
+--R    [0.68,0.63620497,0.6362049697 9573911039,- 0.2042608896 E -9],
+--R    [0.69,0.644455005,0.6444550048 449342957,- 0.1550657043 E -9],
+--R    [0.7,0.652666566,0.6526665660 8235578681,0.8235578681 E -10],
+--R    [0.71,0.660839641,0.6608396407 8958638519,- 0.2104136148 E -9],
+--R    [0.72,0.668974227,0.6689742267 542798277,- 0.2457201723 E -9],
+--R    [0.73,0.677070332,0.6770703319 3083581572,- 0.6916418427 E -10],
+--R    [0.74,0.685127974,0.6851279741 0459912826,0.1045991283 E -9],
+--R    [0.75,0.693147181,0.6931471805 5994530942,- 0.4400546906 E -9],
+--R    [0.76,0.701127988,0.7011279877 5258482712,- 0.2474151729 E -9],
+--R    [0.77,0.709070441,0.7090704409 8638808148,- 0.136119185 E -10],
+--R    [0.78,0.716974594,0.7169745940 9500523361,0.9500523361 E -10],
+--R    [0.79,0.724840509,0.7248405091 2852755291,0.1285275529 E -9],
+--R    [0.8,0.732668256,0.7326682560 4541086415,0.454108642 E -10],
+--R    [0.81,0.740457912,0.7404579124 098567228,0.4098567228 E -9],
+--R    [0.82,0.748209563,0.7482095630 9482316487,0.9482316487 E -10],
+--R    [0.83,0.7559233,0.7559232999 9081426309,- 0.918573691 E -11],
+--R    [0.84,0.763599222,0.7635992217 205762674,- 0.2794237326 E -9],
+--R    [0.85,0.771237433,0.7712374333 5980780282,0.3598078028 E -9],
+--R    [0.86,0.778838046,0.7788380461 6397242546,0.1639724255 E -9],
+--R    [0.87,0.786401177,0.7864011773 0128377712,0.3012837771 E -9],
+--R    [0.88,0.79392695,0.7939269495 9191660771,- 0.4080833923 E -9],
+--R    [0.89,0.801415491,0.8014154912 5348102582,0.2534810258 E -9],
+--R    [0.9,0.808866936,0.8088669356 5278246251,- 0.3472175375 E -9],
+--R    [0.91,0.816281421,0.8162814210 6387596132,0.6387596132 E -10],
+--R    [0.92,0.823659091,0.8236590904 3241050549,- 0.5675894945 2 E -9],
+--R    [0.93,0.831000091,0.8310000911 4624712798,0.146247128 E -9],
+--R    [0.94,0.838304575,0.8383045748 1232348677,- 0.1876765132 E -9],
+--R    [0.95,0.845572697,0.8455726970 3972738992,0.397273899 E -10],
+--R    [0.96,0.852804617,0.8528046172 2893238837,0.2289323884 E -9],
+--R    [0.97,0.860000498,0.8600004983 6713998068,0.3671399807 E -9],
+--R    [0.98,0.867160507,0.8671605068 2966515846,- 0.1703348415 E -9],
+--R    [0.99,0.874284812,0.8742848121 8729492676,0.1872949268 E -9],
+--R    [1.0,0.881373587,0.8813735870 1954302523,0.195430252 E -10]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of atanh, ranging from 0.01 to 0.99. The
+second column is the reference value of atanh from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp221
+
+<<*>>=
+--S 2 of 2
+[[0.00,0.000000000,atanh(0.00),atanh(0.00)-0.000000000],_
+[0.01,0.010000333,atanh(0.01),atanh(0.01)-0.010000333],_
+[0.02,0.020002667,atanh(0.02),atanh(0.02)-0.020002667],_
+[0.03,0.030009004,atanh(0.03),atanh(0.03)-0.030009004],_
+[0.04,0.040021353,atanh(0.04),atanh(0.04)-0.040021353],_
+[0.05,0.050041729,atanh(0.05),atanh(0.05)-0.050041729],_
+[0.06,0.060072156,atanh(0.06),atanh(0.06)-0.060072156],_
+[0.07,0.070114671,atanh(0.07),atanh(0.07)-0.070114671],_
+[0.08,0.080171325,atanh(0.08),atanh(0.08)-0.080171325],_
+[0.09,0.090244188,atanh(0.09),atanh(0.09)-0.090244188],_
+[0.10,0.100335347,atanh(0.10),atanh(0.10)-0.100335347],_
+[0.11,0.110446915,atanh(0.11),atanh(0.11)-0.110446915],_
+[0.12,0.120581028,atanh(0.12),atanh(0.12)-0.120581028],_
+[0.13,0.130739850,atanh(0.13),atanh(0.13)-0.130739850],_
+[0.14,0.140925576,atanh(0.14),atanh(0.14)-0.140925576],_
+[0.15,0.151140436,atanh(0.15),atanh(0.15)-0.151140436],_
+[0.16,0.161386696,atanh(0.16),atanh(0.16)-0.161386696],_
+[0.17,0.171666663,atanh(0.17),atanh(0.17)-0.171666663],_
+[0.18,0.181982689,atanh(0.18),atanh(0.18)-0.181982689],_
+[0.19,0.192337169,atanh(0.19),atanh(0.19)-0.192337169],_
+[0.20,0.202732554,atanh(0.20),atanh(0.20)-0.202732554],_
+[0.21,0.213171346,atanh(0.21),atanh(0.21)-0.213171346],_
+[0.22,0.223656109,atanh(0.22),atanh(0.22)-0.223656109],_
+[0.23,0.234189466,atanh(0.23),atanh(0.23)-0.234189466],_
+[0.24,0.244774112,atanh(0.24),atanh(0.24)-0.244774112],_
+[0.25,0.255412812,atanh(0.25),atanh(0.25)-0.255412812],_
+[0.26,0.266108407,atanh(0.26),atanh(0.26)-0.266108407],_
+[0.27,0.276863823,atanh(0.27),atanh(0.27)-0.276863823],_
+[0.28,0.287682072,atanh(0.28),atanh(0.28)-0.287682072],_
+[0.29,0.298566264,atanh(0.29),atanh(0.29)-0.298566264],_
+[0.30,0.309519604,atanh(0.30),atanh(0.30)-0.309519604],_
+[0.31,0.320545409,atanh(0.31),atanh(0.31)-0.320545409],_
+[0.32,0.331647108,atanh(0.32),atanh(0.32)-0.331647108],_
+[0.33,0.342828254,atanh(0.33),atanh(0.33)-0.342828254],_
+[0.34,0.354092528,atanh(0.34),atanh(0.34)-0.354092528],_
+[0.35,0.365443754,atanh(0.35),atanh(0.35)-0.365443754],_
+[0.36,0.376885901,atanh(0.36),atanh(0.36)-0.376885901],_
+[0.37,0.388423100,atanh(0.37),atanh(0.37)-0.388423100],_
+[0.38,0.400059650,atanh(0.38),atanh(0.38)-0.400059650],_
+[0.39,0.411800034,atanh(0.39),atanh(0.39)-0.411800034],_
+[0.40,0.423648930,atanh(0.40),atanh(0.40)-0.423648930],_
+[0.41,0.435611223,atanh(0.41),atanh(0.41)-0.435611223],_
+[0.42,0.447692023,atanh(0.42),atanh(0.42)-0.447692023],_
+[0.43,0.459896681,atanh(0.43),atanh(0.43)-0.459896681],_
+[0.44,0.472230804,atanh(0.44),atanh(0.44)-0.472230804],_
+[0.45,0.484700279,atanh(0.45),atanh(0.45)-0.484700279],_
+[0.46,0.497311288,atanh(0.46),atanh(0.46)-0.497311288],_
+[0.47,0.510070337,atanh(0.47),atanh(0.47)-0.510070337],_
+[0.48,0.522984278,atanh(0.48),atanh(0.48)-0.522984278],_
+[0.49,0.536060337,atanh(0.49),atanh(0.49)-0.536060337],_
+[0.50,0.549306144,atanh(0.50),atanh(0.50)-0.549306144],_
+[0.51,0.562729769,atanh(0.51),atanh(0.51)-0.562729769],_
+[0.52,0.576339754,atanh(0.52),atanh(0.52)-0.576339754],_
+[0.53,0.590145160,atanh(0.53),atanh(0.53)-0.590145160],_
+[0.54,0.604155603,atanh(0.54),atanh(0.54)-0.604155603],_
+[0.55,0.618381313,atanh(0.55),atanh(0.55)-0.618381313],_
+[0.56,0.632833186,atanh(0.56),atanh(0.56)-0.632833186],_
+[0.57,0.647522844,atanh(0.57),atanh(0.57)-0.647522844],_
+[0.58,0.662462707,atanh(0.58),atanh(0.58)-0.662462707],_
+[0.59,0.677666068,atanh(0.59),atanh(0.59)-0.677666068],_
+[0.60,0.693147180,atanh(0.60),atanh(0.60)-0.693147180],_
+[0.61,0.708921359,atanh(0.61),atanh(0.61)-0.708921359],_
+[0.62,0.725005087,atanh(0.62),atanh(0.62)-0.725005087],_
+[0.63,0.741416144,atanh(0.63),atanh(0.63)-0.741416144],_
+[0.64,0.758173745,atanh(0.64),atanh(0.64)-0.758173745],_
+[0.65,0.775298706,atanh(0.65),atanh(0.65)-0.775298706],_
+[0.66,0.792813631,atanh(0.66),atanh(0.66)-0.792813631],_
+[0.67,0.810743125,atanh(0.67),atanh(0.67)-0.810743125],_
+[0.68,0.829114038,atanh(0.68),atanh(0.68)-0.829114038],_
+[0.69,0.847955755,atanh(0.69),atanh(0.69)-0.847955755],_
+[0.70,0.867300527,atanh(0.70),atanh(0.70)-0.867300527],_
+[0.71,0.887183863,atanh(0.71),atanh(0.71)-0.887183863],_
+[0.72,0.907644983,atanh(0.72),atanh(0.72)-0.907644983],_
+[0.73,0.928727364,atanh(0.73),atanh(0.73)-0.928727364],_
+[0.74,0.950479381,atanh(0.74),atanh(0.74)-0.950479381],_
+[0.75,0.972955074,atanh(0.75),atanh(0.75)-0.972955074],_
+[0.76,0.996215082,atanh(0.76),atanh(0.76)-0.996215082],_
+[0.77,1.020327758,atanh(0.77),atanh(0.77)-1.020327758],_
+[0.78,1.045370548,atanh(0.78),atanh(0.78)-1.045370548],_
+[0.79,1.071431684,atanh(0.79),atanh(0.79)-1.071431684],_
+[0.80,1.098612289,atanh(0.80),atanh(0.80)-1.098612289],_
+[0.81,1.127029026,atanh(0.81),atanh(0.81)-1.127029026],_
+[0.82,1.156817465,atanh(0.82),atanh(0.82)-1.156817465],_
+[0.83,1.188136404,atanh(0.83),atanh(0.83)-1.188136404],_
+[0.84,1.221173518,atanh(0.84),atanh(0.84)-1.221173518],_
+[0.85,1.256152811,atanh(0.85),atanh(0.85)-1.256152811],_
+[0.86,1.293344672,atanh(0.86),atanh(0.86)-1.293344672],_
+[0.87,1.333079629,atanh(0.87),atanh(0.87)-1.333079629],_
+[0.88,1.375767657,atanh(0.88),atanh(0.88)-1.375767657],_
+[0.89,1.421925871,atanh(0.89),atanh(0.89)-1.421925871],_
+[0.90,1.472219490,atanh(0.90),atanh(0.90)-1.472219490],_
+[0.91,1.527524425,atanh(0.91),atanh(0.91)-1.527524425],_
+[0.92,1.589026915,atanh(0.92),atanh(0.92)-1.589026915],_
+[0.93,1.658390020,atanh(0.93),atanh(0.93)-1.658390020],_
+[0.94,1.738049345,atanh(0.94),atanh(0.94)-1.738049345],_
+[0.95,1.831780823,atanh(0.95),atanh(0.95)-1.831780823],_
+[0.96,1.945910149,atanh(0.96),atanh(0.96)-1.945910149],_
+[0.97,2.092295720,atanh(0.97),atanh(0.97)-2.092295720],_
+[0.98,2.297559925,atanh(0.98),atanh(0.98)-2.297559925],_
+[0.99,2.646652412,atanh(0.99),atanh(0.99)-2.646652412]]
+--R 
+--R
+--R   (2)
+--R   [[0.0,0.0,0.0,0.0],
+--R    [0.01,0.010000333,0.0100003333 5333476201 6,0.3533347620 16 E -9],
+--R    [0.02,0.020002667,0.0200026673 0684958071 7,0.3068495807 17 E -9],
+--R    [0.03,0.030009004,0.0300090048 6312647432 6,0.8631264743 26 E -9],
+--R    [0.04,0.040021353,0.0400213538 3676821291 2,0.8367682129 12 E -9],
+--R    [0.05,0.050041729,0.0500417292 7849126824 6,0.2784912682 46 E -9],
+--R    [0.06,0.060072156,0.0600721559 2103162366 2,- 0.7896837633 8 E -10],
+--R    [0.07,0.070114671,0.0701146706 5432511799,- 0.3456748820 1 E -9],
+--R    [0.08,0.080171325,0.0801713250 3758969169,0.3758969169 E -10],
+--R    [0.09,0.090244188,0.0902441878 5614682960 9,- 0.1438531703 9 E -9],
+--R    [0.1,0.100335347,0.1003353477 3107558064,0.7310755806 36 E -9],
+--R    [0.11,0.110446915,0.1104469157 9009714872,0.7900971487 22 E -9],
+--R    [0.12,0.120581028,0.1205810284 0844403523,0.4084440352 3 E -9],
+--R    [0.13,0.13073985,0.1307398500 28878425,0.28878425 E -10],
+--R    [0.14,0.140925576,0.1409255760 7049386396,0.7049386396 E -10],
+--R    [0.15,0.151140436,0.1511404359 3646680528,- 0.6353319472 E -10],
+--R    [0.16,0.161386696,0.1613866961 3152551534,0.1315255153 4 E -9],
+--R    [0.17,0.171666663,0.1716666635 0057909768,0.5005790976 8 E -9],
+--R    [0.18,0.181982689,0.1819826886 0070582337,- 0.3992941766 3 E -9],
+--R    [0.19,0.192337169,0.1923371692 1954530989,0.2195453098 9 E -9],
+--R    [0.2,0.202732554,0.2027325540 5408219099,0.5408219099 E -10],
+--R    [0.21,0.213171346,0.2131713465 6485979698,0.5648597969 8 E -9],
+--R    [0.22,0.223656109,0.2236561090 2183241065,0.2183241065 E -10],
+--R    [0.23,0.234189466,0.2341894667 5936682305,0.7593668230 5 E -9],
+--R    [0.24,0.244774112,0.2447741126 5935289296,0.6593528929 6 E -9],
+--R    [0.25,0.255412812,0.2554128118 829953416,- 0.1170046584 E -9],
+--R    [0.26,0.266108407,0.2661084068 7365412176,- 0.1263458782 E -9],
+--R    [0.27,0.276863823,0.2768638226 5510007198,- 0.3448999280 2 E -9],
+--R    [0.28,0.287682072,0.2876820724 5178092744,0.4517809274 4 E -9],
+--R    [0.29,0.298566264,0.2985662636 6017834677,- 0.3398216532 3 E -9],
+--R    [0.3,0.309519604,0.3095196042 0311171547,0.2031117155 E -9],
+--R    [0.31,0.320545409,0.3205454093 0194608097,0.3019460809 7 E -9],
+--R    [0.32,0.331647108,0.3316471087 051320776,0.7051320776 E -9],
+--R    [0.33,0.342828254,0.3428282544 1539385272,0.4153938527 2 E -9],
+--R    [0.34,0.354092528,0.3540925289 6224291211,0.9622429121 E -9],
+--R    [0.35,0.365443754,0.3654437542 7139616907,0.2713961690 7 E -9],
+--R    [0.36,0.376885901,0.3768859011 88190076,0.188190076 E -9],
+--R    [0.37,0.3884231,0.3884230997 1829611371,- 0.2817038862 9 E -9],
+--R    [0.38,0.40005965,0.4000596500 5605656814,0.5605656814 E -10],
+--R    [0.39,0.411800034,0.4118000344 7869025422,0.4786902542 2 E -9],
+--R    [0.4,0.42364893,0.4236489301 9360180685,0.1936018069 E -9],
+--R    [0.41,0.435611223,0.4356112232 362244138,0.2362244138 E -9],
+--R    [0.42,0.447692023,0.4476920235 2742069707,0.5274206970 7 E -9],
+--R    [0.43,0.459896681,0.4598966812 1267856435,0.2126785644 E -9],
+--R    [0.44,0.472230804,0.4722308044 2042569355,0.4204256935 5 E -9],
+--R    [0.45,0.484700279,0.4847002785 9405174156,- 0.4059482584 4 E -9],
+--R    [0.46,0.497311288,0.4973112875 7203102745,- 0.4279689725 5 E -9],
+--R    [0.47,0.510070337,0.5100703366 1330723373,- 0.3866927663 E -9],
+--R    [0.48,0.522984278,0.5229842775 9134385416,- 0.4086561458 E -9],
+--R    [0.49,0.536060337,0.5360603366 1056668467,- 0.3894333153 E -9],
+--R    [0.5,0.549306144,0.5493061443 340548457,0.3340548457 E -9],
+--R    [0.51,0.562729769,0.5627297693 521488593,0.3521488593 E -9],
+--R    [0.52,0.576339754,0.5763397549 691927296,0.9691927296 E -9],
+--R    [0.53,0.59014516,0.5901451598 4118843811,- 0.1588115619 E -9],
+--R    [0.54,0.604155603,0.6041556029 6226707918,- 0.377329208 E -10],
+--R    [0.55,0.618381313,0.6183813135 7446343157,0.5744634315 7 E -9],
+--R    [0.56,0.632833186,0.6328331866 6563794169,0.6656379416 9 E -9],
+--R    [0.57,0.647522844,0.6475228448 2737281698,0.8273728169 8 E -9],
+--R    [0.58,0.662462707,0.6624627073 7179924883,0.3717992488 E -9],
+--R    [0.59,0.677666068,0.6776660677 5796186084,- 0.2420381392 E -9],
+--R    [0.6,0.69314718,0.6931471805 5994530942,0.5599453094 2 E -9],
+--R    [0.61,0.708921359,0.7089213594 2740828423,0.4274082842 E -9],
+--R    [0.62,0.725005087,0.7250050877 5299915279,0.7529991527 9 E -9],
+--R    [0.63,0.741416144,0.7414161440 8126894485,0.8126894484 E -10],
+--R    [0.64,0.758173745,0.7581737446 8404421054,- 0.3159557895 E -9],
+--R    [0.65,0.775298706,0.7752987062 0558346517,0.2055834652 E -9],
+--R    [0.66,0.792813631,0.7928136318 7019092161,0.8701909216 1 E -9],
+--R    [0.67,0.810743125,0.8107431254 7513743591,0.4751374359 1 E -9],
+--R    [0.68,0.829114038,0.8291140383 0176618883,0.3017661888 E -9],
+--R    [0.69,0.847955755,0.8479557552 1896361309,0.2189636131 E -9],
+--R    [0.7,0.867300527,0.8673005276 9405319443,0.6940531944 3 E -9],
+--R    [0.71,0.887183863,0.8871838632 5809290781,0.2580929078 E -9],
+--R    [0.72,0.907644983,0.9076449833 1912455918,0.3191245592 E -9],
+--R    [0.73,0.928727364,0.9287273642 4672493638,0.2467249364 E -9],
+--R    [0.74,0.950479381,0.9504793805 9652349126,- 0.4034765087 E -9],
+--R    [0.75,0.972955074,0.9729550745 2765665255,0.5276566525 5 E -9],
+--R    [0.76,0.996215082,0.9962150823 4510308105,0.345103081 E -9],
+--R    [0.77,1.020327758,1.0203277583 223397256,0.3223397256 E -9],
+--R    [0.78,1.045370548,1.0453705484 668846471,0.4668846471 E -9],
+--R    [0.79,1.071431684,1.0714316840 586659998,0.586659998 E -10],
+--R    [0.8,1.098612289,1.0986122886 681096914,- 0.3318903086 E -9],
+--R    [0.81,1.127029026,1.1270290260 496926434,0.496926434 E -10],
+--R    [0.82,1.156817465,1.1568174645 903153292,- 0.4096846708 E -9],
+--R    [0.83,1.188136404,1.1881364043 926024299,0.3926024299 E -9],
+--R    [0.84,1.221173518,1.2211735176 846021907,- 0.3153978093 E -9],
+--R    [0.85,1.256152811,1.2561528119 880573765,0.9880573764 9 E -9],
+--R    [0.86,1.293344672,1.2933446720 489713161,0.489713161 E -10],
+--R    [0.87,1.333079629,1.3330796296 965249441,0.6965249441 E -9],
+--R    [0.88,1.375767657,1.3757676565 209744477,- 0.4790255523 E -9],
+--R    [0.89,1.421925871,1.4219258711 306359176,0.1306359176 E -9],
+--R    [0.9,1.47221949,1.4722194895 8322023,- 0.41677977 E -9],
+--R    [0.91,1.527524425,1.5275244253 55205245,0.355205245 E -9],
+--R    [0.92,1.589026915,1.5890269151 739728098,0.1739728098 E -9],
+--R    [0.93,1.65839002,1.6583900199 247861234,- 0.752138766 E -10],
+--R    [0.94,1.738049345,1.7380493449 176365654,- 0.823634346 E -10],
+--R    [0.95,1.831780823,1.8317808230 648232137,0.648232137 E -10],
+--R    [0.96,1.945910149,1.9459101490 553133051,0.553133051 E -10],
+--R    [0.97,2.09229572,2.0922957200 349394077,0.349394077 E -10],
+--R    [0.98,2.297559925,2.2975599250 672949634,0.672949634 E -10],
+--R    [0.99,2.646652412,2.6466524123 622461976,0.3622461976 E -9]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf Digital Library of Mathematical Functions},
+http://dlmf.nist.gov
+\bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp221
+\end{thebibliography}
+\end{document}
diff --git a/src/input/besselk.input.pamphlet b/src/input/besselk.input.pamphlet
index 0cd01ea..7a31a2c 100644
--- a/src/input/besselk.input.pamphlet
+++ b/src/input/besselk.input.pamphlet
@@ -9,6 +9,11 @@
 \eject
 \tableofcontents
 \eject
+In the following table there are 4 columns. The first column
+is the argument of besselK, ranging from 0.01 to 10.0. The
+second column is the reference value of Gamma from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp417-419
 <<*>>=
 )spool besselk.output
 )set message test on
@@ -45,13 +50,122 @@ limit(D(besselK(a,x),a),a=1/2)
 --R   (4)  "failed"
 --R                                                    Type: Union("failed",...)
 --E 4
+
+--S 5
+stegun(x)== %e^x * besselK(1,x)
+--E 5
+
+--S 6
+[[0.1, 10.890182683 , stegun(0.1),  stegun(0.1)- 10.890182683 ],_
+ [0.2,  5.833386037 , stegun(0.2),  stegun(0.2)-  5.833386037 ],_
+ [0.3,  4.125157762 , stegun(0.3),  stegun(0.3)-  4.125157762 ],_
+ [0.4,  3.258673880 , stegun(0.4),  stegun(0.4)-  3.258673880 ],_
+ [0.5,  2.7310097082, stegun(0.5),  stegun(0.5)-  2.7310097082],_
+ [0.6,  2.3739200376, stegun(0.6),  stegun(0.6)-  2.3739200376],_
+ [0.7,  2.1150113128, stegun(0.7),  stegun(0.7)-  2.1150113128],_
+ [0.8,  1.9179302990, stegun(0.8),  stegun(0.8)-  1.9179302990],_
+ [0.9,  1.7623882197, stegun(0.9),  stegun(0.9)-  1.7623882197],_
+ [1.0,  1.6361534863, stegun(1.0),  stegun(1.0)-  1.6361534863],_
+ [1.1,  1.5314037541, stegun(1.1),  stegun(1.1)-  1.5314037541],_
+ [1.2,  1.4428975522, stegun(1.2),  stegun(1.2)-  1.4428975522],_
+ [1.3,  1.3669872841, stegun(1.3),  stegun(1.3)-  1.3669872841],_
+ [1.4,  1.3010537400, stegun(1.4),  stegun(1.4)-  1.3010537400],_
+ [1.5,  1.2431658736, stegun(1.5),  stegun(1.5)-  1.2431658736],_
+ [1.6,  1.1918675654, stegun(1.6),  stegun(1.6)-  1.1918675654],_
+ [1.7,  1.1460392462, stegun(1.7),  stegun(1.7)-  1.1460392462],_
+ [1.8,  1.1048053726, stegun(1.8),  stegun(1.8)-  1.1048053726],_
+ [1.9,  1.0674709298, stegun(1.9),  stegun(1.9)-  1.0674709298],_
+ [2.0,  1.0334768471, stegun(2.0),  stegun(2.0)-  1.0334768471],_
+ [2.1,  1.0023680527, stegun(2.1),  stegun(2.1)-  1.0023680527],_
+ [2.2,  0.9737701679, stegun(2.2),  stegun(2.2)-  0.9737701679],_
+ [2.3,  0.9473722250, stegun(2.3),  stegun(2.3)-  0.9473722250],_
+ [2.4,  0.9229136650, stegun(2.4),  stegun(2.4)-  0.9229136650],_
+ [2.5,  0.9001744239, stegun(2.5),  stegun(2.5)-  0.9001744239],_
+ [2.6,  0.8789672806, stegun(2.6),  stegun(2.6)-  0.8789672806],_
+ [2.7,  0.8591318867, stegun(2.7),  stegun(2.7)-  0.8591318867],_
+ [2.8,  0.8405300604, stegun(2.8),  stegun(2.8)-  0.8405300604],_
+ [2.9,  0.8230420403, stegun(2.9),  stegun(2.9)-  0.8230420403],_
+ [3.0,  0.8065634800, stegun(3.0),  stegun(3.0)-  0.8065634800],_
+ [3.1,  0.7910030157, stegun(3.1),  stegun(3.1)-  0.7910030157],_
+ [3.2,  0.7762802824, stegun(3.2),  stegun(3.2)-  0.7762802824],_
+ [3.3,  0.7623242864, stegun(3.3),  stegun(3.3)-  0.7623242864],_
+ [3.4,  0.7490720613, stegun(3.4),  stegun(3.4)-  0.7490720613],_
+ [3.5,  0.7364675480, stegun(3.5),  stegun(3.5)-  0.7364675480],_
+ [3.6,  0.7244606608, stegun(3.6),  stegun(3.6)-  0.7244606608],_
+ [3.7,  0.7130065010, stegun(3.7),  stegun(3.7)-  0.7130065010],_
+ [3.8,  0.7020646931, stegun(3.8),  stegun(3.8)-  0.7020646931],_
+ [3.9,  0.6915988206, stegun(3.9),  stegun(3.9)-  0.6915988206],_
+ [4.0,  0.6815759452, stegun(4.0),  stegun(4.0)-  0.6815759452],_
+ [4.1,  0.6719661952, stegun(4.1),  stegun(4.1)-  0.6719661952],_
+ [4.2,  0.6627424110, stegun(4.2),  stegun(4.2)-  0.6627424110],_
+ [4.3,  0.6538798395, stegun(4.3),  stegun(4.3)-  0.6538798395],_
+ [4.4,  0.6453558689, stegun(4.4),  stegun(4.4)-  0.6453558689],_
+ [4.5,  0.6371497988, stegun(4.5),  stegun(4.5)-  0.6371497988],_
+ [4.6,  0.6292426383, stegun(4.6),  stegun(4.6)-  0.6292426383],_
+ [4.7,  0.6216169312, stegun(4.7),  stegun(4.7)-  0.6216169312],_
+ [4.8,  0.6142566003, stegun(4.8),  stegun(4.8)-  0.6142566003],_
+ [4.9,  0.6071468131, stegun(4.9),  stegun(4.9)-  0.6071468131],_
+ [5.0,  0.6002738587, stegun(5.0),  stegun(5.0)-  0.6002738587],_
+ [5.1,  0.5936250463, stegun(5.1),  stegun(5.1)-  0.5936250463],_
+ [5.2,  0.5871886062, stegun(5.2),  stegun(5.2)-  0.5871886062],_
+ [5.3,  0.5809536085, stegun(5.3),  stegun(5.3)-  0.5809536085],_
+ [5.4,  0.5749098871, stegun(5.4),  stegun(5.4)-  0.5749098871],_
+ [5.5,  0.5690479741, stegun(5.5),  stegun(5.5)-  0.5690479741],_
+ [5.6,  0.5633590393, stegun(5.6),  stegun(5.6)-  0.5633590393],_
+ [5.7,  0.5578348348, stegun(5.7),  stegun(5.7)-  0.5578348348],_
+ [5.8,  0.5524676495, stegun(5.8),  stegun(5.8)-  0.5524676495],_
+ [5.9,  0.5472502639, stegun(5.9),  stegun(5.9)-  0.5472502639],_
+ [6.0,  0.5421759104, stegun(6.0),  stegun(6.0)-  0.5421759104],_
+ [6.1,  0.5372382386, stegun(6.1),  stegun(6.1)-  0.5372382386],_
+ [6.2,  0.5324312833, stegun(6.2),  stegun(6.2)-  0.5324312833],_
+ [6.3,  0.5277494344, stegun(6.3),  stegun(6.3)-  0.5277494344],_
+ [6.4,  0.5231874101, stegun(6.4),  stegun(6.4)-  0.5231874101],_
+ [6.5,  0.5187402336, stegun(6.5),  stegun(6.5)-  0.5187402336],_
+ [6.6,  0.5144032108, stegun(6.6),  stegun(6.6)-  0.5144032108],_
+ [6.7,  0.5101719097, stegun(6.7),  stegun(6.7)-  0.5101719097],_
+ [6.8,  0.5060421421, stegun(6.8),  stegun(6.8)-  0.5060421421],_
+ [6.9,  0.5020099471, stegun(6.9),  stegun(6.9)-  0.5020099471],_
+ [7.0,  0.4980715749, stegun(7.0),  stegun(7.0)-  0.4980715749],_
+ [7.1,  0.4942234737, stegun(7.1),  stegun(7.1)-  0.4942234737],_
+ [7.2,  0.4904622755, stegun(7.2),  stegun(7.2)-  0.4904622755],_
+ [7.3,  0.4867847842, stegun(7.3),  stegun(7.3)-  0.4867847842],_
+ [7.4,  0.4831879648, stegun(7.4),  stegun(7.4)-  0.4831879648],_
+ [7.5,  0.4796689336, stegun(7.5),  stegun(7.5)-  0.4796689336],_
+ [7.6,  0.4762249486, stegun(7.6),  stegun(7.6)-  0.4762249486],_
+ [7.7,  0.4728533995, stegun(7.7),  stegun(7.7)-  0.4728533995],_
+ [7.8,  0.4695518010, stegun(7.8),  stegun(7.8)-  0.4695518010],_
+ [7.9,  0.4663177847, stegun(7.9),  stegun(7.9)-  0.4663177847],_
+ [8.0,  0.4631490928, stegun(8.0),  stegun(8.0)-  0.4631490928],_
+ [8.1,  0.4600435709, stegun(8.1),  stegun(8.1)-  0.4600435709],_
+ [8.2,  0.4569991615, stegun(8.2),  stegun(8.2)-  0.4569991615],_
+ [8.3,  0.4540139001, stegun(8.3),  stegun(8.3)-  0.4540139001],_
+ [8.4,  0.4510859089, stegun(8.4),  stegun(8.4)-  0.4510859089],_
+ [8.5,  0.4482133915, stegun(8.5),  stegun(8.5)-  0.4482133915],_
+ [8.6,  0.4453946295, stegun(8.6),  stegun(8.6)-  0.4453946295],_
+ [8.7,  0.4426279775, stegun(8.7),  stegun(8.7)-  0.4426279775],_
+ [8.8,  0.4399118594, stegun(8.8),  stegun(8.8)-  0.4399118594],_
+ [8.9,  0.4372447648, stegun(8.9),  stegun(8.9)-  0.4372447648],_
+ [9.0,  0.4346252454, stegun(9.0),  stegun(9.0)-  0.4346252454],_
+ [9.1,  0.4320519116, stegun(9.1),  stegun(9.1)-  0.4320519116],_
+ [9.2,  0.4295234301, stegun(9.2),  stegun(9.2)-  0.4295234301],_
+ [9.3,  0.4270385204, stegun(9.3),  stegun(9.3)-  0.4270385204],_
+ [9.4,  0.4245959520, stegun(9.4),  stegun(9.4)-  0.4245959520],_
+ [9.5,  0.4221945430, stegun(9.5),  stegun(9.5)-  0.4221945430],_
+ [9.6,  0.4198331565, stegun(9.6),  stegun(9.6)-  0.4198331565],_
+ [9.7,  0.4175106989, stegun(9.7),  stegun(9.7)-  0.4175106989],_
+ [9.8,  0.4152261179, stegun(9.8),  stegun(9.8)-  0.4152261179],_
+ [9.9,  0.4129784003, stegun(9.9),  stegun(9.9)-  0.4129784003],_
+ [10.0, 0.4107665704, stegun(10.0), stegun(10.0)- 0.4107665704]]
+--E 6
+
 )spool 
 )lisp (bye)
  
 @
 \eject
 \begin{thebibliography}{99}
-\bibitem{1} nothing
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp417-419
 \end{thebibliography}
 \end{document}
 
diff --git a/src/input/exp.input.pamphlet b/src/input/exp.input.pamphlet
new file mode 100644
index 0000000..88b3759
--- /dev/null
+++ b/src/input/exp.input.pamphlet
@@ -0,0 +1,460 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input exp.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's exp function with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of exp(x), ranging from 0.0 to 10.0. The
+second column is the reference value of exp from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp136-137
+
+<<*>>=
+)spool exp.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.0,     1.000000000000000,  exp(0.0), exp(0.0)-     1.000000000000000],_
+ [0.1,     1.105170918075648,  exp(0.1), exp(0.1)-     1.105170918075648],_
+ [0.2,     1.221402758160170,  exp(0.2), exp(0.2)-     1.221402758160170],_
+ [0.3,     1.349858807576003,  exp(0.3), exp(0.3)-     1.349858807576003],_
+ [0.4,     1.491824697641270,  exp(0.4), exp(0.4)-     1.491824697641270],_
+ [0.5,     1.648721270700128,  exp(0.5), exp(0.5)-     1.648721270700128],_
+ [0.6,     1.822118800390509,  exp(0.6), exp(0.6)-     1.822118800390509],_
+ [0.7,     2.013752707470477,  exp(0.7), exp(0.7)-     2.013752707470477],_
+ [0.8,     2.225540928492468,  exp(0.8), exp(0.8)-     2.225540928492468],_
+ [0.9,     2.459603111156950,  exp(0.9), exp(0.9)-     2.459603111156950],_
+ [1.0,     2.718281828459045,  exp(1.0), exp(1.0)-     2.718281828459045],_
+ [1.1,     3.004166023946433,  exp(1.1), exp(1.1)-     3.004166023946433],_
+ [1.2,     3.320116922736547,  exp(1.2), exp(1.2)-     3.320116922736547],_
+ [1.3,     3.669296667619244,  exp(1.3), exp(1.3)-     3.669296667619244],_
+ [1.4,     4.055199966844675,  exp(1.4), exp(1.4)-     4.055199966844675],_
+ [1.5,     4.481689070338065,  exp(1.5), exp(1.5)-     4.481689070338065],_
+ [1.6,     4.953032424395115,  exp(1.6), exp(1.6)-     4.953032424395115],_
+ [1.7,     5.473947391727200,  exp(1.7), exp(1.7)-     5.473947391727200],_
+ [1.8,     6.049647464412946,  exp(1.8), exp(1.8)-     6.049647464412946],_
+ [1.9,     6.685894442279269,  exp(1.9), exp(1.9)-     6.685894442279269],_
+ [2.0,     7.389056098930650,  exp(2.0), exp(2.0)-     7.389056098930650],_
+ [2.1,     8.166169912567650,  exp(2.1), exp(2.1)-     8.166169912567650],_
+ [2.2,     9.025013499434121,  exp(2.2), exp(2.2)-     9.025013499434121],_
+ [2.3,     9.974182454814721,  exp(2.3), exp(2.3)-     9.974182454814721],_
+ [2.4,    11.023176380641602,  exp(2.4), exp(2.4)-    11.023176380641602],_
+ [2.5,    12.182493960703473,  exp(2.5), exp(2.5)-    12.182493960703473],_
+ [2.6,    13.463738035001690,  exp(2.6), exp(2.6)-    13.463738035001690],_
+ [2.7,    14.879731724872834,  exp(2.7), exp(2.7)-    14.879731724872834],_
+ [2.8,    16.444646771097050,  exp(2.8), exp(2.8)-    16.444646771097050],_
+ [2.9,    18.174145369443061,  exp(2.9), exp(2.9)-    18.174145369443061],_
+ [3.0,    20.085536923187668,  exp(3.0), exp(3.0)-    20.085536923187668],_
+ [3.1,    22.197951281441633,  exp(3.1), exp(3.1)-    22.197951281441633],_
+ [3.2,    24.532530197109349,  exp(3.2), exp(3.2)-    24.532530197109349],_
+ [3.3,    27.112638920657887,  exp(3.3), exp(3.3)-    27.112638920657887],_
+ [3.4,    29.964100047397013,  exp(3.4), exp(3.4)-    29.964100047397013],_
+ [3.5,    33.115451958692314,  exp(3.5), exp(3.5)-    33.115451958692314],_
+ [3.6,    36.598234443677988,  exp(3.6), exp(3.6)-    36.598234443677988],_
+ [3.7,    40.447304360067391,  exp(3.7), exp(3.7)-    40.447304360067391],_
+ [3.8,    44.701184493300823,  exp(3.8), exp(3.8)-    44.701184493300823],_
+ [3.9,    49.402449105530174,  exp(3.9), exp(3.9)-    49.402449105530174],_
+ [4.0,    54.598150033144239,  exp(4.0), exp(4.0)-    54.598150033144239],_
+ [4.1,    60.340287597361969,  exp(4.1), exp(4.1)-    60.340287597361969],_
+ [4.2,    66.686331040925142,  exp(4.2), exp(4.2)-    66.686331040925142],_
+ [4.3,    73.699793699595797,  exp(4.3), exp(4.3)-    73.699793699595797],_
+ [4.4,    81.450868664968117,  exp(4.4), exp(4.4)-    81.450868664968117],_
+ [4.5,    90.017131300521814,  exp(4.5), exp(4.5)-    90.017131300521814],_
+ [4.6,    99.484315641933809,  exp(4.6), exp(4.6)-    99.484315641933809],_
+ [4.7,   109.947172452123499,  exp(4.7), exp(4.7)-   109.947172452123499],_
+ [4.8,   121.510417518734881,  exp(4.8), exp(4.8)-   121.510417518734881],_
+ [4.9,   134.289779684935485,  exp(4.9), exp(4.9)-   134.289779684935485],_
+ [5.0,   148.413159102577,     exp(5.0), exp(5.0)-   148.413159102577],_
+ [5.1,   164.021907299902,     exp(5.1), exp(5.1)-   164.021907299902],_
+ [5.2,   181.272241875151,     exp(5.2), exp(5.2)-   181.272241875151],_
+ [5.3,   200.336809974792,     exp(5.3), exp(5.3)-   200.336809974792],_
+ [5.4,   221.406416204187,     exp(5.4), exp(5.4)-   221.406416204187],_
+ [5.5,   244.691932264220,     exp(5.5), exp(5.5)-   244.691932264220],_
+ [5.6,   270.426407426153,     exp(5.6), exp(5.6)-   270.426407426153],_
+ [5.7,   298.867400967060,     exp(5.7), exp(5.7)-   298.867400967060],_
+ [5.8,   330.299559909649,     exp(5.8), exp(5.8)-   330.299559909649],_
+ [5.9,   365.037467865329,     exp(5.9), exp(5.9)-   365.037467865329],_
+ [6.0,   403.428793492735,     exp(6.0), exp(6.0)-   403.428793492735],_
+ [6.1,   445.857770082517,     exp(6.1), exp(6.1)-   445.857770082517],_
+ [6.2,   492.749041093256,     exp(6.2), exp(6.2)-   492.749041093256],_
+ [6.3,   544.571910125929,     exp(6.3), exp(6.3)-   544.571910125929],_
+ [6.4,   601.845037872082,     exp(6.4), exp(6.4)-   601.845037872082],_
+ [6.5,   665.141633044362,     exp(6.5), exp(6.5)-   665.141633044362],_
+ [6.6,   735.095189241973,     exp(6.6), exp(6.6)-   735.095189241973],_
+ [6.7,   812.405825167543,     exp(6.7), exp(6.7)-   812.405825167543],_
+ [6.8,   897.847291650418,     exp(6.8), exp(6.8)-   897.847291650418],_
+ [6.9,   992.274715605026,     exp(6.9), exp(6.9)-   992.274715605026],_
+ [7.0,  1096.633158428459,     exp(7.0), exp(7.0)-  1096.633158428459],_
+ [7.1,  1211.967074492577,     exp(7.1), exp(7.1)-  1211.967074492577],_
+ [7.2,  1339.430764394418,     exp(7.2), exp(7.2)-  1339.430764394418],_
+ [7.3,  1480.299927584545,     exp(7.3), exp(7.3)-  1480.299927584545],_
+ [7.4,  1635.984429995927,     exp(7.4), exp(7.4)-  1635.984429995927],_
+ [7.5,  1808.042414456063,     exp(7.5), exp(7.5)-  1808.042414456063],_
+ [7.6,  1998.195895104118,     exp(7.6), exp(7.6)-  1998.195895104118],_
+ [7.7,  2208.347991887209,     exp(7.7), exp(7.7)-  2208.347991887209],_
+ [7.8,  2440.601977624499,     exp(7.8), exp(7.8)-  2440.601977624499],_
+ [7.9,  2697.282328268509,     exp(7.9), exp(7.9)-  2697.282328268509],_
+ [8.0,  2980.957987041728,     exp(8.0), exp(8.0)-  2980.957987041728],_
+ [8.1,  3294.468075283841,     exp(8.1), exp(8.1)-  3294.468075283841],_
+ [8.2,  3640.950307332355,     exp(8.2), exp(8.2)-  3640.950307332355],_
+ [8.3,  4023.872393822310,     exp(8.3), exp(8.3)-  4023.872393822310],_
+ [8.4,  4447.066747699856,     exp(8.4), exp(8.4)-  4447.066747699856],_
+ [8.5,  4914.768840299134,     exp(8.5), exp(8.5)-  4914.768840299134],_
+ [8.6,  5431.659591362980,     exp(8.6), exp(8.6)-  5431.659591362980],_
+ [8.7,  6002.912217261022,     exp(8.7), exp(8.7)-  6002.912217261022],_
+ [8.8,  6634.244006277885,     exp(8.8), exp(8.8)-  6634.244006277885],_
+ [8.9,  7331.973539155993,     exp(8.9), exp(8.9)-  7331.973539155993],_
+ [9.0,  8103.083927575384,     exp(9.0), exp(9.0)-  8103.083927575384],_
+ [9.1,  8955.292703482512,     exp(9.1), exp(9.1)-  8955.292703482512],_
+ [9.2,  9897.129058743916,     exp(9.2), exp(9.2)-  9897.129058743916],_
+ [9.3, 10938.019208165184,     exp(9.3), exp(9.3)- 10938.019208165184],_
+ [9.4, 12088.380730216984,     exp(9.4), exp(9.4)- 12088.380730216984],_
+ [9.5, 13359.726829661872,     exp(9.5), exp(9.5)- 13359.726829661872],_
+ [9.6, 14764.781565577273,     exp(9.6), exp(9.6)- 14764.781565577273],_
+ [9.7, 16317.607198015432,     exp(9.7), exp(9.7)- 16317.607198015432],_
+ [9.8, 18033.744927828511,     exp(9.8), exp(9.8)- 18033.744927828511],_
+ [9.9, 19930.370438230289,     exp(9.9), exp(9.9)- 19930.370438230289],_
+[10.0, 22026.465794806717,    exp(10.0), exp(10.0)-22026.465794806717]]
+--R 
+--R
+--R   (1)
+--R   [[0.0,1.0,1.0,0.0],
+--R    [0.1,1.1051709180 75648,1.1051709180 756476248,- 0.3752 E -15],
+--R    [0.2,1.2214027581 6017,1.2214027581 601698339,- 0.1661 E -15],
+--R    [0.3,1.3498588075 76003,1.3498588075 76003104,0.104 E -15],
+--R    [0.4,1.4918246976 4127,1.4918246976 412703178,0.3178 E -15],
+--R    [0.5,1.6487212707 00128,1.6487212707 001281468,0.1468 E -15],
+--R    [0.6,1.8221188003 90509,1.8221188003 905089749,- 0.251 E -16],
+--R    [0.7,2.0137527074 70477,2.0137527074 704765216,- 0.4784 E -15],
+--R    [0.8,2.2255409284 92468,2.2255409284 924676046,- 0.3954 E -15],
+--R    [0.9,2.4596031111 5695,2.4596031111 569496638,- 0.3362 E -15],
+--R    [1.0,2.7182818284 59045,2.7182818284 590452354,0.2354 E -15],
+--R    [1.1,3.0041660239 46433,3.0041660239 464331121,0.112 E -15],
+--R    [1.2,3.3201169227 36547,3.3201169227 365474895,0.4895 E -15],
+--R    [1.3,3.6692966676 19244,3.6692966676 192442204,0.22 E -15],
+--R    [1.4,4.0551999668 44675,4.0551999668 446745872,- 0.413 E -15],
+--R    [1.5,4.4816890703 38065,4.4816890703 380648226,- 0.177 E -15],
+--R    [1.6,4.9530324243 95115,4.9530324243 951148037,- 0.196 E -15],
+--R    [1.7,5.4739473917 272,5.4739473917 271997608,- 0.239 E -15],
+--R    [1.8,6.0496474644 12946,6.0496474644 129460837,0.837 E -16],
+--R    [1.9,6.6858944422 79269,6.6858944422 792694161,0.416 E -15],
+--R    [2.0,7.3890560989 3065,7.3890560989 306502272,0.227 E -15],
+--R    [2.1,8.1661699125 6765,8.1661699125 676500734,0.734 E -16],
+--R    [2.2,9.0250134994 34121,9.0250134994 341209265,- 0.735 E -16],
+--R    [2.3,9.9741824548 14721,9.9741824548 1472074,- 0.26 E -15],
+--R    [2.4,11.0231763806 41602,11.0231763806 41601652,- 0.348 E -15],
+--R    [2.5,12.1824939607 03473,12.1824939607 03473438,0.438 E -15],
+--R    [2.6,13.4637380350 0169,13.4637380350 01690398,0.398 E -15],
+--R    [2.7,14.8797317248 72834,14.8797317248 72834112,0.112 E -15],
+--R    [2.8,16.4446467710 9705,16.4446467710 97049871,- 0.129 E -15],
+--R    [2.9,18.1741453694 43061,18.1741453694 43060943,- 0.57 E -16],
+--R    [3.0,20.0855369231 87668,20.0855369231 87667741,- 0.259 E -15],
+--R    [3.1,22.1979512814 41633,22.1979512814 41633405,0.405 E -15],
+--R    [3.2,24.5325301971 09349,24.5325301971 09348644,- 0.356 E -15],
+--R    [3.3,27.1126389206 57887,27.1126389206 57887427,0.427 E -15],
+--R    [3.4,29.9641000473 97013,29.9641000473 97013348,0.348 E -15],
+--R    [3.5,33.1154519586 92314,33.1154519586 92313751,- 0.249 E -15],
+--R    [3.6,36.5982344436 77988,36.5982344436 77987752,- 0.248 E -15],
+--R    [3.7,40.4473043600 67391,40.4473043600 67390529,- 0.471 E -15],
+--R    [3.8,44.7011844933 00823,44.7011844933 00823038,0.38 E -16],
+--R    [3.9,49.4024491055 30174,49.4024491055 3017388,- 0.12 E -15],
+--R    [4.0,54.5981500331 44239,54.5981500331 44239078,0.78 E -16],
+--R    [4.1,60.3402875973 61969,60.3402875973 61969497,0.497 E -15],
+--R    [4.2,66.6863310409 25142,66.6863310409 25141644,- 0.36 E -15],
+--R    [4.3,73.6997936995 95797,73.6997936995 95796913,- 0.88 E -16],
+--R    [4.4,81.4508686649 68117,81.4508686649 68117445,0.445 E -15],
+--R    [4.5,90.0171313005 21814,90.0171313005 2181355,- 0.45 E -15],
+--R    [4.6,99.4843156419 33809,99.4843156419 33808735,- 0.26 E -15],
+--R    [4.7,109.9471724521 23499,109.9471724521 2349888,- 0.12 E -15],
+--R    [4.8,121.5104175187 34881,121.5104175187 3488076,- 0.24 E -15],
+--R    [4.9,134.2897796849 35485,134.2897796849 3548484,- 0.16 E -15],
+--R    [5.0,148.4131591025 77,148.4131591025 7660342,- 0.39658 E -12],
+--R    [5.1,164.0219072999 02,164.0219072999 0174394,- 0.25606 E -12],
+--R    [5.2,181.2722418751 51,181.2722418751 5117937,0.17937 E -12],
+--R    [5.3,200.3368099747 92,200.3368099747 9168484,- 0.31516 E -12],
+--R    [5.4,221.4064162041 87,221.4064162041 8708703,0.8703 E -13],
+--R    [5.5,244.6919322642 2,244.6919322642 2038792,0.38792 E -12],
+--R    [5.6,270.4264074261 53,270.4264074261 5262815,- 0.37185 E -12],
+--R    [5.7,298.8674009670 6,298.8674009670 6023267,0.23267 E -12],
+--R    [5.8,330.2995599096 49,330.2995599096 4865412,- 0.34588 E -12],
+--R    [5.9,365.0374678653 29,365.0374678653 2877732,- 0.2227 E -12],
+--R    [6.0,403.4287934927 35,403.4287934927 3512261,0.1226 E -12],
+--R    [6.1,445.8577700825 17,445.8577700825 1693179,- 0.6821 E -13],
+--R    [6.2,492.7490410932 56,492.7490410932 5625456,0.25456 E -12],
+--R    [6.3,544.5719101259 29,544.5719101259 2903306,0.331 E -13],
+--R    [6.4,601.8450378720 82,601.8450378720 8205661,0.566 E -13],
+--R    [6.5,665.1416330443 62,665.1416330443 6184069,- 0.1593 E -12],
+--R    [6.6,735.0951892419 73,735.0951892419 728949,- 0.1051 E -12],
+--R    [6.7,812.4058251675 43,812.4058251675 4311346,0.1135 E -12],
+--R    [6.8,897.8472916504 18,897.8472916504 1769759,- 0.3024 E -12],
+--R    [6.9,992.2747156050 26,992.2747156050 2587698,- 0.123 E -12],
+--R    [7.0,1096.6331584284 59,1096.6331584284 585993,- 0.4007 E -12],
+--R    [7.1,1211.9670744925 77,1211.9670744925 767212,- 0.2788 E -12],
+--R    [7.2,1339.4307643944 18,1339.4307643944 178297,- 0.1703 E -12],
+--R    [7.3,1480.2999275845 45,1480.2999275845 452229,0.2228 E -12],
+--R    [7.4,1635.9844299959 27,1635.9844299959 265401,- 0.4599 E -12],
+--R    [7.5,1808.0424144560 63,1808.0424144560 632069,0.2069 E -12],
+--R    [7.6,1998.1958951041 18,1998.1958951041 179592,- 0.408 E -13],
+--R    [7.7,2208.3479918872 09,2208.3479918872 08524,- 0.476 E -12],
+--R    [7.8,2440.6019776244 99,2440.6019776244 990773,0.773 E -13],
+--R    [7.9,2697.2823282685 09,2697.2823282685 088472,- 0.153 E -12],
+--R    [8.0,2980.9579870417 28,2980.9579870417 282747,0.2747 E -12],
+--R    [8.1,3294.4680752838 41,3294.4680752838 413332,0.3332 E -12],
+--R    [8.2,3640.9503073323 55,3640.9503073323 547215,- 0.2785 E -12],
+--R    [8.3,4023.8723938223 1,4023.8723938223 098416,- 0.158 E -12],
+--R    [8.4,4447.0667476998 56,4447.0667476998 560855,0.855 E -13],
+--R    [8.5,4914.7688402991 34,4914.7688402991 343754,0.375 E -12],
+--R    [8.6,5431.6595913629 8,5431.6595913629 803217,0.322 E -12],
+--R    [8.7,6002.9122172610 22,6002.9122172610 2198,- 0.2 E -13],
+--R    [8.8,6634.2440062778 85,6634.2440062778 851586,0.159 E -12],
+--R    [8.9,7331.9735391559 93,7331.9735391559 929051,- 0.949 E -13],
+--R    [9.0,8103.0839275753 84,8103.0839275753 840077,0.77 E -14],
+--R    [9.1,8955.2927034825 12,8955.2927034825 11711,- 0.289 E -12],
+--R    [9.2,9897.1290587439 16,9897.1290587439 158867,- 0.113 E -12],
+--R    [9.3,10938.0192081651 84,10938.0192081651 83753,- 0.247 E -12],
+--R    [9.4,12088.3807302169 84,12088.3807302169 84397,0.397 E -12],
+--R    [9.5,13359.7268296618 72,13359.7268296618 72276,0.276 E -12],
+--R    [9.6,14764.7815655772 73,14764.7815655772 72616,- 0.384 E -12],
+--R    [9.7,16317.6071980154 32,16317.6071980154 32233,0.233 E -12],
+--R    [9.8,18033.7449278285 11,18033.7449278285 11246,0.246 E -12],
+--R    [9.9,19930.3704382302 89,19930.3704382302 8949,0.49 E -12],
+--R    [10.0,22026.4657948067 17,22026.4657948067 16517,- 0.483 E -12]]
+--R                                                        Type: List List Float
+--E 1
+
+@ 
+In the following table there are 4 columns. The first column
+is the argument of exp(-x), ranging from 0.0 to 10.0. The
+second column is the reference value of exp from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp136-137
+
+<<*>>=
+--S 2 of 2
+[[0.1,0.90483741803595957316,exp(-0.1),exp(-0.1)-0.90483741803595957316],_
+ [0.2,0.81873075307798185867,exp(-0.2),exp(-0.2)-0.81873075307798185867],_
+ [0.3,0.74081822068171786607,exp(-0.3),exp(-0.3)-0.74081822068171786607],_
+ [0.4,0.67032004603563930074,exp(-0.4),exp(-0.4)-0.67032004603563930074],_
+ [0.5,0.60653065971263342360,exp(-0.5),exp(-0.5)-0.60653065971263342360],_
+ [0.6,0.54881163609402643263,exp(-0.6),exp(-0.6)-0.54881163609402643263],_
+ [0.7,0.49658530379140951470,exp(-0.7),exp(-0.7)-0.49658530379140951470],_
+ [0.8,0.44932896411722159143,exp(-0.8),exp(-0.8)-0.44932896411722159143],_
+ [0.9,0.40656965974059911188,exp(-0.9),exp(-0.9)-0.40656965974059911188],_
+ [1.0,0.36787944117144232160,exp(-1.0),exp(-1.0)-0.36787944117144232160],_
+ [1.1,0.33287108369807955329,exp(-1.1),exp(-1.1)-0.33287108369807955329],_
+ [1.2,0.30119421191220209664,exp(-1.2),exp(-1.2)-0.30119421191220209664],_
+ [1.3,0.27253179303401260312,exp(-1.3),exp(-1.3)-0.27253179303401260312],_
+ [1.4,0.24659696394160647694,exp(-1.4),exp(-1.4)-0.24659696394160647694],_
+ [1.5,0.22313016014842982893,exp(-1.5),exp(-1.5)-0.22313016014842982893],_
+ [1.6,0.20189651799465540849,exp(-1.6),exp(-1.6)-0.20189651799465540849],_
+ [1.7,0.18268352405273465022,exp(-1.7),exp(-1.7)-0.18268352405273465022],_
+ [1.8,0.16529888822158653830,exp(-1.8),exp(-1.8)-0.16529888822158653830],_
+ [1.9,0.14956861922263505264,exp(-1.9),exp(-1.9)-0.14956861922263505264],_
+ [2.0,0.13533528323661269189,exp(-2.0),exp(-2.0)-0.13533528323661269189],_
+ [2.1,0.12245642825298191022,exp(-2.1),exp(-2.1)-0.12245642825298191022],_
+ [2.2,0.11080315836233388333,exp(-2.2),exp(-2.2)-0.11080315836233388333],_
+ [2.3,0.10025884372280373373,exp(-2.3),exp(-2.3)-0.10025884372280373373],_
+ [2.4,0.09071795328941250338,exp(-2.4),exp(-2.4)-0.09071795328941250338],_
+ [2.5,0.08208499862389879517,exp(-2.5),exp(-2.5)-0.08208499862389879517],_
+ [2.6,0.07427357821433388043,exp(-2.6),exp(-2.6)-0.07427357821433388043],_
+ [2.7,0.06720551273974976513,exp(-2.7),exp(-2.7)-0.06720551273974976513],_
+ [2.8,0.06081006262521796500,exp(-2.8),exp(-2.8)-0.06081006262521796500],_
+ [2.9,0.05502322005640722903,exp(-2.9),exp(-2.9)-0.05502322005640722903],_
+ [3.0,0.04978706836786394298,exp(-3.0),exp(-3.0)-0.04978706836786394298],_
+ [3.1,0.04504920239355780607,exp(-3.1),exp(-3.1)-0.04504920239355780607],_
+ [3.2,0.04076220397836621517,exp(-3.2),exp(-3.2)-0.04076220397836621517],_
+ [3.3,0.03688316740124000545,exp(-3.3),exp(-3.3)-0.03688316740124000545],_
+ [3.4,0.03337326996032607948,exp(-3.4),exp(-3.4)-0.03337326996032607948],_
+ [3.5,0.03019738342231850074,exp(-3.5),exp(-3.5)-0.03019738342231850074],_
+ [3.6,0.02732372244729256080,exp(-3.6),exp(-3.6)-0.02732372244729256080],_
+ [3.7,0.02472352647033939120,exp(-3.7),exp(-3.7)-0.02472352647033939120],_
+ [3.8,0.02237077185616559578,exp(-3.8),exp(-3.8)-0.02237077185616559578],_
+ [3.9,0.02024191144580438847,exp(-3.9),exp(-3.9)-0.02024191144580438847],_
+ [4.0,0.01831563888873418029,exp(-4.0),exp(-4.0)-0.01831563888873418029],_
+ [4.1,0.01657267540176124754,exp(-4.1),exp(-4.1)-0.01657267540176124754],_
+ [4.2,0.01499557682047770621,exp(-4.2),exp(-4.2)-0.01499557682047770621],_
+ [4.3,0.01356855901220093176,exp(-4.3),exp(-4.3)-0.01356855901220093176],_
+ [4.4,0.01227733990306844118,exp(-4.4),exp(-4.4)-0.01227733990306844118],_
+ [4.5,0.01110899653824230650,exp(-4.5),exp(-4.5)-0.01110899653824230650],_
+ [4.6,0.01005183574463358164,exp(-4.6),exp(-4.6)-0.01005183574463358164],_
+ [4.7,0.00909527710169581709,exp(-4.7),exp(-4.7)-0.00909527710169581709],_
+ [4.8,0.00822974704902002884,exp(-4.8),exp(-4.8)-0.00822974704902002884],_
+ [4.9,0.00744658307092434052,exp(-4.9),exp(-4.9)-0.00744658307092434052],_
+ [5.0,0.00673794699908546710,exp(-5.0),exp(-5.0)-0.00673794699908546710],_
+ [5.1,0.00609674656551563611,exp(-5.1),exp(-5.1)-0.00609674656551563611],_
+ [5.2,0.00551656442076077242,exp(-5.2),exp(-5.2)-0.00551656442076077242],_
+ [5.3,0.00499159390691021621,exp(-5.3),exp(-5.3)-0.00499159390691021621],_
+ [5.4,0.00451658094261266798,exp(-5.4),exp(-5.4)-0.00451658094261266798],_
+ [5.5,0.00408677143846406699,exp(-5.5),exp(-5.5)-0.00408677143846406699],_
+ [5.6,0.00369786371648293082,exp(-5.6),exp(-5.6)-0.00369786371648293082],_
+ [5.7,0.00334596545747127277,exp(-5.7),exp(-5.7)-0.00334596545747127277],_
+ [5.8,0.00302755474537581475,exp(-5.8),exp(-5.8)-0.00302755474537581475],_
+ [5.9,0.00273944481876836923,exp(-5.9),exp(-5.9)-0.00273944481876836923],_
+ [6.0,0.00247875217666635842,exp(-6.0),exp(-6.0)-0.00247875217666635842],_
+ [6.1,0.00224286771948580247,exp(-6.1),exp(-6.1)-0.00224286771948580247],_
+ [6.2,0.00202943063629573436,exp(-6.2),exp(-6.2)-0.00202943063629573436],_
+ [6.3,0.00183630477702890683,exp(-6.3),exp(-6.3)-0.00183630477702890683],_
+ [6.4,0.00166155727317393450,exp(-6.4),exp(-6.4)-0.00166155727317393450],_
+ [6.5,0.00150343919297757245,exp(-6.5),exp(-6.5)-0.00150343919297757245],_
+ [6.6,0.00136036803754789342,exp(-6.6),exp(-6.6)-0.00136036803754789342],_
+ [6.7,0.00123091190267348118,exp(-6.7),exp(-6.7)-0.00123091190267348118],_
+ [6.8,0.00111377514784480308,exp(-6.8),exp(-6.8)-0.00111377514784480308],_
+ [6.9,0.00100778542904851076,exp(-6.9),exp(-6.9)-0.00100778542904851076],_
+ [7.0,0.00091188196555451621,exp(-7.0),exp(-7.0)-0.00091188196555451621],_
+ [7.1,0.00082510492326590427,exp(-7.1),exp(-7.1)-0.00082510492326590427],_
+ [7.2,0.00074658580837667937,exp(-7.2),exp(-7.2)-0.00074658580837667937],_
+ [7.3,0.00067553877519384424,exp(-7.3),exp(-7.3)-0.00067553877519384424],_
+ [7.4,0.00061125276112957256,exp(-7.4),exp(-7.4)-0.00061125276112957256],_
+ [7.5,0.00055308437014783358,exp(-7.5),exp(-7.5)-0.00055308437014783358],_
+ [7.6,0.00050045143344061070,exp(-7.6),exp(-7.6)-0.00050045143344061070],_
+ [7.7,0.00045282718288679706,exp(-7.7),exp(-7.7)-0.00045282718288679706],_
+ [7.8,0.00040973497897978671,exp(-7.8),exp(-7.8)-0.00040973497897978671],_
+ [7.9,0.00037074354045908837,exp(-7.9),exp(-7.9)-0.00037074354045908837],_
+ [8.0,0.00033546262790251184,exp(-8.0),exp(-8.0)-0.00033546262790251184],_
+ [8.1,0.00030353913807886666,exp(-8.1),exp(-8.1)-0.00030353913807886666],_
+ [8.2,0.00027465356997214233,exp(-8.2),exp(-8.2)-0.00027465356997214233],_
+ [8.3,0.00024851682710795202,exp(-8.3),exp(-8.3)-0.00024851682710795202],_
+ [8.4,0.00022486732417884827,exp(-8.4),exp(-8.4)-0.00022486732417884827],_
+ [8.5,0.00020346836901064417,exp(-8.5),exp(-8.5)-0.00020346836901064417],_
+ [8.6,0.00018410579366757912,exp(-8.6),exp(-8.6)-0.00018410579366757912],_
+ [8.7,0.00016658581098763341,exp(-8.7),exp(-8.7)-0.00016658581098763341],_
+ [8.8,0.00015073307509547660,exp(-8.8),exp(-8.8)-0.00015073307509547660],_
+ [8.9,0.00013638892648201145,exp(-8.9),exp(-8.9)-0.00013638892648201145],_
+ [9.0,0.00012340980408667955,exp(-9.0),exp(-9.0)-0.00012340980408667955],_
+ [9.1,0.00011166580849011474,exp(-9.1),exp(-9.1)-0.00011166580849011474],_
+ [9.2,0.00010103940183709335,exp(-9.2),exp(-9.2)-0.00010103940183709335],_
+ [9.3,0.00009142423147817334,exp(-9.3),exp(-9.3)-0.00009142423147817334],_
+ [9.4,0.00008272406555663226,exp(-9.4),exp(-9.4)-0.00008272406555663226],_
+ [9.5,0.00007485182988770059,exp(-9.5),exp(-9.5)-0.00007485182988770059],_
+ [9.6,0.00006772873649085387,exp(-9.6),exp(-9.6)-0.00006772873649085387],_
+ [9.7,0.00006128349505322210,exp(-9.7),exp(-9.7)-0.00006128349505322210],_
+ [9.8,0.00005545159943217698,exp(-9.8),exp(-9.8)-0.00005545159943217698],_
+ [9.9,0.00005017468205617530,exp(-9.9),exp(-9.9)-0.00005017468205617530],_
+[10.0,0.00004539992976248485,exp(-10.0),exp(-10.0)-0.00004539992976248485]]
+--R 
+--R
+--R   (2)
+--R   [[0.1,0.9048374180 3595957316,0.9048374180 3595957316,0.3 E -20],
+--R    [0.2,0.8187307530 7798185867,0.8187307530 7798185867,0.0],
+--R    [0.3,0.7408182206 8171786607,0.7408182206 8171786606,- 0.7 E -20],
+--R    [0.4,0.6703200460 3563930074,0.6703200460 3563930075,0.3 E -20],
+--R    [0.5,0.6065306597 126334236,0.6065306597 126334236,0.3 E -20],
+--R    [0.6,0.5488116360 9402643263,0.5488116360 9402643263,- 0.3 E -20],
+--R    [0.7,0.4965853037 914095147,0.4965853037 9140951471,0.5 E -20],
+--R    [0.8,0.4493289641 1722159143,0.4493289641 1722159143,0.0],
+--R    [0.9,0.4065696597 4059911188,0.4065696597 4059911188,0.3 E -20],
+--R    [1.0,0.3678794411 714423216,0.3678794411 7144232159,- 0.5 E -20],
+--R    [1.1,0.3328710836 9807955329,0.3328710836 9807955329,- 0.2 E -20],
+--R    [1.2,0.3011942119 1220209664,0.3011942119 1220209664,0.3 E -20],
+--R    [1.3,0.2725317930 3401260312,0.2725317930 3401260312,0.3 E -20],
+--R    [1.4,0.2465969639 4160647694,0.2465969639 4160647694,0.0],
+--R    [1.5,0.2231301601 4842982893,0.2231301601 4842982893,0.3 E -20],
+--R    [1.6,0.2018965179 9465540849,0.2018965179 9465540848,- 0.5 E -20],
+--R    [1.7,0.1826835240 5273465022,0.1826835240 5273465022,0.3 E -20],
+--R    [1.8,0.1652988882 215865383,0.1652988882 215865383,- 0.3 E -20],
+--R    [1.9,0.1495686192 2263505264,0.1495686192 2263505264,0.8 E -21],
+--R    [2.0,0.1353352832 3661269189,0.1353352832 3661269189,0.4 E -20],
+--R    [2.1,0.1224564282 5298191022,0.1224564282 5298191022,- 0.8 E -21],
+--R    [2.2,0.1108031583 6233388333,0.1108031583 6233388333,0.4 E -20],
+--R    [2.3,0.1002588437 2280373373,0.1002588437 2280373373,0.0],
+--R    [2.4,0.0907179532 8941250338,0.0907179532 8941250337 5,- 0.6 E -20],
+--R    [2.5,0.0820849986 2389879517,0.0820849986 2389879516 9,- 0.4 E -21],
+--R    [2.6,0.0742735782 1433388043,0.0742735782 1433388042 9,- 0.1 E -20],
+--R    [2.7,0.0672055127 3974976513,0.0672055127 3974976512 6,- 0.4 E -20],
+--R    [2.8,0.0608100626 25217965,0.0608100626 2521796499 6,- 0.4 E -20],
+--R    [2.9,0.0550232200 5640722903,0.0550232200 5640722903,- 0.4 E -21],
+--R    [3.0,0.0497870683 6786394298,0.0497870683 6786394297 9,- 0.6 E -21],
+--R    [3.1,0.0450492023 9355780607,0.0450492023 9355780606 9,- 0.1 E -20],
+--R    [3.2,0.0407622039 7836621517,0.0407622039 7836621516 6,- 0.4 E -20],
+--R    [3.3,0.0368831674 0124000545,0.0368831674 0124000544 6,- 0.4 E -20],
+--R    [3.4,0.0333732699 6032607948,0.0333732699 6032607948 2,0.2 E -20],
+--R    [3.5,0.0301973834 2231850074,0.0301973834 2231850074,- 0.2 E -21],
+--R    [3.6,0.0273237224 472925608,0.0273237224 4729256080 2,0.2 E -20],
+--R    [3.7,0.0247235264 703393912,0.0247235264 7033939120 3,0.3 E -20],
+--R    [3.8,0.0223707718 5616559578,0.0223707718 5616559577 9,- 0.1 E -20],
+--R    [3.9,0.0202419114 4580438847,0.0202419114 4580438847 2,0.2 E -20],
+--R    [4.0,0.0183156388 8873418029,0.0183156388 8873418029 4,0.4 E -20],
+--R    [4.1,0.0165726754 0176124754,0.0165726754 0176124754 2,0.2 E -20],
+--R    [4.2,0.0149955768 2047770621,0.0149955768 2047770621 2,0.2 E -20],
+--R    [4.3,0.0135685590 1220093176,0.0135685590 1220093175 7,- 0.3 E -20],
+--R    [4.4,0.0122773399 0306844118,0.0122773399 0306844117 9,- 0.1 E -20],
+--R    [4.5,0.0111089965 382423065,0.0111089965 3824230649 6,- 0.4 E -20],
+--R    [4.6,0.0100518357 4463358164,0.0100518357 4463358164 2,0.2 E -20],
+--R    [4.7,0.0090952771 0169581709,0.0090952771 0169581709 21,0.2 E -20],
+--R    [4.8,0.0082297470 4902002884,0.0082297470 4902002884 13,0.1 E -20],
+--R    [4.9,0.0074465830 7092434052,0.0074465830 7092434051 82,- 0.2 E -20],
+--R    [5.0,0.0067379469 990854671,0.0067379469 9908546709 66,- 0.3 E -20],
+--R    [5.1,0.0060967465 6551563611,0.0060967465 6551563610 72,- 0.3 E -20],
+--R    [5.2,0.0055165644 2076077242,0.0055165644 2076077241 81,- 0.2 E -20],
+--R    [5.3,0.0049915939 0691021621,0.0049915939 0691021621 22,0.2 E -20],
+--R    [5.4,0.0045165809 4261266798,0.0045165809 4261266798 16,0.2 E -20],
+--R    [5.5,0.0040867714 3846406699,0.0040867714 3846406699 35,0.35 E -20],
+--R    [5.6,0.0036978637 1648293082,0.0036978637 1648293082 07,0.7 E -21],
+--R    [5.7,0.0033459654 5747127277,0.0033459654 5747127276 58,- 0.42 E -20],
+--R    [5.8,0.0030275547 4537581475,0.0030275547 4537581474 82,- 0.18 E -20],
+--R    [5.9,0.0027394448 1876836923,0.0027394448 1876836923 28,0.28 E -20],
+--R    [6.0,0.0024787521 7666635842,0.0024787521 7666635842 3,0.3 E -20],
+--R    [6.1,0.0022428677 1948580247,0.0022428677 1948580247 32,0.32 E -20],
+--R    [6.2,0.0020294306 3629573436,0.0020294306 3629573436 34,0.34 E -20],
+--R    [6.3,0.0018363047 7702890683,0.0018363047 7702890682 52,- 0.48 E -20],
+--R    [6.4,0.0016615572 731739345,0.0016615572 7317393449 91,- 0.93 E -21],
+--R    [6.5,0.0015034391 9297757245,0.0015034391 9297757244 74,- 0.26 E -20],
+--R    [6.6,0.0013603680 3754789342,0.0013603680 3754789341 69,- 0.31 E -20],
+--R    [6.7,0.0012309119 0267348118,0.0012309119 0267348118 46,0.46 E -20],
+--R    [6.8,0.0011137751 4784480308,0.0011137751 4784480307 88,- 0.12 E -20],
+--R    [6.9,0.0010077854 2904851076,0.0010077854 2904851076 14,0.14 E -20],
+--R    [7.0,0.0009118819 6555451621,0.0009118819 6555451620 8,- 0.2 E -20],
+--R    [7.1,0.0008251049 2326590427,0.0008251049 2326590427 015,0.1 E -21],
+--R    [7.2,0.0007465858 0837667937,0.0007465858 0837667936 81,- 0.19 E -20],
+--R    [7.3,0.0006755387 7519384424,0.0006755387 7519384423 783,- 0.22 E -20],
+--R    [7.4,0.0006112527 6112957256,0.0006112527 6112957255 567,- 0.433 E -20],
+--R    [7.5,0.0005530843 7014783358,0.0005530843 7014783358 31,0.31 E -20],
+--R    [7.6,0.0005004514 334406107,0.0005004514 3344061069 551,- 0.449 E -20],
+--R    [7.7,0.0004528271 8288679706,0.0004528271 8288679705 8,- 0.2 E -20],
+--R    [7.8,0.0004097349 7897978671,0.0004097349 7897978670 846,- 0.15 E -20],
+--R    [7.9,0.0003707435 4045908837,0.0003707435 4045908837 443,0.443 E -20],
+--R    [8.0,0.0003354626 2790251184,0.0003354626 2790251183 882,- 0.12 E -20],
+--R    [8.1,0.0003035391 3807886666,0.0003035391 3807886666 086,0.86 E -21],
+--R    [8.2,0.0002746535 6997214233,0.0002746535 6997214232 763,- 0.237 E -20],
+--R    [8.3,0.0002485168 2710795202,0.0002485168 2710795202 08,0.8 E -21],
+--R    [8.4,0.0002248673 2417884827,0.0002248673 2417884827 28,0.28 E -20],
+--R    [8.5,0.0002034683 6901064417,0.0002034683 6901064417 437,0.437 E -20],
+--R    [8.6,0.0001841057 9366757912,0.0001841057 9366757912 495,0.495 E -20],
+--R    [8.7,0.0001665858 1098763341,0.0001665858 1098763341 149,0.149 E -20],
+--R    [8.8,0.0001507330 750954766,0.0001507330 7509547660 064,0.64 E -21],
+--R    [8.9,0.0001363889 2648201145,0.0001363889 2648201144 785,- 0.215 E -20],
+--R    [9.0,0.0001234098 0408667955,0.0001234098 0408667954 95,- 0.5 E -21],
+--R    [9.1,0.0001116658 0849011474,0.0001116658 0849011473 564,- 0.436 E -20],
+--R    [9.2,0.0001010394 0183709335,0.0001010394 0183709335 073,0.732 E -21],
+--R    [9.3,0.0000914242 3147817334,0.0000914242 3147817333 7862,- 0.214 E -20],
+--R    [9.4,0.0000827240 6555663226,0.0000827240 6555663226 2731,0.273 E -20],
+--R    [9.5,0.0000748518 2988770059,0.0000748518 2988770059 1471,0.147 E -20],
+--R    [9.6,0.0000677287 3649085387,0.0000677287 3649085387 2996,0.3 E -20],
+--R    [9.7,0.0000612834 950532221,0.0000612834 9505322209 5514,- 0.449 E -20],
+--R    [9.8,0.0000554515 9943217698,0.0000554515 9943217698 1808,0.181 E -20],
+--R    [9.9,0.0000501746 820561753,0.0000501746 8205617530 2187,0.219 E -20],
+--R    [10.0,0.0000453999 2976248485,0.0000453999 2976248485 1536,0.154 E -20]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp136-137
+\end{thebibliography}
+\end{document}
+
diff --git a/src/input/gamma.input.pamphlet b/src/input/gamma.input.pamphlet
index 608cf75..6b5f38b 100644
--- a/src/input/gamma.input.pamphlet
+++ b/src/input/gamma.input.pamphlet
@@ -17,19 +17,13 @@ second column is the reference value of Gamma from the book
 Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp267-270
 
-Notice that certain entries have reasonably large differences:
-\begin{verbatim}
-[1.215, 0.9143261399876792,  0.91432614997006778, 9.9823885824079639E-9]
-[1.455, 0.88562206999631599, 0.88562207999314335, 9.9968273659101214E-9]
-\end{verbatim}
-
 <<*>>=
 )spool gamma.output
 )set message test on
 )set message auto off
 )clear all
  
---S 1 of 3
+--S 1 of 4
 [[1.000,1.0000000000,Gamma(1.000),Gamma(1.000)-1.0000000000],_
  [1.005,0.9971385354,Gamma(1.005),Gamma(1.005)-0.9971385354],_
  [1.010,0.9943258512,Gamma(1.010),Gamma(1.010)-0.9943258512],_
@@ -73,7 +67,7 @@ Notice that certain entries have reasonably large differences:
  [1.200,0.9181687424,Gamma(1.200),Gamma(1.200)-0.9181687424],_
  [1.205,0.9168572606,Gamma(1.205),Gamma(1.205)-0.9168572606],_
  [1.210,0.9155764930,Gamma(1.210),Gamma(1.210)-0.9155764930],_
- [1.215,0.9143261400,Gamma(1.215),Gamma(1.215)-0.9143261400],_
+ [1.215,0.9143261500,Gamma(1.215),Gamma(1.215)-0.9143261500],_
  [1.220,0.9131059475,Gamma(1.220),Gamma(1.220)-0.9131059475],_
  [1.225,0.9119156071,Gamma(1.225),Gamma(1.225)-0.9119156071],_
  [1.230,0.9107548564,Gamma(1.230),Gamma(1.230)-0.9107548564],_
@@ -121,7 +115,7 @@ Notice that certain entries have reasonably large differences:
  [1.440,0.8858050635,Gamma(1.440),Gamma(1.440)-0.8858050635],_
  [1.445,0.8857223397,Gamma(1.445),Gamma(1.445)-0.8857223397],_
  [1.450,0.8856613803,Gamma(1.450),Gamma(1.450)-0.8856613803],_
- [1.455,0.8856220700,Gamma(1.455),Gamma(1.455)-0.8856220700],_
+ [1.455,0.8856220800,Gamma(1.455),Gamma(1.455)-0.8856220800],_
  [1.460,0.8856043364,Gamma(1.460),Gamma(1.460)-0.8856043364],_
  [1.465,0.8856080495,Gamma(1.465),Gamma(1.465)-0.8856080495],_
  [1.470,0.8856331217,Gamma(1.470),Gamma(1.470)-0.8856331217],_
@@ -349,8 +343,8 @@ Notice that certain entries have reasonably large differences:
 --R     ,
 --R    [1.21,0.91557649299999999,0.91557649300335908,3.3590907833058736E-12],
 --R
---R     [1.2150000000000001, 0.91432614000000001, 0.91432614997639472,
---R      9.9763947103426176E-9]
+--R     [1.2150000000000001, 0.91432614999999995, 0.91432614997639472,
+--R      - 2.3605228882672691E-11]
 --R     ,
 --R    [1.22,0.91310594749999996,0.9131059475104113,1.041133845802733E-11],
 --R
@@ -475,8 +469,8 @@ Notice that certain entries have reasonably large differences:
 --R     ,
 --R    [1.45,0.88566138029999997,0.88566138027107244,- 2.8927527040423229E-11],
 --R
---R     [1.4550000000000001, 0.88562207000000004, 0.88562207999323628,
---R      9.9932362385146689E-9]
+--R     [1.4550000000000001, 0.88562207999999998, 0.88562207999323628,
+--R      - 6.7637007106213787E-12]
 --R     ,
 --R    [1.46,0.88560433640000003,0.88560433642514391,2.5143886972500695E-11],
 --R
@@ -762,7 +756,7 @@ Notice that certain entries have reasonably large differences:
 --R                                                  Type: List List DoubleFloat
 --E 1
 
---S 2 of 3
+--S 2 of 4
 Psi(x:DFLOAT):DFLOAT==polygamma(0,x)
 --R 
 --R   Function declaration Psi : DoubleFloat -> DoubleFloat has been added
@@ -778,7 +772,7 @@ Dover Publications, Inc. New York 1965. pp267-270
 
 <<*>>=
 
---S 3 of 3
+--S 3 of 4
 [[1.000, -0.5772156649, Psi(1.000), Psi(1.000)- -0.5772156649],_
  [1.005, -0.5690209113, Psi(1.005), Psi(1.005)- -0.5690209113],_
  [1.010, -0.5608854579, Psi(1.010), Psi(1.010)- -0.5608854579],_
@@ -1540,6 +1534,836 @@ Dover Publications, Inc. New York 1965. pp267-270
 --R    [2.,0.42278433510000002,0.42278433509846725,- 1.5327739077974911E-12]]
 --R                                                  Type: List List DoubleFloat
 --E 3
+@
+In the following table there are 4 columns. The first column
+is the argument of complex Gamma, ranging from 1.0 + 0i to 1.0+10i
+second column is the reference value of complex Gamma from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp227
+
+Note that at the value 1.0+4.6i there is a radical departure between
+the table and the computed values in the imaginary part of the value
+even though the real part is exact.
+<<*>>=
+--S 4 of 4
+[[1. + 0.0 * %i,0.,log(Gamma(1. + 0.0 * %i)),log(Gamma(1. + 0.0 * %i))-0.0],_
+[1. + 0.1 * %i, -0.008197780565 - 0.057322940417 * %i,_
+log(Gamma(1. + 0.1 * %i)),_
+log(Gamma(1. + 0.1 * %i))-( -0.008197780565 - 0.057322940417 * %i)],_
+[1. + 0.2 * %i, -0.032476292318 - 0.112302222644 * %i,_
+log(Gamma(1. + 0.2 * %i)),_
+log(Gamma(1. + 0.2 * %i))-( -0.032476292318 - 0.112302222644 * %i)],_
+[1. + 0.3 * %i, -0.071946250900 - 0.162820672168 * %i,_
+log(Gamma(1. + 0.3 * %i)),_
+log(Gamma(1. + 0.3 * %i))-( -0.071946250900 - 0.162820672168 * %i)],_
+[1. + 0.4 * %i, -0.125289374821 - 0.207155826316 * %i,_
+log(Gamma(1. + 0.4 * %i)),_
+log(Gamma(1. + 0.4 * %i))-( -0.125289374821 - 0.207155826316 * %i)],_
+[1. + 0.5 * %i,- 0.190945499187 - 0.244058298905 * %i,_
+log(Gamma(1. + 0.5 * %i)),_
+log(Gamma(1. + 0.5 * %i))-(- 0.190945499187 - 0.244058298905 * %i)],_
+[1. + 0.6 * %i,- 0.267290068214 - 0.272743810491 * %i,_
+log(Gamma(1. + 0.6 * %i)),_
+log(Gamma(1. + 0.6 * %i))-(- 0.267290068214 - 0.272743810491 * %i)],_
+[1. + 0.7 * %i,- 0.352768690860 - 0.292826351187 * %i,_
+log(Gamma(1. + 0.7 * %i)),_
+log(Gamma(1. + 0.7 * %i))-(- 0.352768690860 - 0.292826351187 * %i)],_
+[1. + 0.8 * %i,- 0.445978783549 - 0.304225602976 * %i,_
+log(Gamma(1. + 0.8 * %i)),_
+log(Gamma(1. + 0.8 * %i))-(- 0.445978783549 - 0.304225602976 * %i)],_
+[1. + 0.9 * %i,- 0.545705128605 - 0.307074375642 * %i,_
+log(Gamma(1. + 0.9 * %i)),_
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+log(Gamma(1. + 9.9 * %i)),_
+log(Gamma(1. + 9.9 * %i))-(- 13.485677723495 + 13.573071879455 * %i)],_
+[1. + 10.0 * %i,- 13.637732188247 + 13.802912974230 * %i,_
+log(Gamma(1. + 10.0 * %i)),_
+log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 + 13.802912974230 * %i)]]
+--R 
+--R
+--R   (4)
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+--R     ,
+--R
+--R     [1. + 7. %i, - 9.1036806798319994 + 7.3948562984360002 %i,
+--R      - 9.1036806798319478 + 1.1116709912566856 %i,
+--R      5.1514348342607263E-14 - 6.2831853071793144 %i]
+--R     ,
+--R
+--R     [1. + 7.0999999999999996 %i, - 9.2536679950150003 + 7.5903262351840004 %i,
+--R      - 9.2536679950154568 + 1.3071409280042976 %i,
+--R      - 4.5652370772586437E-13 - 6.2831853071797026 %i]
+--R     ,
+--R
+--R     [1. + 7.2000000000000002 %i, - 9.4037545067079993 + 7.7871999928770004 %i,
+--R      - 9.4037545067075801 + 1.5040146856973393 %i,
+--R      4.1922021409845911E-13 - 6.2831853071796608 %i]
+--R     ,
+--R
+--R     [1. + 7.2999999999999998 %i, - 9.553937478321 + 7.9854582004679999 %i,
+--R      - 9.5539374783209006 + 1.7022728932880065 %i,
+--R      9.9475983006414026E-14 - 6.283185307179993 %i]
+--R     ,
+--R
+--R     [1. + 7.4000000000000004 %i, - 9.7042142849720001 + 8.1850820125030008 %i,
+--R      - 9.7042142849724993 + 1.9018967053232116 %i,
+--R      - 4.9915627187147038E-13 - 6.2831853071797887 %i]
+--R     ,
+--R
+--R     [1. + 7.5 %i, - 9.8545824074859993 + 8.3860530880889996 %i,
+--R      - 9.8545824074859194 + 2.1028677809095977 %i,
+--R      7.9936057773011271E-14 - 6.2831853071794015 %i]
+--R     ,
+--R
+--R     [1. + 7.5999999999999996 %i, - 10.005039426790001 + 8.5883535709619991 %i,
+--R      - 10.005039426790399 + 2.3051682637825208 %i,
+--R      - 3.979039320256561E-13 - 6.2831853071794779 %i]
+--R     ,
+--R
+--R     [1. + 7.7000000000000002 %i, - 10.155583018686 + 8.7919660705869997 %i,
+--R      - 10.155583018686212 + 2.5087807634076618 %i,
+--R      - 2.1138646388862981E-13 - 6.2831853071793375 %i]
+--R     ,
+--R
+--R     [1. + 7.7999999999999998 %i, - 10.306210948947999 + 8.9968736442289998 %i,
+--R      - 10.306210948947749 + 2.7136883370494931 %i,
+--R      2.5046631435543532E-13 - 6.2831853071795063 %i]
+--R     ,
+--R
+--R     [1. + 7.9000000000000004 %i, - 10.456921068739 + 9.2030597799250007 %i,
+--R      - 10.456921068738524 + 2.9198744727458439 %i,
+--R      4.7606363295926712E-13 - 6.2831853071791564 %i]
+--R     ,
+--R
+--R     [1. + 8. %i, - 10.607711310315 + 9.4105083803120007 %i,
+--R      - 10.607711310314581 + 3.1273230731320214 %i,
+--R      4.1922021409845911E-13 - 6.2831853071799788 %i]
+--R     ,
+--R
+--R     [1. + 8.0999999999999996 %i, - 10.758579682995 + 9.6192037472420004 %i,
+--R      - 10.758579682994794 - 2.9471668671170055 %i,
+--R      2.0605739337042905E-13 - 12.566370614359005 %i]
+--R     ,
+--R
+--R     [1. + 8.1999999999999993 %i, - 10.909524269378 + 9.8291305671620002 %i,
+--R      - 10.909524269378375 - 2.7372400471974712 %i,
+--R      - 3.7481129311345285E-13 - 12.566370614359471 %i]
+--R     ,
+--R
+--R     [1. + 8.3000000000000007 %i, - 11.060543221792001 + 10.040273897180001 %i,
+--R      - 11.060543221791693 - 2.5260967171793411 %i,
+--R      3.0730973321624333E-13 - 12.566370614359341 %i]
+--R     ,
+--R
+--R     [1. + 8.4000000000000004 %i, - 11.211634758948 + 10.252619151809 %i,
+--R      - 11.211634758947826 - 2.3137514625505582 %i,
+--R      1.7408297026122455E-13 - 12.566370614359558 %i]
+--R     ,
+--R
+--R     [1. + 8.5 %i, - 11.362797162804 + 10.466152090324 %i,
+--R      - 11.362797162803814 - 2.100218524035578 %i,
+--R      1.865174681370263E-13 - 12.566370614359577 %i]
+--R     ,
+--R
+--R     [1. + 8.5999999999999996 %i, - 11.514028775602 + 10.680858804712001 %i,
+--R      - 11.514028775601707 - 1.8855118096468773 %i,
+--R      2.9309887850104133E-13 - 12.566370614358878 %i]
+--R     ,
+--R
+--R     [1. + 8.6999999999999993 %i, - 11.665327997081 + 10.896725708177 %i,
+--R      - 11.665327997080658 - 1.6696449061826013 %i,
+--R      3.4283687000424834E-13 - 12.566370614359601 %i]
+--R     ,
+--R
+--R     [1. + 8.8000000000000007 %i, - 11.816693281848 + 11.113739524156999 %i,
+--R      - 11.816693281848337 - 1.452631090201765 %i,
+--R      - 3.3750779948604759E-13 - 12.566370614358764 %i]
+--R     ,
+--R
+--R     [1. + 8.9000000000000004 %i, - 11.968123136900999 + 11.331887275852999 %i,
+--R      - 11.968123136900861 - 1.2344833385062608 %i,
+--R      1.3855583347321954E-13 - 12.56637061435926 %i]
+--R     ,
+--R
+--R     [1. + 9. %i, - 12.119616119281 + 11.551156276202001 %i,
+--R      - 12.119616119281286 - 1.0152143381569982 %i,
+--R      - 2.8599345114344032E-13 - 12.566370614358998 %i]
+--R     ,
+--R
+--R     [1. + 9.0999999999999996 %i, - 12.271170833867 + 11.771534118309001 %i,
+--R      - 12.271170833867483 - 0.79483649604973161 %i,
+--R      - 4.8316906031686813E-13 - 12.566370614358732 %i]
+--R     ,
+--R
+--R     [1. + 9.1999999999999993 %i, - 12.422785931281 + 11.993008666285 %i,
+--R      - 12.422785931280877 - 0.5733619480744393 %i,
+--R      1.2256862191861728E-13 - 12.566370614359439 %i]
+--R     ,
+--R
+--R     [1. + 9.3000000000000007 %i, - 12.574460105908001 + 12.215568046479 %i,
+--R      - 12.574460105908262 - 0.35080256788055886 %i,
+--R      - 2.6112445539183682E-13 - 12.566370614359558 %i]
+--R     ,
+--R
+--R     [1. + 9.4000000000000004 %i, - 12.726192094029001 + 12.43920063909 %i,
+--R      - 12.726192094029377 - 0.12716997526913024 %i,
+--R      - 3.765876499528531E-13 - 12.56637061435913 %i]
+--R     ,
+--R
+--R     [1. + 9.5 %i, - 12.877980672044 + 12.663895070128 %i,
+--R      - 12.877980672043599 + 9.7524455768741289E-2 %i,
+--R      4.0145664570445661E-13 - 12.566370614359258 %i]
+--R     ,
+--R
+--R     [1. + 9.5999999999999996 %i, - 13.029824654789 + 12.889640203708 %i,
+--R      - 13.02982465478944 + 0.32326958934851951 %i,
+--R      - 4.4053649617126212E-13 - 12.56637061435948 %i]
+--R     ,
+--R
+--R     [1. + 9.6999999999999993 %i, - 13.181722893950999 + 13.116425134666001 %i,
+--R      - 13.181722893951155 + 0.55005452030683832 %i,
+--R      - 1.5631940186722204E-13 - 12.566370614359162 %i]
+--R     ,
+--R
+--R     [1. + 9.8000000000000007 %i, - 13.333674276547001 + 13.344239181477 %i,
+--R      - 13.333674276547052 + 0.77786856711780805 %i,
+--R      - 5.1514348342607263E-14 - 12.566370614359192 %i]
+--R     ,
+--R
+--R     [1. + 9.9000000000000004 %i, - 13.485677723495 + 13.573071879455 %i,
+--R      - 13.485677723494533 + 1.0067012650958465 %i,
+--R      4.6718184876226587E-13 - 12.566370614359153 %i]
+--R     ,
+--R
+--R     [1. + 10. %i, - 13.637732188247 + 13.802912974230001 %i,
+--R      - 13.637732188247268 + 1.2365423598707301 %i,
+--R      - 2.6822988274943782E-13 - 12.56637061435927 %i]
+--R     ]
+--R                                          Type: List List Complex DoubleFloat
+--E 4
 
 )spool 
 )lisp (bye)
@@ -1592,5 +2416,7 @@ close(viewport)
 http://dlmf.nist.gov
 \bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp267-270
+\bibitem{3} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp277
 \end{thebibliography}
 \end{document}
diff --git a/src/input/log.input.pamphlet b/src/input/log.input.pamphlet
new file mode 100644
index 0000000..8c6f78e
--- /dev/null
+++ b/src/input/log.input.pamphlet
@@ -0,0 +1,442 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input log.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's log function with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of log(x), ranging from 0.01 to 2.00. The
+second column is the reference value of log from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp100-113
+
+<<*>>=
+)spool log.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 1
+[[0.01, -4.6051701859880914, log(0.01), log(0.01)-(-4.6051701859880914)], _
+[0.02, -3.9120230054281461, log(0.02), log(0.02)-(-3.9120230054281461)], _
+[0.03, -3.5065578973199817, log(0.03), log(0.03)-(-3.5065578973199817)], _
+[0.04, -3.2188758248682007, log(0.04), log(0.04)-(-3.2188758248682007)], _
+[0.05, -2.9957322735539910, log(0.05), log(0.05)-(-2.9957322735539910)], _
+[0.06, -2.8134107167600364, log(0.06), log(0.06)-(-2.8134107167600364)], _
+[0.07, -2.6592600369327781, log(0.07), log(0.07)-(-2.6592600369327781)], _
+[0.08, -2.5257286443082554, log(0.08), log(0.08)-(-2.5257286443082554)], _
+[0.09, -2.4079456086518720, log(0.09), log(0.09)-(-2.4079456086518720)], _
+[0.10, -2.3025850929940457, log(0.10), log(0.10)-(-2.3025850929940457)], _
+[0.11, -2.2072749131897208, log(0.11), log(0.11)-(-2.2072749131897208)], _
+[0.12, -2.1202635362000911, log(0.12), log(0.12)-(-2.1202635362000911)], _
+[0.13, -2.0402208285265546, log(0.13), log(0.13)-(-2.0402208285265546)], _
+[0.14, -1.9661128563728328, log(0.14), log(0.14)-(-1.9661128563728328)], _
+[0.15, -1.8971199848858813, log(0.15), log(0.15)-(-1.8971199848858813)], _
+[0.16, -1.8325814637483101, log(0.16), log(0.16)-(-1.8325814637483101)], _
+[0.17, -1.7719568419318753, log(0.17), log(0.17)-(-1.7719568419318753)], _
+[0.18, -1.7147984280919267, log(0.18), log(0.18)-(-1.7147984280919267)], _
+[0.19, -1.6607312068216509, log(0.19), log(0.19)-(-1.6607312068216509)], _
+[0.20, -1.6094379124341004, log(0.20), log(0.20)-(-1.6094379124341004)], _
+[0.21, -1.5606477482646684, log(0.21), log(0.21)-(-1.5606477482646684)], _
+[0.22, -1.5141277326297755, log(0.22), log(0.22)-(-1.5141277326297755)], _
+[0.23, -1.4696759700589417, log(0.23), log(0.23)-(-1.4696759700589417)], _
+[0.24, -1.4271163556401457, log(0.24), log(0.24)-(-1.4271163556401457)], _
+[0.25, -1.3862943611198906, log(0.25), log(0.25)-(-1.3862943611198906)], _
+[0.26, -1.3470736479666093, log(0.26), log(0.26)-(-1.3470736479666093)], _
+[0.27, -1.3093333199837623, log(0.27), log(0.27)-(-1.3093333199837623)], _
+[0.28, -1.2729656758128874, log(0.28), log(0.28)-(-1.2729656758128874)], _
+[0.29, -1.2378743560016173, log(0.29), log(0.29)-(-1.2378743560016173)], _
+[0.30, -1.2039728043259360, log(0.30), log(0.30)-(-1.2039728043259360)], _
+[0.31, -1.1711829815029451, log(0.31), log(0.31)-(-1.1711829815029451)], _
+[0.32, -1.1394342831883648, log(0.32), log(0.32)-(-1.1394342831883648)], _
+[0.33, -1.1086626245216111, log(0.33), log(0.33)-(-1.1086626245216111)], _
+[0.34, -1.0788096613719300, log(0.34), log(0.34)-(-1.0788096613719300)], _
+[0.35, -1.0498221244986777, log(0.35), log(0.35)-(-1.0498221244986777)], _
+[0.36, -1.0216512475319814, log(0.36), log(0.36)-(-1.0216512475319814)], _
+[0.37, -0.9942522733438669, log(0.37), log(0.37)-(-0.9942522733438669)], _
+[0.38, -0.9675840262617056, log(0.38), log(0.38)-(-0.9675840262617056)], _
+[0.39, -0.9416085398584449, log(0.39), log(0.39)-(-0.9416085398584449)], _
+[0.40, -0.9162907318741551, log(0.40), log(0.40)-(-0.9162907318741551)], _
+[0.41, -0.8915981192837836, log(0.41), log(0.41)-(-0.8915981192837836)], _
+[0.42, -0.8675005677047231, log(0.42), log(0.42)-(-0.8675005677047231)], _
+[0.43, -0.8439700702945289, log(0.43), log(0.43)-(-0.8439700702945289)], _
+[0.44, -0.8209805520698302, log(0.44), log(0.44)-(-0.8209805520698302)], _
+[0.45, -0.7985076962177716, log(0.45), log(0.45)-(-0.7985076962177716)], _
+[0.46, -0.7765287894989964, log(0.46), log(0.46)-(-0.7765287894989964)], _
+[0.47, -0.7550225842780328, log(0.47), log(0.47)-(-0.7550225842780328)], _
+[0.48, -0.7339691750802004, log(0.48), log(0.48)-(-0.7339691750802004)], _
+[0.49, -0.7133498878774648, log(0.49), log(0.49)-(-0.7133498878774648)], _
+[0.50, -0.6931471805599453, log(0.50), log(0.50)-(-0.6931471805599453)], _
+[0.51, -0.6733445532637656, log(0.51), log(0.51)-(-0.6733445532637656)], _
+[0.52, -0.6539264674066640, log(0.52), log(0.52)-(-0.6539264674066640)], _
+[0.53, -0.6348782724359695, log(0.53), log(0.53)-(-0.6348782724359695)], _
+[0.54, -0.6161861394238170, log(0.54), log(0.54)-(-0.6161861394238170)], _
+[0.55, -0.5978370007556204, log(0.55), log(0.55)-(-0.5978370007556204)], _
+[0.56, -0.5798184952529421, log(0.56), log(0.56)-(-0.5798184952529421)], _
+[0.57, -0.5621189181535412, log(0.57), log(0.57)-(-0.5621189181535412)], _
+[0.58, -0.5447271754416720, log(0.58), log(0.58)-(-0.5447271754416720)], _
+[0.59, -0.5276327420823719, log(0.59), log(0.59)-(-0.5276327420823719)], _
+[0.60, -0.5108256237659907, log(0.60), log(0.60)-(-0.5108256237659907)], _
+[0.61, -0.4942963218147801, log(0.61), log(0.61)-(-0.4942963218147801)], _
+[0.62, -0.4780358009429998, log(0.62), log(0.62)-(-0.4780358009429998)], _
+[0.63, -0.4620354595965587, log(0.63), log(0.63)-(-0.4620354595965587)], _
+[0.64, -0.4462871026284195, log(0.64), log(0.64)-(-0.4462871026284195)], _
+[0.65, -0.4307829160924543, log(0.65), log(0.65)-(-0.4307829160924543)], _
+[0.66, -0.4155154439616658, log(0.66), log(0.66)-(-0.4155154439616658)], _
+[0.67, -0.4004775665971253, log(0.67), log(0.67)-(-0.4004775665971253)], _
+[0.68, -0.3856624808119847, log(0.68), log(0.68)-(-0.3856624808119847)], _
+[0.69, -0.3710636813908320, log(0.69), log(0.69)-(-0.3710636813908320)], _
+[0.70, -0.3566749439387324, log(0.70), log(0.70)-(-0.3566749439387324)], _
+[0.71, -0.3424903089467759, log(0.71), log(0.71)-(-0.3424903089467759)], _
+[0.72, -0.3285040669720361, log(0.72), log(0.72)-(-0.3285040669720361)], _
+[0.73, -0.3147107448397002, log(0.73), log(0.73)-(-0.3147107448397002)], _
+[0.74, -0.3011050927839216, log(0.74), log(0.74)-(-0.3011050927839216)], _
+[0.75, -0.2876820724517809, log(0.75), log(0.75)-(-0.2876820724517809)], _
+[0.76, -0.2744368457017603, log(0.76), log(0.76)-(-0.2744368457017603)], _
+[0.77, -0.2613647641344075, log(0.77), log(0.77)-(-0.2613647641344075)], _
+[0.78, -0.2484613592984996, log(0.78), log(0.78)-(-0.2484613592984996)], _
+[0.79, -0.2357223335210699, log(0.79), log(0.79)-(-0.2357223335210699)], _
+[0.80, -0.2231435513142098, log(0.80), log(0.80)-(-0.2231435513142098)], _
+[0.81, -0.2107210313156526, log(0.81), log(0.81)-(-0.2107210313156526)], _
+[0.82, -0.1984509387238383, log(0.82), log(0.82)-(-0.1984509387238383)], _
+[0.83, -0.1863295781914934, log(0.83), log(0.83)-(-0.1863295781914934)], _
+[0.84, -0.1743533871447778, log(0.84), log(0.84)-(-0.1743533871447778)], _
+[0.85, -0.1625189294977749, log(0.85), log(0.85)-(-0.1625189294977749)], _
+[0.86, -0.1508228897345836, log(0.86), log(0.86)-(-0.1508228897345836)], _
+[0.87, -0.1392620673335076, log(0.87), log(0.87)-(-0.1392620673335076)], _
+[0.88, -0.1278333715098849, log(0.88), log(0.88)-(-0.1278333715098849)], _
+[0.89, -0.1165338162559515, log(0.89), log(0.89)-(-0.1165338162559515)], _
+[0.90, -0.1053605156578263, log(0.90), log(0.90)-(-0.1053605156578263)], _
+[0.91, -0.0943106794712413, log(0.91), log(0.91)-(-0.0943106794712413)], _
+[0.92, -0.0833816089390511, log(0.92), log(0.92)-(-0.0833816089390511)], _
+[0.93, -0.0725706928348354, log(0.93), log(0.93)-(-0.0725706928348354)], _
+[0.94, -0.0618754037180875, log(0.94), log(0.94)-(-0.0618754037180875)], _
+[0.95, -0.0512932943875505, log(0.95), log(0.95)-(-0.0512932943875505)], _
+[0.96, -0.0408219945202551, log(0.96), log(0.96)-(-0.0408219945202551)], _
+[0.97, -0.0304592074847085, log(0.97), log(0.97)-(-0.0304592074847085)], _
+[0.98, -0.0202027073175194, log(0.98), log(0.98)-(-0.0202027073175194)], _
+[0.99, -0.0100503358535014, log(0.99), log(0.99)-(-0.0100503358535014)], _
+[1.00, 0.0000000000000000, log(1.00), log(1.00)-(0.0000000000000000)], _
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+[2.00, 0.6931471805599453, log(2.00), log(2.00)-(0.6931471805599453)]]
+--R 
+--R
+--R   (1)
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+--R    [1.0,0.0,0.0,0.0],
+--R    [1.01,0.0099503308 531681,0.0099503308 5316808284 64,- 0.17154 E -16],
+--R    [1.02,0.0198026272 961797,0.0198026272 9617971302 9,0.1303 E -16],
+--R    [1.03,0.0295588022 415444,0.0295588022 4154440273 4,0.2734 E -17],
+--R    [1.04,0.0392207131 532813,0.0392207131 5328129626 8,- 0.3732 E -17],
+--R    [1.05,0.0487901641 69432,0.0487901641 6943200306 3,0.306 E -17],
+--R    [1.06,0.0582689081 239758,0.0582689081 2397577552 8,- 0.2447 E -16],
+--R    [1.07,0.0676586484 738148,0.0676586484 7381480526 9,0.527 E -17],
+--R    [1.08,0.0769610411 361283,0.0769610411 3612832498 3,0.2498 E -16],
+--R    [1.09,0.0861776962 410523,0.0861776962 4105233234 4,0.3234 E -16],
+--R    [1.1,0.0953101798 043249,0.0953101798 0432486004 5,- 0.3995 E -16],
+--R    [1.11,0.1043600153 242428,0.1043600153 2424276773,- 0.3227 E -16],
+--R    [1.12,0.1133286853 070032,0.1133286853 0700317474,- 0.2526 E -16],
+--R    [1.13,0.1222176327 242492,0.1222176327 2424920055,0.548 E -18],
+--R    [1.14,0.1310282624 064041,0.1310282624 0640409279,- 0.721 E -17],
+--R    [1.15,0.1397619423 751587,0.1397619423 7515869737,- 0.263 E -17],
+--R    [1.16,0.1484200051 182733,0.1484200051 1827327798,- 0.2202 E -16],
+--R    [1.17,0.1570037488 096648,0.1570037488 0966475081,- 0.4919 E -16],
+--R    [1.18,0.1655144384 775734,0.1655144384 77573392,- 0.8 E -17],
+--R    [1.19,0.1739533071 23438,0.1739533071 2343801732,0.1732 E -16],
+--R    [1.2,0.1823215567 939546,0.1823215567 9395462621,0.2621 E -16],
+--R    [1.21,0.1906203596 086497,0.1906203596 0864972009,0.2009 E -16],
+--R    [1.22,0.1988508587 451652,0.1988508587 4516519013,- 0.987 E -17],
+--R    [1.23,0.2070141693 843261,0.2070141693 8432612722,0.2722 E -16],
+--R    [1.24,0.2151113796 169455,0.2151113796 1694549673,- 0.327 E -17],
+--R    [1.25,0.2231435513 142098,0.2231435513 1420975577,- 0.4423 E -16],
+--R    [1.26,0.2311117209 633866,0.2311117209 6338662928,0.2928 E -16],
+--R    [1.27,0.2390169004 704999,0.2390169004 7049990501,0.501 E -17],
+--R    [1.28,0.2468600779 315258,0.2468600779 3152579789,- 0.211 E -17],
+--R    [1.29,0.2546422183 735807,0.2546422183 7358074683,0.4683 E -16],
+--R    [1.3,0.2623642644 674911,0.2623642644 6749105203,- 0.4797 E -16],
+--R    [1.31,0.2700271372 130602,0.2700271372 1306017612,- 0.239 E -16],
+--R    [1.32,0.2776317365 982795,0.2776317365 9827948626,- 0.137 E -16],
+--R    [1.33,0.2851789422 336624,0.2851789422 3366239708,- 0.292 E -17],
+--R    [1.34,0.2926696139 6282,0.2926696139 6282000105,0.11 E -17],
+--R    [1.35,0.3001045924 503381,0.3001045924 5033808075,- 0.192 E -16],
+--R    [1.36,0.3074846997 479606,0.3074846997 4796064047,0.4046 E -16],
+--R    [1.37,0.3148107398 400335,0.3148107398 4003354728,0.4728 E -16],
+--R    [1.38,0.3220834991 691133,0.3220834991 6911332359,0.236 E -16],
+--R    [1.39,0.3293037471 426004,0.3293037471 4260038915,- 0.108 E -16],
+--R    [1.4,0.3364722366 212129,0.3364722366 212129305,0.305 E -16],
+--R    [1.41,0.3435897043 900769,0.3435897043 9007691018,0.102 E -16],
+--R    [1.42,0.3506568716 131694,0.3506568716 1316936271,- 0.3729 E -16],
+--R    [1.43,0.3576744442 718159,0.3576744442 7181591208,0.121 E -16],
+--R    [1.44,0.3646431135 879093,0.3646431135 8790925242,- 0.4758 E -16],
+--R    [1.45,0.3715635564 32483,0.3715635564 3248303375,0.3375 E -16],
+--R    [1.46,0.3784364357 202451,0.3784364357 2024507047,- 0.2953 E -16],
+--R    [1.47,0.3852624007 906449,0.3852624007 9064493357,0.3357 E -16],
+--R    [1.48,0.3920420877 760237,0.3920420877 7602369517,- 0.483 E -17],
+--R    [1.49,0.3987761199 573678,0.3987761199 5736777296,- 0.27 E -16],
+--R    [1.5,0.4054651081 081644,0.4054651081 0816438198,- 0.18 E -16],
+--R    [1.51,0.4121096508 26833,0.4121096508 2683296076,- 0.3924 E -16],
+--R    [1.52,0.4187103348 58185,0.4187103348 5818502023,0.202 E -16],
+--R    [1.53,0.4252677354 043441,0.4252677354 0434409501,- 0.5 E -17],
+--R    [1.54,0.4317824164 255378,0.4317824164 2553779055,- 0.945 E -17],
+--R    [1.55,0.4382549309 311553,0.4382549309 3115525249,- 0.4751 E -16],
+--R    [1.56,0.4446858212 614457,0.4446858212 6144567825,- 0.218 E -16],
+--R    [1.57,0.4510756193 602167,0.4510756193 6021668939,- 0.106 E -16],
+--R    [1.58,0.4574248470 388754,0.4574248470 3887543555,0.3555 E -16],
+--R    [1.59,0.4637340162 321402,0.4637340162 3214015751,- 0.4249 E -16],
+--R    [1.6,0.4700036292 457356,0.4700036292 4573555365,- 0.4635 E -16],
+--R    [1.61,0.4762341789 963716,0.4762341789 9637162788,0.2788 E -16],
+--R    [1.62,0.4824261492 442927,0.4824261492 4429270696,0.696 E -17],
+--R    [1.63,0.4885800148 18671,0.4885800148 1867096603,- 0.3397 E -16],
+--R    [1.64,0.4946962418 361071,0.4946962418 3610705467,- 0.4533 E -16],
+--R    [1.65,0.5007752879 124892,0.5007752879 1248924202,0.42 E -16],
+--R    [1.66,0.5068176023 684519,0.5068176023 6845186486,- 0.351 E -16],
+--R    [1.67,0.5128236264 286637,0.5128236264 2866373922,0.392 E -16],
+--R    [1.68,0.5187937934 151676,0.5187937934 1516755672,- 0.433 E -16],
+--R    [1.69,0.5247285289 349821,0.5247285289 3498210407,0.407 E -17],
+--R    [1.7,0.5306282510 621704,0.5306282510 6217039623,- 0.377 E -17],
+--R    [1.71,0.5364933705 145685,0.5364933705 1456847476,- 0.252 E -16],
+--R    [1.72,0.5423242908 253617,0.5423242908 2536167427,- 0.257 E -16],
+--R    [1.73,0.5481214085 096876,0.5481214085 096875789,- 0.211 E -16],
+--R    [1.74,0.5538851132 264377,0.5538851132 2643765996,- 0.4 E -16],
+--R    [1.75,0.5596157879 354227,0.5596157879 3542268627,- 0.137 E -16],
+--R    [1.76,0.5653138090 500604,0.5653138090 5006041369,0.137 E -16],
+--R    [1.77,0.5709795465 857378,0.5709795465 8573777398,- 0.26 E -16],
+--R    [1.78,0.5766133643 039938,0.5766133643 039937797,- 0.203 E -16],
+--R    [1.79,0.5822156198 526636,0.5822156198 5266362814,0.281 E -16],
+--R    [1.8,0.5877866649 02119,0.5877866649 0211900819,0.819 E -17],
+--R    [1.81,0.5933268452 777344,0.5933268452 777343788,- 0.212 E -16],
+--R    [1.82,0.5988365010 88704,0.5988365010 8870398254,- 0.175 E -16],
+--R    [1.83,0.6043159668 533296,0.6043159668 5332957211,- 0.279 E -16],
+--R    [1.84,0.6097655716 208943,0.6097655716 2089425102,- 0.49 E -16],
+--R    [1.85,0.6151856390 902335,0.6151856390 9023345093,- 0.491 E -16],
+--R    [1.86,0.6205764877 251099,0.6205764877 2510987871,- 0.213 E -16],
+--R    [1.87,0.6259384308 664953,0.6259384308 6649525628,- 0.437 E -16],
+--R    [1.88,0.6312717768 418578,0.6312717768 4185783762,0.376 E -16],
+--R    [1.89,0.6365768290 71551,0.6365768290 7155101126,0.113 E -16],
+--R    [1.9,0.6418538861 723948,0.6418538861 7239477599,- 0.24 E -16],
+--R    [1.91,0.6471032420 585385,0.6471032420 5853850481,0.481 E -17],
+--R    [1.92,0.6523251860 396902,0.6523251860 3969017986,- 0.201 E -16],
+--R    [1.93,0.6575200029 167942,0.6575200029 1679418382,- 0.162 E -16],
+--R    [1.94,0.6626879730 752368,0.6626879730 752367635,- 0.365 E -16],
+--R    [1.95,0.6678293725 756554,0.6678293725 7565543401,0.34 E -16],
+--R    [1.96,0.6729444732 424259,0.6729444732 4242586101,- 0.39 E -16],
+--R    [1.97,0.6780335427 498971,0.6780335427 4989713874,0.387 E -16],
+--R    [1.98,0.6830968447 064439,0.6830968447 0644386823,- 0.318 E -16],
+--R    [1.99,0.6881346387 36401,0.6881346387 3640102737,0.274 E -16],
+--R    [2.0,0.6931471805 599453,0.6931471805 5994530942,0.942 E -17]]
+--R                                                        Type: List List Float
+--E 1
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp100-113
+\end{thebibliography}
+\end{document}
+
diff --git a/src/input/seccsc.input.pamphlet b/src/input/seccsc.input.pamphlet
new file mode 100644
index 0000000..05d789e
--- /dev/null
+++ b/src/input/seccsc.input.pamphlet
@@ -0,0 +1,695 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input seccsc.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's sec and csc functions with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of sec(x), ranging from 0.01 to 1.60. The
+second column is the reference value of sec from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+<<*>>=
+)spool seccsc.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.01,1.0000500,sec(0.01),sec(0.01)-1.0000500],_
+[0.02,1.0002000,sec(0.02),sec(0.02)-1.0002000],_
+[0.03,1.0004502,sec(0.03),sec(0.03)-1.0004502],_
+[0.04,1.0008005,sec(0.04),sec(0.04)-1.0008005],_
+[0.05,1.0012513,sec(0.05),sec(0.05)-1.0012513],_
+[0.06,1.0018027,sec(0.06),sec(0.06)-1.0018027],_
+[0.07,1.0024550,sec(0.07),sec(0.07)-1.0024550],_
+[0.08,1.0032086,sec(0.08),sec(0.08)-1.0032086],_
+[0.09,1.0040637,sec(0.09),sec(0.09)-1.0040637],_
+[0.10,1.0050209,sec(0.10),sec(0.10)-1.0050209],_
+[0.11,1.0060807,sec(0.11),sec(0.11)-1.0060807],_
+[0.12,1.0072435,sec(0.12),sec(0.12)-1.0072435],_
+[0.13,1.0085099,sec(0.13),sec(0.13)-1.0085099],_
+[0.14,1.0098807,sec(0.14),sec(0.14)-1.0098807],_
+[0.15,1.0113564,sec(0.15),sec(0.15)-1.0113564],_
+[0.16,1.0129380,sec(0.16),sec(0.16)-1.0129380],_
+[0.17,1.0146261,sec(0.17),sec(0.17)-1.0146261],_
+[0.18,1.0164216,sec(0.18),sec(0.18)-1.0164216],_
+[0.19,1.0183255,sec(0.19),sec(0.19)-1.0183255],_
+[0.20,1.0203388,sec(0.20),sec(0.20)-1.0203388],_
+[0.21,1.0224626,sec(0.21),sec(0.21)-1.0224626],_
+[0.22,1.0246978,sec(0.22),sec(0.22)-1.0246978],_
+[0.23,1.0270458,sec(0.23),sec(0.23)-1.0270458],_
+[0.24,1.0295078,sec(0.24),sec(0.24)-1.0295078],_
+[0.25,1.0320850,sec(0.25),sec(0.25)-1.0320850],_
+[0.26,1.0347789,sec(0.26),sec(0.26)-1.0347789],_
+[0.27,1.0375910,sec(0.27),sec(0.27)-1.0375910],_
+[0.28,1.0405227,sec(0.28),sec(0.28)-1.0405227],_
+[0.29,1.0435757,sec(0.29),sec(0.29)-1.0435757],_
+[0.30,1.0467516,sec(0.30),sec(0.30)-1.0467516],_
+[0.31,1.0500522,sec(0.31),sec(0.31)-1.0500522],_
+[0.32,1.0534794,sec(0.32),sec(0.32)-1.0534794],_
+[0.33,1.0570351,sec(0.33),sec(0.33)-1.0570351],_
+[0.34,1.0607213,sec(0.34),sec(0.34)-1.0607213],_
+[0.35,1.0645402,sec(0.35),sec(0.35)-1.0645402],_
+[0.36,1.0684938,sec(0.36),sec(0.36)-1.0684938],_
+[0.37,1.0725847,sec(0.37),sec(0.37)-1.0725847],_
+[0.38,1.0768150,sec(0.38),sec(0.38)-1.0768150],_
+[0.39,1.0811874,sec(0.39),sec(0.39)-1.0811874],_
+[0.40,1.0857044,sec(0.40),sec(0.40)-1.0857044],_
+[0.41,1.0903689,sec(0.41),sec(0.41)-1.0903689],_
+[0.42,1.0951836,sec(0.42),sec(0.42)-1.0951836],_
+[0.43,1.1001515,sec(0.43),sec(0.43)-1.1001515],_
+[0.44,1.1052757,sec(0.44),sec(0.44)-1.1052757],_
+[0.45,1.1105594,sec(0.45),sec(0.45)-1.1105594],_
+[0.46,1.1160060,sec(0.46),sec(0.46)-1.1160060],_
+[0.47,1.1216191,sec(0.47),sec(0.47)-1.1216191],_
+[0.48,1.1274022,sec(0.48),sec(0.48)-1.1274022],_
+[0.49,1.1333591,sec(0.49),sec(0.49)-1.1333591],_
+[0.50,1.1394939,sec(0.50),sec(0.50)-1.1394939],_
+[0.51,1.1458107,sec(0.51),sec(0.51)-1.1458107],_
+[0.52,1.1523138,sec(0.52),sec(0.52)-1.1523138],_
+[0.53,1.1590077,sec(0.53),sec(0.53)-1.1590077],_
+[0.54,1.1658970,sec(0.54),sec(0.54)-1.1658970],_
+[0.55,1.1729868,sec(0.55),sec(0.55)-1.1729868],_
+[0.56,1.1802821,sec(0.56),sec(0.56)-1.1802821],_
+[0.57,1.1877881,sec(0.57),sec(0.57)-1.1877881],_
+[0.58,1.1955106,sec(0.58),sec(0.58)-1.1955106],_
+[0.59,1.2034553,sec(0.59),sec(0.59)-1.2034553],_
+[0.60,1.2116283,sec(0.60),sec(0.60)-1.2116283],_
+[0.61,1.2200359,sec(0.61),sec(0.61)-1.2200359],_
+[0.62,1.2286847,sec(0.62),sec(0.62)-1.2286847],_
+[0.63,1.2375816,sec(0.63),sec(0.63)-1.2375816],_
+[0.64,1.2467339,sec(0.64),sec(0.64)-1.2467339],_
+[0.65,1.2561492,sec(0.65),sec(0.65)-1.2561492],_
+[0.66,1.2658352,sec(0.66),sec(0.66)-1.2658352],_
+[0.67,1.2758004,sec(0.67),sec(0.67)-1.2758004],_
+[0.68,1.2860534,sec(0.68),sec(0.68)-1.2860534],_
+[0.69,1.2966031,sec(0.69),sec(0.69)-1.2966031],_
+[0.70,1.3074593,sec(0.70),sec(0.70)-1.3074593],_
+[0.71,1.3186317,sec(0.71),sec(0.71)-1.3186317],_
+[0.72,1.3301309,sec(0.72),sec(0.72)-1.3301309],_
+[0.73,1.3419677,sec(0.73),sec(0.73)-1.3419677],_
+[0.74,1.3541538,sec(0.74),sec(0.74)-1.3541538],_
+[0.75,1.3667011,sec(0.75),sec(0.75)-1.3667011],_
+[0.76,1.3796224,sec(0.76),sec(0.76)-1.3796224],_
+[0.77,1.3929310,sec(0.77),sec(0.77)-1.3929310],_
+[0.78,1.4066408,sec(0.78),sec(0.78)-1.4066408],_
+[0.79,1.4207667,sec(0.79),sec(0.79)-1.4207667],_
+[0.80,1.4353242,sec(0.80),sec(0.80)-1.4353242],_
+[0.81,1.4503296,sec(0.81),sec(0.81)-1.4503296],_
+[0.82,1.4658002,sec(0.82),sec(0.82)-1.4658002],_
+[0.83,1.4817542,sec(0.83),sec(0.83)-1.4817542],_
+[0.84,1.4982108,sec(0.84),sec(0.84)-1.4982108],_
+[0.85,1.5151902,sec(0.85),sec(0.85)-1.5151902],_
+[0.86,1.5327139,sec(0.86),sec(0.86)-1.5327139],_
+[0.87,1.5508046,sec(0.87),sec(0.87)-1.5508046],_
+[0.88,1.5694863,sec(0.88),sec(0.88)-1.5694863],_
+[0.89,1.5887844,sec(0.89),sec(0.89)-1.5887844],_
+[0.90,1.6087258,sec(0.90),sec(0.90)-1.6087258],_
+[0.91,1.6293392,sec(0.91),sec(0.91)-1.6293392],_
+[0.92,1.6506549,sec(0.92),sec(0.92)-1.6506549],_
+[0.93,1.6727052,sec(0.93),sec(0.93)-1.6727052],_
+[0.94,1.6955244,sec(0.94),sec(0.94)-1.6955244],_
+[0.95,1.7191492,sec(0.95),sec(0.95)-1.7191492],_
+[0.96,1.7436184,sec(0.96),sec(0.96)-1.7436184],_
+[0.97,1.7689737,sec(0.97),sec(0.97)-1.7689737],_
+[0.98,1.7952595,sec(0.98),sec(0.98)-1.7952595],_
+[0.99,1.8225232,sec(0.99),sec(0.99)-1.8225232],_
+[1.00,1.8508157,sec(1.00),sec(1.00)-1.8508157],_
+[1.01,1.8801915,sec(1.01),sec(1.01)-1.8801915],_
+[1.02,1.9107089,sec(1.02),sec(1.02)-1.9107089],_
+[1.03,1.9424308,sec(1.03),sec(1.03)-1.9424308],_
+[1.04,1.9754247,sec(1.04),sec(1.04)-1.9754247],_
+[1.05,2.0097632,sec(1.05),sec(1.05)-2.0097632],_
+[1.06,2.0455249,sec(1.06),sec(1.06)-2.0455249],_
+[1.07,2.0827943,sec(1.07),sec(1.07)-2.0827943],_
+[1.08,2.1216631,sec(1.08),sec(1.08)-2.1216631],_
+[1.09,2.1622306,sec(1.09),sec(1.09)-2.1622306],_
+[1.10,2.2046044,sec(1.10),sec(1.10)-2.2046044],_
+[1.11,2.2489016,sec(1.11),sec(1.11)-2.2489016],_
+[1.12,2.2952497,sec(1.12),sec(1.12)-2.2952497],_
+[1.13,2.3437877,sec(1.13),sec(1.13)-2.3437877],_
+[1.14,2.3946675,sec(1.14),sec(1.14)-2.3946675],_
+[1.15,2.4480557,sec(1.15),sec(1.15)-2.4480557],_
+[1.16,2.5041348,sec(1.16),sec(1.16)-2.5041348],_
+[1.17,2.5631057,sec(1.17),sec(1.17)-2.5631057],_
+[1.18,2.6251899,sec(1.18),sec(1.18)-2.6251899],_
+[1.19,2.6906321,sec(1.19),sec(1.19)-2.6906321],_
+[1.20,2.7597036,sec(1.20),sec(1.20)-2.7597036],_
+[1.21,2.8327055,sec(1.21),sec(1.21)-2.8327055],_
+[1.22,2.9099735,sec(1.22),sec(1.22)-2.9099735],_
+[1.23,2.9918825,sec(1.23),sec(1.23)-2.9918825],_
+[1.24,3.0788530,sec(1.24),sec(1.24)-3.0788530],_
+[1.25,3.1713577,sec(1.25),sec(1.25)-3.1713577],_
+[1.26,3.2699304,sec(1.26),sec(1.26)-3.2699304],_
+[1.27,3.3751757,sec(1.27),sec(1.27)-3.3751757],_
+[1.28,3.4877815,sec(1.28),sec(1.28)-3.4877815],_
+[1.29,3.6085336,sec(1.29),sec(1.29)-3.6085336],_
+[1.30,3.7383341,sec(1.30),sec(1.30)-3.7383341],_
+[1.31,3.8782233,sec(1.31),sec(1.31)-3.8782233],_
+[1.32,4.0294074,sec(1.32),sec(1.32)-4.0294074],_
+[1.33,4.1932931,sec(1.33),sec(1.33)-4.1932931],_
+[1.34,4.3715310,sec(1.34),sec(1.34)-4.3715310],_
+[1.35,4.5660706,sec(1.35),sec(1.35)-4.5660706],_
+[1.36,4.7792314,sec(1.36),sec(1.36)-4.7792314],_
+[1.37,5.0137949,sec(1.37),sec(1.37)-5.0137949],_
+[1.38,5.2731260,sec(1.38),sec(1.38)-5.2731260],_
+[1.39,5.5613339,sec(1.39),sec(1.39)-5.5613339],_
+[1.40,5.8834901,sec(1.40),sec(1.40)-5.8834901],_
+[1.41,6.2459280,sec(1.41),sec(1.41)-6.2459280],_
+[1.42,6.6566608,sec(1.42),sec(1.42)-6.6566608],_
+[1.43,7.1259785,sec(1.43),sec(1.43)-7.1259785],_
+[1.44,7.6673176,sec(1.44),sec(1.44)-7.6673176],_
+[1.45,8.2985645,sec(1.45),sec(1.45)-8.2985645],_
+[1.46,9.0440625,sec(1.46),sec(1.46)-9.0440625],_
+[1.47,9.9378158,sec(1.47),sec(1.47)-9.9378158],_
+[1.48,11.0288087,sec(1.48),sec(1.48)-11.0288087],_
+[1.49,12.3902766,sec(1.49),sec(1.49)-12.3902766],_
+[1.50,14.1368329,sec(1.50),sec(1.50)-14.1368329],_
+[1.51,16.4584992,sec(1.51),sec(1.51)-16.4584992],_
+[1.52,19.6949314,sec(1.52),sec(1.52)-19.6949314],_
+[1.53,24.5188114,sec(1.53),sec(1.53)-24.5188114],_
+[1.54,32.4765383,sec(1.54),sec(1.54)-32.4765383],_
+[1.55,48.0888810,sec(1.55),sec(1.55)-48.0888810],_
+[1.56,92.6258945,sec(1.56),sec(1.56)-92.6258945],_
+[1.57,1255.7659897,sec(1.57),sec(1.57)-1255.7659897],_
+[1.58,-108.6538055,sec(1.58),sec(1.58)-(-108.6538055)],_
+[1.59,-52.0765718,sec(1.59),sec(1.59)-(-52.0765718)],_
+[1.60,-34.2471356,sec(1.60),sec(1.60)-(-34.2471356)]]
+--R 
+--R
+--R   (1)
+--R   [[0.01,1.00005,1.0000500020 83418059,0.2083418059 E -8],
+--R    [0.02,1.0002,1.0002000333 387564351,0.3333875643 51 E -7],
+--R    [0.03,1.0004502,1.0004501688 117850454,- 0.3118821495 46 E -7],
+--R    [0.04,1.0008005,1.0008005336 805808191,0.3368058081 91 E -7],
+--R    [0.05,1.0012513,1.0012513034 084612212,0.3408461221 2 E -8],
+--R    [0.06,1.0018027,1.0018027039 585779448,0.3958577944 8 E -8],
+--R    [0.07,1.002455,1.0024550120 706596665,0.1207065966 65 E -7],
+--R    [0.08,1.0032086,1.0032085556 00535503,- 0.4439946449 7 E -7],
+--R    [0.09,1.0040637,1.0040637139 232158779,0.1392321587 79 E -7],
+--R    [0.1,1.0050209,1.0050209184 004554285,0.1840045542 85 E -7],
+--R    [0.11,1.0060807,1.0060806529 138739295,- 0.4708612607 05 E -7],
+--R    [0.12,1.0072435,1.0072434544 648665414,- 0.4553513345 86 E -7],
+--R    [0.13,1.0085099,1.0085099138 426946141,0.1384269461 41 E -7],
+--R    [0.14,1.0098807,1.0098806763 623134089,- 0.2363768659 11 E -7],
+--R    [0.15,1.0113564,1.0113564426 736640909,0.4267366409 09 E -7],
+--R    [0.16,1.012938,1.0129379696 44334881,- 0.3035566511 9 E -7],
+--R    [0.17,1.0146261,1.0146260713 176810517,- 0.2868231894 83 E -7],
+--R    [0.18,1.0164216,1.0164216199 486862747,0.1994868627 47 E -7],
+--R    [0.19,1.0183255,1.0183255471 200494754,0.4712004947 54 E -7],
+--R    [0.2,1.0203388,1.0203388449 411926898,0.4494119268 98 E -7],
+--R    [0.21,1.0224626,1.0224625673 331073524,- 0.3266689264 76 E -7],
+--R    [0.22,1.0246978,1.0246978314 021899569,0.3140218995 69 E -7],
+--R    [0.23,1.0270458,1.0270458189 064641552,0.1890646415 52 E -7],
+--R    [0.24,1.0295078,1.0295077778 178462266,- 0.2218215377 34 E -7],
+--R    [0.25,1.032085,1.0320850239 843856444,0.2398438564 44 E -7],
+--R    [0.26,1.0347789,1.0347789428 967034948,0.4289670349 48 E -7],
+--R    [0.27,1.037591,1.0375909915 631601515,- 0.8436839848 51 E -8],
+--R    [0.28,1.0405227,1.0405227004 986113757,0.4986113757 E -9],
+--R    [0.29,1.0435757,1.0435756758 319605292,- 0.2416803947 08 E -7],
+--R    [0.3,1.0467516,1.0467516015 380856009,0.1538085600 9 E -8],
+--R    [0.31,1.0500522,1.0500522418 001151664,0.4180011516 64 E -7],
+--R    [0.32,1.0534794,1.0534794435 08449282,0.4350844928 2 E -7],
+--R    [0.33,1.0570351,1.0570351389 033719014,0.3890337190 14 E -7],
+--R    [0.34,1.0607213,1.0607213483 685831203,0.4836858312 03 E -7],
+--R    [0.35,1.0645402,1.0645401833 834950584,- 0.1661650494 16 E -7],
+--R    [0.36,1.0684938,1.0684938496 426873472,0.4964268734 72 E -7],
+--R    [0.37,1.0725847,1.0725846503 515101556,- 0.4964848984 44 E -7],
+--R    [0.38,1.076815,1.0768149897 074578722,- 0.1029254212 78 E -7],
+--R    [0.39,1.0811874,1.0811873765 776187103,- 0.2342238128 97 E -7],
+--R    [0.4,1.0857044,1.0857044283 832387031,0.2838323870 31 E -7],
+--R    [0.41,1.0903689,1.0903688752 032272686,- 0.2479677273 14 E -7],
+--R    [0.42,1.0951836,1.0951835641 092806502,- 0.3589071934 98 E -7],
+--R    [0.43,1.1001515,1.1001514637 462144293,- 0.3625378557 07 E -7],
+--R    [0.44,1.1052757,1.1052756691 720828447,- 0.3082791715 53 E -7],
+--R    [0.45,1.1105594,1.1105594069 737272884,0.6973727288 4 E -8],
+--R    [0.46,1.116006,1.1160060406 745461642,0.4067454616 42 E -7],
+--R    [0.47,1.1216191,1.1216190764 525210869,- 0.2354747891 32 E -7],
+--R    [0.48,1.1274022,1.1274021691 878787428,- 0.3081212125 72 E -7],
+--R    [0.49,1.1333591,1.1333591288 612230659,0.2886122306 59 E -7],
+--R    [0.5,1.1394939,1.1394939273 245491223,0.2732454912 23 E -7],
+--R    [0.51,1.1458107,1.1458107054 692597334,0.5469259733 4 E -8],
+--R    [0.52,1.1523138,1.1523137808 171610842,- 0.1918283891 58 E -7],
+--R    [0.53,1.1590077,1.1590076555 624283961,- 0.4443757160 39 E -7],
+--R    [0.54,1.165897,1.1658970250 947227087,0.2509472270 87 E -7],
+--R    [0.55,1.1729868,1.1729867870 360221145,- 0.1296397788 55 E -7],
+--R    [0.56,1.1802821,1.1802820508 263244778,- 0.4917367552 22 E -7],
+--R    [0.57,1.1877881,1.1877881478 962049114,0.4789620491 14 E -7],
+--R    [0.58,1.1955106,1.1955106424 672935752,0.4246729357 52 E -7],
+--R    [0.59,1.2034553,1.2034553430 251038492,0.4302510384 92 E -7],
+--R    [0.6,1.2116283,1.2116283145 123167046,0.1451231670 46 E -7],
+--R    [0.61,1.2200359,1.2200358912 946465932,- 0.8705353406 83 E -8],
+--R    [0.62,1.2286847,1.2286846909 558135602,- 0.9044186439 82 E -8],
+--R    [0.63,1.2375816,1.2375816289 829659823,0.2898296598 23 E -7],
+--R    [0.64,1.2467339,1.2467339344 091835044,0.3440918350 44 E -7],
+--R    [0.65,1.2561492,1.2561491664 854909887,- 0.3351450901 13 E -7],
+--R    [0.66,1.2658352,1.2658352324 611882698,0.3246118826 98 E -7],
+--R    [0.67,1.2758004,1.2758004065 583108419,0.6558310841 9 E -8],
+--R    [0.68,1.2860534,1.2860533502 337547427,- 0.4976624525 73 E -7],
+--R    [0.69,1.2966031,1.2966031338 311052265,0.3383110522 65 E -7],
+--R    [0.7,1.3074593,1.3074592597 335938699,- 0.4026640613 01 E -7],
+--R    [0.71,1.3186317,1.3186316871 399746265,- 0.1286002537 35 E -7],
+--R    [0.72,1.3301309,1.3301308585 965713491,- 0.4140342865 09 E -7],
+--R    [0.73,1.3419677,1.3419677284 314378096,0.2843143780 96 E -7],
+--R    [0.74,1.3541538,1.3541537932 506339213,- 0.6749366078 7 E -8],
+--R    [0.75,1.3667011,1.3667011246 722261352,0.2467222613 52 E -7],
+--R    [0.76,1.3796224,1.3796224044 909559952,0.4490955995 2 E -8],
+--R    [0.77,1.392931,1.3929309624 858048639,- 0.3751419513 61 E -7],
+--R    [0.78,1.4066408,1.4066408171 041612211,0.1710416122 11 E -7],
+--R    [0.79,1.4207667,1.4207667192 802507471,0.1928025074 71 E -7],
+--R    [0.8,1.4353242,1.4353241996 722398005,- 0.3277601995 E -9],
+--R    [0.81,1.4503296,1.4503296196 323374816,0.1963233748 16 E -7],
+--R    [0.82,1.4658002,1.4658002262 577216995,0.2625772169 95 E -7],
+--R    [0.83,1.4817542,1.4817542119 076845659,0.1190768456 59 E -7],
+--R    [0.84,1.4982108,1.4982107786 145889242,- 0.2138541107 58 E -7],
+--R    [0.85,1.5151902,1.5151902078 636927634,0.7863692763 37 E -8],
+--R    [0.86,1.5327139,1.5327139362 703728857,0.3627037288 57 E -7],
+--R    [0.87,1.5508046,1.5508046377 436210518,0.3774362105 18 E -7],
+--R    [0.88,1.5694863,1.5694863127 928890316,0.1279288903 16 E -7],
+--R    [0.89,1.5887844,1.5887843857 125782891,- 0.1428742171 09 E -7],
+--R    [0.9,1.6087258,1.6087258104 66049513,0.1046604951 3 E -7],
+--R    [0.91,1.6293392,1.6293391861 905347268,- 0.1380946527 32 E -7],
+--R    [0.92,1.6506549,1.6506548833 576031068,- 0.1664239689 32 E -7],
+--R    [0.93,1.6727052,1.6727051817 530083925,- 0.1824699160 75 E -7],
+--R    [0.94,1.6955244,1.6955244215 873528318,0.2158735283 18 E -7],
+--R    [0.95,1.7191492,1.7191491692 180088055,- 0.3078199119 45 E -7],
+--R    [0.96,1.7436184,1.7436183991 566481012,- 0.8433518988 E -9],
+--R    [0.97,1.7689737,1.7689736942 596852048,- 0.5740314795 2 E -8],
+--R    [0.98,1.7952595,1.7952594662 558616387,- 0.3374413836 13 E -7],
+--R    [0.99,1.8225232,1.8225231990 619311821,- 0.9380688178 8 E -9],
+--R    [1.0,1.8508157,1.8508157176 809256179,0.1768092561 79 E -7],
+--R    [1.01,1.8801915,1.8801914858 76130416,- 0.1412386958 4 E -7],
+--R    [1.02,1.9107089,1.9107089362 776886454,0.3627768864 54 E -7],
+--R    [1.03,1.9424308,1.9424308371 197244332,0.3711972443 32 E -7],
+--R    [1.04,1.9754247,1.9754247004 385756762,0.4385756762 E -9],
+--R    [1.05,2.0097632,2.0097632373 047619814,0.3730476198 15 E -7],
+--R    [1.06,2.0455249,2.0455248665 340933506,- 0.3346590664 94 E -7],
+--R    [1.07,2.0827943,2.0827942843 529550117,- 0.1564704498 83 E -7],
+--R    [1.08,2.1216631,2.1216631037 112423848,0.3711242384 8 E -8],
+--R    [1.09,2.1622306,2.1622305733 829374458,- 0.2661706255 42 E -7],
+--R    [1.1,2.2046044,2.2046043887 17359034,- 0.1128264096 6 E -7],
+--R    [1.11,2.2489016,2.2489016079 636949635,0.7963694963 5 E -8],
+--R    [1.12,2.2952497,2.2952496905 621863932,- 0.9437813606 7 E -8],
+--R    [1.13,2.3437877,2.3437876767 70567607,- 0.2322943239 29 E -7],
+--R    [1.14,2.3946675,2.3946675315 911689408,0.3159116894 08 E -7],
+--R    [1.15,2.4480557,2.4480556803 301802962,- 0.1966981970 38 E -7],
+--R    [1.16,2.5041348,2.5041347684 432143166,- 0.3155678568 34 E -7],
+--R    [1.17,2.5631057,2.5631056848 391647153,- 0.1516083528 47 E -7],
+--R    [1.18,2.6251899,2.6251898958 332672918,- 0.4166732708 2 E -8],
+--R    [1.19,2.6906321,2.6906321468 544274307,0.4685442743 07 E -7],
+--R    [1.2,2.7597036,2.7597036013 324064569,0.1332406457 E -8],
+--R    [1.21,2.8327055,2.8327055015 843215452,0.1584321545 E -8],
+--R    [1.22,2.9099735,2.9099734558 632119939,- 0.4413678800 61 E -7],
+--R    [1.23,2.9918825,2.9918824801 830430638,- 0.1981695693 62 E -7],
+--R    [1.24,3.078853,3.0788529546 387066657,- 0.4536129333 43 E -7],
+--R    [1.25,3.1713577,3.1713576937 701033609,- 0.6229896639 1 E -8],
+--R    [1.26,3.2699304,3.2699303818 839873852,- 0.1811601261 48 E -7],
+--R    [1.27,3.3751757,3.3751756909 804379951,- 0.9019562004 9 E -8],
+--R    [1.28,3.4877815,3.4877814863 175858326,- 0.1368241416 7 E -7],
+--R    [1.29,3.6085336,3.6085336400 439333502,0.4004393335 02 E -7],
+--R    [1.3,3.7383341,3.7383341270 754411719,0.2707544117 19 E -7],
+--R    [1.31,3.8782233,3.8782232842 002831965,- 0.1579971680 35 E -7],
+--R    [1.32,4.0294074,4.0294073944 135738582,- 0.5586426141 8 E -8],
+--R    [1.33,4.1932931,4.1932931445 375416679,0.4453754166 79 E -7],
+--R    [1.34,4.371531,4.3715310407 788541477,0.4077885414 77 E -7],
+--R    [1.35,4.5660706,4.5660706221 959586487,0.2219595864 9 E -7],
+--R    [1.36,4.7792314,4.7792313898 566530439,- 0.1014334695 6 E -7],
+--R    [1.37,5.0137949,5.0137949303 632701112,0.3036327011 12 E -7],
+--R    [1.38,5.273126,5.2731260096 064537537,0.9606453753 8 E -8],
+--R    [1.39,5.5613339,5.5613338533 248644648,- 0.4667513553 52 E -7],
+--R    [1.4,5.8834901,5.8834900848 27344827,- 0.1517265517 3 E -7],
+--R    [1.41,6.245928,6.2459279846 284984955,- 0.1537150150 4 E -7],
+--R    [1.42,6.6566608,6.6566608218 158868243,0.2181588682 4 E -7],
+--R    [1.43,7.1259785,7.1259784562 869696006,- 0.4371303039 94 E -7],
+--R    [1.44,7.6673176,7.6673176194 346817373,0.1943468173 7 E -7],
+--R    [1.45,8.2985645,8.2985644666 089360285,- 0.3339106397 1 E -7],
+--R    [1.46,9.0440625,9.0440625062 951398291,0.6295139829 E -8],
+--R    [1.47,9.9378158,9.9378157688 028373544,- 0.3119716264 6 E -7],
+--R    [1.48,11.0288087,11.0288086918 92599526,- 0.8107400473 9 E -8],
+--R    [1.49,12.3902766,12.3902765961 36436494,- 0.3863563506 E -8],
+--R    [1.5,14.1368329,14.1368329029 69903082,0.2969903082 E -8],
+--R    [1.51,16.4584992,16.4584992338 68991507,0.3386899150 7 E -7],
+--R    [1.52,19.6949314,19.6949314465 35659932,0.4653565993 2 E -7],
+--R    [1.53,24.5188114,24.5188114348 30172969,0.3483017296 9 E -7],
+--R    [1.54,32.4765383,32.4765382933 55461787,- 0.6644538213 E -8],
+--R    [1.55,48.088881,48.0888810173 88679831,0.1738867983 E -7],
+--R    [1.56,92.6258945,92.6258945325 36624847,0.3253662485 E -7],
+--R    [1.57,1255.7659897,1255.7659896641 100942,- 0.358899058 E -7],
+--R    [1.58,- 108.6538055,- 108.6538054739 3097269,0.2606902731 E -7],
+--R    [1.59,- 52.0765718,- 52.0765717826 07291365,0.1739270863 E -7],
+--R    [1.6,- 34.2471356,- 34.2471356100 18689204,- 0.100186892 E -7]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of csc(x), ranging from 0.01 to 1.60. The
+second column is the reference value of csc from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+<<*>>=
+
+--S 2 of 2
+[[0.01,100.0016667,csc(0.01),csc(0.01)-100.0016667],_
+[0.02,50.0033335,csc(0.02),csc(0.02)-50.0033335],_
+[0.03,33.3383339,csc(0.03),csc(0.03)-33.3383339],_
+[0.04,25.0066679,csc(0.04),csc(0.04)-25.0066679],_
+[0.05,20.0083358,csc(0.05),csc(0.05)-20.0083358],_
+[0.06,16.6766709,csc(0.06),csc(0.06)-16.6766709],_
+[0.07,14.2973876,csc(0.07),csc(0.07)-14.2973876],_
+[0.08,12.5133432,csc(0.08),csc(0.08)-12.5133432],_
+[0.09,11.1261253,csc(0.09),csc(0.09)-11.1261253],_
+[0.10,10.0166861,csc(0.10),csc(0.10)-10.0166861],_
+[0.11,9.1092683,csc(0.11),csc(0.11)-9.1092683],_
+[0.12,8.3533670,csc(0.12),csc(0.12)-8.3533670],_
+[0.13,7.7140172,csc(0.13),csc(0.13)-7.7140172],_
+[0.14,7.1662439,csc(0.14),csc(0.14)-7.1662439],_
+[0.15,6.6917324,csc(0.15),csc(0.15)-6.6917324],_
+[0.16,6.2767465,csc(0.16),csc(0.16)-6.2767465],_
+[0.17,5.9107821,csc(0.17),csc(0.17)-5.9107821],_
+[0.18,5.5856693,csc(0.18),csc(0.18)-5.5856693],_
+[0.19,5.2949584,csc(0.19),csc(0.19)-5.2949584],_
+[0.20,5.0334895,csc(0.20),csc(0.20)-5.0334895],_
+[0.21,4.7970857,csc(0.21),csc(0.21)-4.7970857],_
+[0.22,4.5823293,csc(0.22),csc(0.22)-4.5823293],_
+[0.23,4.3863973,csc(0.23),csc(0.23)-4.3863973],_
+[0.24,4.2069371,csc(0.24),csc(0.24)-4.2069371],_
+[0.25,4.0419725,csc(0.25),csc(0.25)-4.0419725],_
+[0.26,3.8898314,csc(0.26),csc(0.26)-3.8898314],_
+[0.27,3.7490894,csc(0.27),csc(0.27)-3.7490894],_
+[0.28,3.6185256,csc(0.28),csc(0.28)-3.6185256],_
+[0.29,3.4970877,csc(0.29),csc(0.29)-3.4970877],_
+[0.30,3.3838634,csc(0.30),csc(0.30)-3.3838634],_
+[0.31,3.2780583,csc(0.31),csc(0.31)-3.2780583],_
+[0.32,3.1789774,csc(0.32),csc(0.32)-3.1789774],_
+[0.33,3.0860099,csc(0.33),csc(0.33)-3.0860099],_
+[0.34,2.9986168,csc(0.34),csc(0.34)-2.9986168],_
+[0.35,2.9163208,csc(0.35),csc(0.35)-2.9163208],_
+[0.36,2.8386975,csc(0.36),csc(0.36)-2.8386975],_
+[0.37,2.7653687,csc(0.37),csc(0.37)-2.7653687],_
+[0.38,2.6959957,csc(0.38),csc(0.38)-2.6959957],_
+[0.39,2.6302748,csc(0.39),csc(0.39)-2.6302748],_
+[0.40,2.5679325,csc(0.40),csc(0.40)-2.5679325],_
+[0.41,2.5087220,csc(0.41),csc(0.41)-2.5087220],_
+[0.42,2.4524203,csc(0.42),csc(0.42)-2.4524203],_
+[0.43,2.3988248,csc(0.43),csc(0.43)-2.3988248],_
+[0.44,2.3477515,csc(0.44),csc(0.44)-2.3477515],_
+[0.45,2.2990327,csc(0.45),csc(0.45)-2.2990327],_
+[0.46,2.2525155,csc(0.46),csc(0.46)-2.2525155],_
+[0.47,2.2080598,csc(0.47),csc(0.47)-2.2080598],_
+[0.48,2.1655372,csc(0.48),csc(0.48)-2.1655372],_
+[0.49,2.1248300,csc(0.49),csc(0.49)-2.1248300],_
+[0.50,2.0858296,csc(0.50),csc(0.50)-2.0858296],_
+[0.51,2.0484363,csc(0.51),csc(0.51)-2.0484363],_
+[0.52,2.0125578,csc(0.52),csc(0.52)-2.0125578],_
+[0.53,1.9781089,csc(0.53),csc(0.53)-1.9781089],_
+[0.54,1.9450107,csc(0.54),csc(0.54)-1.9450107],_
+[0.55,1.9131900,csc(0.55),csc(0.55)-1.9131900],_
+[0.56,1.8825790,csc(0.56),csc(0.56)-1.8825790],_
+[0.57,1.8531145,csc(0.57),csc(0.57)-1.8531145],_
+[0.58,1.8247378,csc(0.58),csc(0.58)-1.8247378],_
+[0.59,1.7973941,csc(0.59),csc(0.59)-1.7973941],_
+[0.60,1.7710322,csc(0.60),csc(0.60)-1.7710322],_
+[0.61,1.7456045,csc(0.61),csc(0.61)-1.7456045],_
+[0.62,1.7210662,csc(0.62),csc(0.62)-1.7210662],_
+[0.63,1.6973757,csc(0.63),csc(0.63)-1.6973757],_
+[0.64,1.6744937,csc(0.64),csc(0.64)-1.6744937],_
+[0.65,1.6523834,csc(0.65),csc(0.65)-1.6523834],_
+[0.66,1.6310105,csc(0.66),csc(0.66)-1.6310105],_
+[0.67,1.6103423,csc(0.67),csc(0.67)-1.6103423],_
+[0.68,1.5903484,csc(0.68),csc(0.68)-1.5903484],_
+[0.69,1.5710001,csc(0.69),csc(0.69)-1.5710001],_
+[0.70,1.5522703,csc(0.70),csc(0.70)-1.5522703],_
+[0.71,1.5341335,csc(0.71),csc(0.71)-1.5341335],_
+[0.72,1.5165654,csc(0.72),csc(0.72)-1.5165654],_
+[0.73,1.4995435,csc(0.73),csc(0.73)-1.4995435],_
+[0.74,1.4830460,csc(0.74),csc(0.74)-1.4830460],_
+[0.75,1.4670527,csc(0.75),csc(0.75)-1.4670527],_
+[0.76,1.4515443,csc(0.76),csc(0.76)-1.4515443],_
+[0.77,1.4365025,csc(0.77),csc(0.77)-1.4365025],_
+[0.78,1.4219099,csc(0.78),csc(0.78)-1.4219099],_
+[0.79,1.4077503,csc(0.79),csc(0.79)-1.4077503],_
+[0.80,1.3940078,csc(0.80),csc(0.80)-1.3940078],_
+[0.81,1.3806678,csc(0.81),csc(0.81)-1.3806678],_
+[0.82,1.3677162,csc(0.82),csc(0.82)-1.3677162],_
+[0.83,1.3551396,csc(0.83),csc(0.83)-1.3551396],_
+[0.84,1.3429252,csc(0.84),csc(0.84)-1.3429252],_
+[0.85,1.3310609,csc(0.85),csc(0.85)-1.3310609],_
+[0.86,1.3195353,csc(0.86),csc(0.86)-1.3195353],_
+[0.87,1.3083372,csc(0.87),csc(0.87)-1.3083372],_
+[0.88,1.2974563,csc(0.88),csc(0.88)-1.2974563],_
+[0.89,1.2868825,csc(0.89),csc(0.89)-1.2868825],_
+[0.90,1.2766062,csc(0.90),csc(0.90)-1.2766062],_
+[0.91,1.2666184,csc(0.91),csc(0.91)-1.2666184],_
+[0.92,1.2569105,csc(0.92),csc(0.92)-1.2569105],_
+[0.93,1.2474740,csc(0.93),csc(0.93)-1.2474740],_
+[0.94,1.2383010,csc(0.94),csc(0.94)-1.2383010],_
+[0.95,1.2293840,csc(0.95),csc(0.95)-1.2293840],_
+[0.96,1.2207157,csc(0.96),csc(0.96)-1.2207157],_
+[0.97,1.2122891,csc(0.97),csc(0.97)-1.2122891],_
+[0.98,1.2040977,csc(0.98),csc(0.98)-1.2040977],_
+[0.99,1.1961351,csc(0.99),csc(0.99)-1.1961351],_
+[1.00,1.1883951,csc(1.00),csc(1.00)-1.1883951],_
+[1.01,1.1808720,csc(1.01),csc(1.01)-1.1808720],_
+[1.02,1.1735601,csc(1.02),csc(1.02)-1.1735601],_
+[1.03,1.1664542,csc(1.03),csc(1.03)-1.1664542],_
+[1.04,1.1595490,csc(1.04),csc(1.04)-1.1595490],_
+[1.05,1.1528398,csc(1.05),csc(1.05)-1.1528398],_
+[1.06,1.1463217,csc(1.06),csc(1.06)-1.1463217],_
+[1.07,1.1399902,csc(1.07),csc(1.07)-1.1399902],_
+[1.08,1.1338411,csc(1.08),csc(1.08)-1.1338411],_
+[1.09,1.1278701,csc(1.09),csc(1.09)-1.1278701],_
+[1.10,1.1220733,csc(1.10),csc(1.10)-1.1220733],_
+[1.11,1.1164469,csc(1.11),csc(1.11)-1.1164469],_
+[1.12,1.1109871,csc(1.12),csc(1.12)-1.1109871],_
+[1.13,1.1056905,csc(1.13),csc(1.13)-1.1056905],_
+[1.14,1.1005537,csc(1.14),csc(1.14)-1.1005537],_
+[1.15,1.0955735,csc(1.15),csc(1.15)-1.0955735],_
+[1.16,1.0907467,csc(1.16),csc(1.16)-1.0907467],_
+[1.17,1.0860704,csc(1.17),csc(1.17)-1.0860704],_
+[1.18,1.0815417,csc(1.18),csc(1.18)-1.0815417],_
+[1.19,1.0771579,csc(1.19),csc(1.19)-1.0771579],_
+[1.20,1.0729164,csc(1.20),csc(1.20)-1.0729164],_
+[1.21,1.0688146,csc(1.21),csc(1.21)-1.0688146],_
+[1.22,1.0648501,csc(1.22),csc(1.22)-1.0648501],_
+[1.23,1.0610206,csc(1.23),csc(1.23)-1.0610206],_
+[1.24,1.0573239,csc(1.24),csc(1.24)-1.0573239],_
+[1.25,1.0537579,csc(1.25),csc(1.25)-1.0537579],_
+[1.26,1.0503205,csc(1.26),csc(1.26)-1.0503205],_
+[1.27,1.0470098,csc(1.27),csc(1.27)-1.0470098],_
+[1.28,1.0438241,csc(1.28),csc(1.28)-1.0438241],_
+[1.29,1.0407614,csc(1.29),csc(1.29)-1.0407614],_
+[1.30,1.0378200,csc(1.30),csc(1.30)-1.0378200],_
+[1.31,1.0349985,csc(1.31),csc(1.31)-1.0349985],_
+[1.32,1.0322953,csc(1.32),csc(1.32)-1.0322953],_
+[1.33,1.0297088,csc(1.33),csc(1.33)-1.0297088],_
+[1.34,1.0272377,csc(1.34),csc(1.34)-1.0272377],_
+[1.35,1.0248807,csc(1.35),csc(1.35)-1.0248807],_
+[1.36,1.0226365,csc(1.36),csc(1.36)-1.0226365],_
+[1.37,1.0205039,csc(1.37),csc(1.37)-1.0205039],_
+[1.38,1.0184818,csc(1.38),csc(1.38)-1.0184818],_
+[1.39,1.0165693,csc(1.39),csc(1.39)-1.0165693],_
+[1.40,1.0147651,csc(1.40),csc(1.40)-1.0147651],_
+[1.41,1.0130685,csc(1.41),csc(1.41)-1.0130685],_
+[1.42,1.0114785,csc(1.42),csc(1.42)-1.0114785],_
+[1.43,1.0099943,csc(1.43),csc(1.43)-1.0099943],_
+[1.44,1.0086152,csc(1.44),csc(1.44)-1.0086152],_
+[1.45,1.0073405,csc(1.45),csc(1.45)-1.0073405],_
+[1.46,1.0061695,csc(1.46),csc(1.46)-1.0061695],_
+[1.47,1.0051015,csc(1.47),csc(1.47)-1.0051015],_
+[1.48,1.0041362,csc(1.48),csc(1.48)-1.0041362],_
+[1.49,1.0032729,csc(1.49),csc(1.49)-1.0032729],_
+[1.50,1.0025113,csc(1.50),csc(1.50)-1.0025113],_
+[1.51,1.0018509,csc(1.51),csc(1.51)-1.0018509],_
+[1.52,1.0012915,csc(1.52),csc(1.52)-1.0012915],_
+[1.53,1.0008327,csc(1.53),csc(1.53)-1.0008327],_
+[1.54,1.0004744,csc(1.54),csc(1.54)-1.0004744],_
+[1.55,1.0002163,csc(1.55),csc(1.55)-1.0002163],_
+[1.56,1.0000583,csc(1.56),csc(1.56)-1.0000583],_
+[1.57,1.0000003,csc(1.57),csc(1.57)-1.0000003],_
+[1.58,1.0000424,csc(1.58),csc(1.58)-1.0000424],_
+[1.59,1.0001844,csc(1.59),csc(1.59)-1.0001844],_
+[1.60,1.0004266,csc(1.60),csc(1.60)-1.0004266]]
+--R 
+--R
+--R   (2)
+--R   [[0.01,100.0016667,100.0016666861 1131614,- 0.1388868386 E -7],
+--R    [0.02,50.0033335,50.0033334888 95450004,- 0.1110455 E -7],
+--R    [0.03,33.3383339,33.3383338583 83159355,- 0.4161684064 5 E -7],
+--R    [0.04,25.0066679,25.0066679113 21092611,0.1132109261 E -7],
+--R    [0.05,20.0083358,20.0083357645 29760655,- 0.3547023934 5 E -7],
+--R    [0.06,16.6766709,16.6766708682 61540425,- 0.3173845957 5 E -7],
+--R    [0.07,14.2973876,14.2973876252 73006648,0.2527300664 8 E -7],
+--R    [0.08,12.5133432,12.5133432956 11602375,0.9561160237 45 E -7],
+--R    [0.09,11.1261253,11.1261252982 27770129,- 0.1772229871 E -8],
+--R    [0.1,10.0166861,10.0166861316 34776649,0.3163477664 9 E -7],
+--R    [0.11,9.1092683,9.1092683378 58666259,0.3785866625 9 E -7],
+--R    [0.12,8.353367,8.3533669844 258284979,- 0.1557417150 2 E -7],
+--R    [0.13,7.7140172,7.7140171546 756812171,- 0.4532431878 29 E -7],
+--R    [0.14,7.1662439,7.1662439422 359760119,0.4223597601 19 E -7],
+--R    [0.15,6.6917324,6.6917324477 182351442,0.4771823514 42 E -7],
+--R    [0.16,6.2767465,6.2767465266 620801848,0.2666208018 5 E -7],
+--R    [0.17,5.9107821,5.9107820970 378013996,- 0.29621986 E -8],
+--R    [0.18,5.5856693,5.5856693442 568113649,0.4425681136 49 E -7],
+--R    [0.19,5.2949584,5.2949584403 976760687,0.4039767606 87 E -7],
+--R    [0.2,5.0334895,5.0334895476 723442024,0.4767234420 24 E -7],
+--R    [0.21,4.7970857,4.7970856780 525825143,- 0.2194741748 6 E -7],
+--R    [0.22,4.5823293,4.5823293184 602630424,0.1846026304 2 E -7],
+--R    [0.23,4.3863973,4.3863973276 543190295,0.2765431903 E -7],
+--R    [0.24,4.2069371,4.2069371089 029182611,0.8902918261 1 E -8],
+--R    [0.25,4.0419725,4.0419725012 210710728,0.1221071073 E -8],
+--R    [0.26,3.8898314,3.8898313880 215197583,- 0.1197848024 2 E -7],
+--R    [0.27,3.7490894,3.7490893927 385091643,- 0.7261490835 7 E -8],
+--R    [0.28,3.6185256,3.6185256396 812993389,0.3968129933 89 E -7],
+--R    [0.29,3.4970877,3.4970876678 206681256,- 0.3217933187 44 E -7],
+--R    [0.3,3.3838634,3.3838633618 241225849,- 0.3817587741 51 E -7],
+--R    [0.31,3.2780583,3.2780583157 998724616,0.1579987246 16 E -7],
+--R    [0.32,3.1789774,3.1789774413 44609289,0.4134460928 89 E -7],
+--R    [0.33,3.0860099,3.0860099195 883096277,0.1958830962 77 E -7],
+--R    [0.34,2.9986168,2.9986168087 661743229,0.8766174322 8 E -8],
+--R    [0.35,2.9163208,2.9163207762 123651255,- 0.2378763487 45 E -7],
+--R    [0.36,2.8386975,2.8386975416 936122006,0.4169361220 06 E -7],
+--R    [0.37,2.7653687,2.7653687083 157807533,0.8315780753 3 E -8],
+--R    [0.38,2.6959957,2.6959957253 979436817,0.2539794368 17 E -7],
+--R    [0.39,2.6302748,2.6302747801 40398548,- 0.1985960145 2 E -7],
+--R    [0.4,2.5679325,2.5679324555 477830703,- 0.4445221692 97 E -7],
+--R    [0.41,2.508722,2.5087220237 8334365,0.2378334365 E -7],
+--R    [0.42,2.4524203,2.4524202690 492617941,- 0.3095073820 59 E -7],
+--R    [0.43,2.3988248,2.3988247537 912204017,- 0.4620877959 83 E -7],
+--R    [0.44,2.3477515,2.3477514576 984533152,- 0.4230154668 48 E -7],
+--R    [0.45,2.2990327,2.2990327315 089707901,0.3150897079 01 E -7],
+--R    [0.46,2.2525155,2.2525155177 149280284,0.1771492802 84 E -7],
+--R    [0.47,2.2080598,2.2080597984 171557468,- 0.1582844253 E -8],
+--R    [0.48,2.1655372,2.1655372372 03038964,0.3720303896 4 E -7],
+--R    [0.49,2.12483,2.1248299873 30230062,- 0.1266976993 8 E -7],
+--R    [0.5,2.0858296,2.0858296429 334881858,0.4293348818 58 E -7],
+--R    [0.51,2.0484363,2.0484363136 241303362,0.1362413033 6 E -7],
+--R    [0.52,2.0125578,2.0125578058 716625444,0.5871662544 4 E -8],
+--R    [0.53,1.9781089,1.9781088970 643983743,- 0.2935601625 7 E -8],
+--R    [0.54,1.9450107,1.9450106902 35237839,- 0.9764762161 01 E -8],
+--R    [0.55,1.91319,1.9131900391 862495448,0.3918624954 48 E -7],
+--R    [0.56,1.882579,1.8825790352 123253874,0.3521232538 73 E -7],
+--R    [0.57,1.8531145,1.8531145478 592311512,0.4785923115 12 E -7],
+--R    [0.58,1.8247378,1.8247378131 947842124,0.1319478421 24 E -7],
+--R    [0.59,1.7973941,1.7973940639 561336125,- 0.3604386638 75 E -7],
+--R    [0.6,1.7710322,1.7710321966 877253734,- 0.3312274626 6 E -8],
+--R    [0.61,1.7456045,1.7456044716 252507363,- 0.2837474926 37 E -7],
+--R    [0.62,1.7210662,1.7210662416 285829803,0.4162858298 03 E -7],
+--R    [0.63,1.6973757,1.6973757069 361725968,0.6936172596 8 E -8],
+--R    [0.64,1.6744937,1.6744936929 168164811,- 0.7083183518 9 E -8],
+--R    [0.65,1.6523834,1.6523834483 423011016,0.4834230110 16 E -7],
+--R    [0.66,1.6310105,1.6310104620 046062096,- 0.3799539379 04 E -7],
+--R    [0.67,1.6103423,1.6103422957 612188444,- 0.4238781155 6 E -8],
+--R    [0.68,1.5903484,1.5903484323 175770922,0.3231757709 22 E -7],
+--R    [0.69,1.5710001,1.5710001362 517238398,0.3625172383 98 E -7],
+--R    [0.7,1.5522703,1.5522703269 57103912,0.2695710391 2 E -7],
+--R    [0.71,1.5341335,1.5341334623 286338699,- 0.3767136613 01 E -7],
+--R    [0.72,1.5165654,1.5165654321 477203357,0.3214772033 57 E -7],
+--R    [0.73,1.4995435,1.4995434602 363547077,- 0.3976364529 23 E -7],
+--R    [0.74,1.483046,1.4830460145 509444868,0.1455094448 68 E -7],
+--R    [0.75,1.4670527,1.4670527244 75010117,0.2447501011 7 E -7],
+--R    [0.76,1.4515443,1.4515443046 478685727,0.4647868572 7 E -8],
+--R    [0.77,1.4365025,1.4365024847 353014963,- 0.1526469850 37 E -7],
+--R    [0.78,1.4219099,1.4219099446 09135172,0.4460913517 2 E -7],
+--R    [0.79,1.4077503,1.4077502544 566480148,- 0.4554335198 52 E -7],
+--R    [0.8,1.3940078,1.3940078193 886361745,0.1938863617 45 E -7],
+--R    [0.81,1.3806678,1.3806678281 575591498,0.2815755914 98 E -7],
+--R    [0.82,1.3677162,1.3677162056 35104192,0.5635104192 E -8],
+--R    [0.83,1.3551396,1.3551395687 323139549,- 0.3126768604 51 E -7],
+--R    [0.84,1.3429252,1.3429251854 756057525,- 0.1452439424 75 E -7],
+--R    [0.85,1.3310609,1.3310609369 789990246,0.3697899902 46 E -7],
+--R    [0.86,1.3195353,1.3195352820 770317831,- 0.1792296821 69 E -7],
+--R    [0.87,1.3083372,1.3083372244 045115315,0.2440451153 15 E -7],
+--R    [0.88,1.2974563,1.2974562817 286954688,- 0.1827130453 12 E -7],
+--R    [0.89,1.2868825,1.2868824573 569776297,- 0.4264302237 03 E -7],
+--R    [0.9,1.2766062,1.2766062134 58895496,0.1345889549 6 E -7],
+--R    [0.91,1.2666184,1.2666184461 554476329,0.4615544763 29 E -7],
+--R    [0.92,1.2569105,1.2569104622 415061901,- 0.3775849380 99 E -7],
+--R    [0.93,1.247474,1.2474739574 186627875,- 0.4258133721 25 E -7],
+--R    [0.94,1.238301,1.2383009959 262950556,- 0.4073704944 4 E -8],
+--R    [0.95,1.229384,1.2293839914 681003518,- 0.8531899648 15 E -8],
+--R    [0.96,1.2207157,1.2207156893 399160175,- 0.1066008398 25 E -7],
+--R    [0.97,1.2122891,1.2122891496 724233695,0.4967242336 95 E -7],
+--R    [0.98,1.2040977,1.2040977317 093965936,0.3170939659 36 E -7],
+--R    [0.99,1.1961351,1.1961350790 485799284,- 0.2095142007 16 E -7],
+--R    [1.0,1.1883951,1.1883951057 781212163,0.5778121216 3 E -8],
+--R    [1.01,1.180872,1.1808719834 468142375,- 0.1655318576 25 E -7],
+--R    [1.02,1.1735601,1.1735601288 11257282,0.2881125728 2 E -7],
+--R    [1.03,1.1664542,1.1664541923 074667313,- 0.7692533268 72 E -8],
+--R    [1.04,1.159549,1.1595490471 9853279,0.4719853279 E -7],
+--R    [1.05,1.1528398,1.1528397793 536064181,- 0.2064639358 19 E -7],
+--R    [1.06,1.1463217,1.1463216776 16894679,- 0.2238310532 1 E -7],
+--R    [1.07,1.1399902,1.1399902247 284454927,0.2472844549 27 E -7],
+--R    [1.08,1.1338411,1.1338410887 613485139,- 0.1123865148 61 E -7],
+--R    [1.09,1.1278701,1.1278701150 42590264,0.1504259026 4 E -7],
+--R    [1.1,1.1220733,1.1220733185 272000494,0.1852720004 94 E -7],
+--R    [1.11,1.1164469,1.1164468765 975278264,- 0.2340247217 36 E -7],
+--R    [1.12,1.1109871,1.1109871222 615233658,0.2226152336 58 E -7],
+--R    [1.13,1.1056905,1.1056905377 257534608,0.3772575346 08 E -7],
+--R    [1.14,1.1005537,1.1005537483 206146725,0.4832061467 25 E -7],
+--R    [1.15,1.0955735,1.0955735167 567859913,0.1675678599 13 E -7],
+--R    [1.16,1.0907467,1.0907467376 934304007,0.3769343040 07 E -7],
+--R    [1.17,1.0860704,1.0860704326 000071521,0.3260000715 21 E -7],
+--R    [1.18,1.0815417,1.0815417448 948071115,0.4489480711 15 E -7],
+--R    [1.19,1.0771579,1.0771579353 444804665,0.3534448046 65 E -7],
+--R    [1.2,1.0729164,1.0729163777 098972287,- 0.2229010277 13 E -7],
+--R    [1.21,1.0688146,1.0688145546 246734829,- 0.4537532651 71 E -7],
+--R    [1.22,1.0648501,1.0648500536 936167054,- 0.4630638329 46 E -7],
+--R    [1.23,1.0610206,1.0610205637 991976338,- 0.3620080236 62 E -7],
+--R    [1.24,1.0573239,1.0573238716 04949513,- 0.2839505048 7 E -7],
+--R    [1.25,1.0537579,1.0537578582 454329877,- 0.4175456701 23 E -7],
+--R    [1.26,1.0503205,1.0503204961 930909767,- 0.3806909023 3 E -8],
+--R    [1.27,1.0470098,1.0470098462 929566457,0.4629295664 57 E -7],
+--R    [1.28,1.0438241,1.0438240549 567728718,- 0.4504322712 82 E -7],
+--R    [1.29,1.0407614,1.0407613515 086368049,- 0.4849136319 51 E -7],
+--R    [1.3,1.03782,1.0378200456 748014427,0.4567480144 27 E -7],
+--R    [1.31,1.0349985,1.0349985252 107504451,0.2521075044 51 E -7],
+--R    [1.32,1.0322953,1.0322952536 591153897,- 0.4634088461 03 E -7],
+--R    [1.33,1.0297088,1.0297087682 324287489,- 0.3176757125 11 E -7],
+--R    [1.34,1.0272377,1.0272376778 151033218,- 0.2218489667 82 E -7],
+--R    [1.35,1.0248807,1.0248806610 794017372,- 0.3892059826 28 E -7],
+--R    [1.36,1.0226365,1.0226364647 10509886,- 0.3528949011 4 E -7],
+--R    [1.37,1.0205039,1.0205039017 361575009,0.1736157500 9 E -8],
+--R    [1.38,1.0184818,1.0184818499 565392157,0.4995653921 57 E -7],
+--R    [1.39,1.0165693,1.0165692504 705818114,- 0.4952941818 86 E -7],
+--R    [1.4,1.0147651,1.0147651062 948794009,0.6294879400 9 E -8],
+--R    [1.41,1.0130685,1.0130684810 718793105,- 0.1892812068 95 E -7],
+--R    [1.42,1.0114785,1.0114784978 641485933,- 0.2135851406 7 E -8],
+--R    [1.43,1.0099943,1.0099943380 317855887,0.3803178558 87 E -7],
+--R    [1.44,1.0086152,1.0086152401 902637594,0.4019026375 94 E -7],
+--R    [1.45,1.0073405,1.0073404992 46207189,- 0.753792811 E -9],
+--R    [1.46,1.0061695,1.0061694655 087995079,- 0.3449120049 21 E -7],
+--R    [1.47,1.0051015,1.0051015438 747215132,0.4387472151 32 E -7],
+--R    [1.48,1.0041362,1.0041361930 846981533,- 0.6915301846 7 E -8],
+--R    [1.49,1.0032729,1.0032729250 49913637,0.2504991363 7 E -7],
+--R    [1.5,1.0025113,1.0025113042 4672491,0.424672491 E -8],
+--R    [1.51,1.0018509,1.0018509471 78269319,0.4717826931 9 E -7],
+--R    [1.52,1.0012915,1.0012915219 017225962,0.2190172259 62 E -7],
+--R    [1.53,1.0008327,1.0008327476 201189712,0.4762011897 12 E -7],
+--R    [1.54,1.0004744,1.0004743943 377968613,- 0.5662203138 7 E -8],
+--R    [1.55,1.0002163,1.0002162825 786817653,- 0.1742131823 47 E -7],
+--R    [1.56,1.0000583,1.0000582831 667632601,- 0.1683323673 99 E -7],
+--R    [1.57,1.0000003,1.0000003170 68265912,0.1706826591 2 E -7],
+--R    [1.58,1.0000424,1.0000423552 951549942,- 0.4470484500 58 E -7],
+--R    [1.59,1.0001844,1.0001844188 697576625,0.1886975766 25 E -7],
+--R    [1.6,1.0004266,1.0004265788 504192126,- 0.2114958078 74 E -7]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+\end{thebibliography}
+\end{document}
diff --git a/src/input/sincos.input.pamphlet b/src/input/sincos.input.pamphlet
new file mode 100644
index 0000000..915ee4b
--- /dev/null
+++ b/src/input/sincos.input.pamphlet
@@ -0,0 +1,696 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input sincos.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's sincos function with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of sin(x), ranging from 0.00 to 1.60. The
+second column is the reference value of sin from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp142-173
+
+<<*>>=
+)spool sincos.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 2
+[[0.01,0.00999983333416666468254,sin(0.01),sin(0.01)-(0.00999983333416666468254)],_
+[0.02,0.01999866669333307936649,sin(0.02),sin(0.02)-(0.01999866669333307936649)],_
+[0.03,0.02999550020249566076853,sin(0.03),sin(0.03)-(0.02999550020249566076853)],_
+[0.04,0.03998933418663415945255,sin(0.04),sin(0.04)-(0.03998933418663415945255)],_
+[0.05,0.04997916927067832879487,sin(0.05),sin(0.05)-(0.04997916927067832879487)],_
+[0.06,0.05996400647944459919909,sin(0.06),sin(0.06)-(0.05996400647944459919909)],_
+[0.07,0.06994284733753276397655,sin(0.07),sin(0.07)-(0.06994284733753276397655)],_
+[0.08,0.07991469396917268730688,sin(0.08),sin(0.08)-(0.07991469396917268730688)],_
+[0.09,0.08987854919801104969125,sin(0.09),sin(0.09)-(0.08987854919801104969125)],_
+[0.10,0.09983341664682815230681,sin(0.10),sin(0.10)-(0.09983341664682815230681)],_
+[0.11,0.10977830083717480866495,sin(0.11),sin(0.11)-(0.10977830083717480866495)],_
+[0.12,0.11971220728891935996735,sin(0.12),sin(0.12)-(0.11971220728891935996735)],_
+[0.13,0.12963414261969485954121,sin(0.13),sin(0.13)-(0.12963414261969485954121)],_
+[0.14,0.13954311464423648171799,sin(0.14),sin(0.14)-(0.13954311464423648171799)],_
+[0.15,0.14943813247359922149773,sin(0.15),sin(0.15)-(0.14943813247359922149773)],_
+[0.16,0.15931820661424596331146,sin(0.16),sin(0.16)-(0.15931820661424596331146)],_
+[0.17,0.16918234906699601015762,sin(0.17),sin(0.17)-(0.16918234906699601015762)],_
+[0.18,0.17902957342582417834180,sin(0.18),sin(0.18)-(0.17902957342582417834180)],_
+[0.19,0.18885889497650057799285,sin(0.19),sin(0.19)-(0.18885889497650057799285)],_
+[0.20,0.19866933079506121545941,sin(0.20),sin(0.20)-(0.19866933079506121545941)],_
+[0.21,0.20845989984609957060871,sin(0.21),sin(0.21)-(0.20845989984609957060871)],_
+[0.22,0.21822962308086931995179,sin(0.22),sin(0.22)-(0.21822962308086931995179)],_
+[0.23,0.22797752353518839540462,sin(0.23),sin(0.23)-(0.22797752353518839540462)],_
+[0.24,0.23770262642713458836079,sin(0.24),sin(0.24)-(0.23770262642713458836079)],_
+[0.25,0.24740395925452292959685,sin(0.25),sin(0.25)-(0.24740395925452292959685)],_
+[0.26,0.25708055189215509735339,sin(0.26),sin(0.26)-(0.25708055189215509735339)],_
+[0.27,0.26673143668883112873229,sin(0.27),sin(0.27)-(0.26673143668883112873229)],_
+[0.28,0.27635564856411373331967,sin(0.28),sin(0.28)-(0.27635564856411373331967)],_
+[0.29,0.28595222510483553268394,sin(0.29),sin(0.29)-(0.28595222510483553268394)],_
+[0.30,0.29552020666133957510532,sin(0.30),sin(0.30)-(0.29552020666133957510532)],_
+[0.31,0.30505863644344350156564,sin(0.31),sin(0.31)-(0.30505863644344350156564)],_
+[0.32,0.31456656061611776666176,sin(0.32),sin(0.32)-(0.31456656061611776666176)],_
+[0.33,0.32404302839486834670020,sin(0.33),sin(0.33)-(0.32404302839486834670020)],_
+[0.34,0.33348709214081439678177,sin(0.34),sin(0.34)-(0.33348709214081439678177)],_
+[0.35,0.34289780745545134918963,sin(0.35),sin(0.35)-(0.34289780745545134918963)],_
+[0.36,0.35227423327508997684991,sin(0.36),sin(0.36)-(0.35227423327508997684991)],_
+[0.37,0.36161543196496197803729,sin(0.37),sin(0.37)-(0.36161543196496197803729)],_
+[0.38,0.37092046941298267184549,sin(0.38),sin(0.38)-(0.37092046941298267184549)],_
+[0.39,0.38018841512316142823118,sin(0.39),sin(0.39)-(0.38018841512316142823118)],_
+[0.40,0.38941834230865049166631,sin(0.40),sin(0.40)-(0.38941834230865049166631)],_
+[0.41,0.39860932798442289359380,sin(0.41),sin(0.41)-(0.39860932798442289359380)],_
+[0.42,0.40776045305957018597279,sin(0.42),sin(0.42)-(0.40776045305957018597279)],_
+[0.43,0.41687080242921076621692,sin(0.43),sin(0.43)-(0.41687080242921076621692)],_
+[0.44,0.42593946506599960276972,sin(0.44),sin(0.44)-(0.42593946506599960276972)],_
+[0.45,0.43496553411123021042084,sin(0.45),sin(0.45)-(0.43496553411123021042084)],_
+[0.46,0.44394810696551976524151,sin(0.46),sin(0.46)-(0.44394810696551976524151)],_
+[0.47,0.45288628537906829070327,sin(0.47),sin(0.47)-(0.45288628537906829070327)],_
+[0.48,0.46177917554148288913664,sin(0.48),sin(0.48)-(0.46177917554148288913664)],_
+[0.49,0.47062588817115803618136,sin(0.49),sin(0.49)-(0.47062588817115803618136)],_
+[0.50,0.47942553860420300027329,sin(0.50),sin(0.50)-(0.47942553860420300027329)],_
+[0.51,0.48817724688290749450013,sin(0.51),sin(0.51)-(0.48817724688290749450013)],_
+[0.52,0.49688013784373671433446,sin(0.52),sin(0.52)-(0.49688013784373671433446)],_
+[0.53,0.50553334120484696181366,sin(0.53),sin(0.53)-(0.50553334120484696181366)],_
+[0.54,0.51413599165311310467728,sin(0.54),sin(0.54)-(0.51413599165311310467728)],_
+[0.55,0.52268722893065916778838,sin(0.55),sin(0.55)-(0.52268722893065916778838)],_
+[0.56,0.53118619792088340385187,sin(0.56),sin(0.56)-(0.53118619792088340385187)],_
+[0.57,0.53963204873396924099446,sin(0.57),sin(0.57)-(0.53963204873396924099446)],_
+[0.58,0.54802393679187355618270,sin(0.58),sin(0.58)-(0.54802393679187355618270)],_
+[0.59,0.55636102291278377572254,sin(0.59),sin(0.59)-(0.55636102291278377572254)],_
+[0.60,0.56464247339503535720095,sin(0.60),sin(0.60)-(0.56464247339503535720095)],_
+[0.61,0.57286746010048126119098,sin(0.61),sin(0.61)-(0.57286746010048126119098)],_
+[0.62,0.58103516053730507584296,sin(0.62),sin(0.62)-(0.58103516053730507584296)],_
+[0.63,0.58914475794226951311811,sin(0.63),sin(0.63)-(0.58914475794226951311811)],_
+[0.64,0.59719544136239205188355,sin(0.64),sin(0.64)-(0.59719544136239205188355)],_
+[0.65,0.60518640573603956037252,sin(0.65),sin(0.65)-(0.60518640573603956037252)],_
+[0.66,0.61311685197343378861515,sin(0.66),sin(0.66)-(0.61311685197343378861515)],_
+[0.67,0.62098598703655968035744,sin(0.67),sin(0.67)-(0.62098598703655968035744)],_
+[0.68,0.62879302401846851370418,sin(0.68),sin(0.68)-(0.62879302401846851370418)],_
+[0.69,0.63653718222196794023743,sin(0.69),sin(0.69)-(0.63653718222196794023743)],_
+[0.70,0.64421768723769105367261,sin(0.70),sin(0.70)-(0.64421768723769105367261)],_
+[0.71,0.65183377102153668121013,sin(0.71),sin(0.71)-(0.65183377102153668121013)],_
+[0.72,0.65938467197147315361800,sin(0.72),sin(0.72)-(0.65938467197147315361800)],_
+[0.73,0.66686963500369787373259,sin(0.73),sin(0.73)-(0.66686963500369787373259)],_
+[0.74,0.67428791162814506748388,sin(0.74),sin(0.74)-(0.67428791162814506748388)],_
+[0.75,0.68163876002333416673324,sin(0.75),sin(0.75)-(0.68163876002333416673324)],_
+[0.76,0.68892144511055133914776,sin(0.76),sin(0.76)-(0.68892144511055133914776)],_
+[0.77,0.69613523862735674701988,sin(0.77),sin(0.77)-(0.69613523862735674701988)],_
+[0.78,0.70327941920041018436790,sin(0.78),sin(0.78)-(0.70327941920041018436790)],_
+[0.79,0.71035327241760780981403,sin(0.79),sin(0.79)-(0.71035327241760780981403)],_
+[0.80,0.71735609089952276162718,sin(0.80),sin(0.80)-(0.71735609089952276162718)],_
+[0.81,0.72428717437014251092818,sin(0.81),sin(0.81)-(0.72428717437014251092818)],_
+[0.82,0.73114582972689587938131,sin(0.82),sin(0.82)-(0.73114582972689587938131)],_
+[0.83,0.73793137110996271872858,sin(0.83),sin(0.83)-(0.73793137110996271872858)],_
+[0.84,0.74464311997085932125657,sin(0.84),sin(0.84)-(0.74464311997085932125657)],_
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+[0.87,0.76432893702550507814480,sin(0.87),sin(0.87)-(0.76432893702550507814480)],_
+[0.88,0.77073887889896929120965,sin(0.88),sin(0.88)-(0.77073887889896929120965)],_
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+[0.93,0.80161994088377715208432,sin(0.93),sin(0.93)-(0.80161994088377715208432)],_
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+[1.40,0.98544972998846018065947,sin(1.40),sin(1.40)-(0.98544972998846018065947)],_
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+[1.43,0.99010456033717779485729,sin(1.43),sin(1.43)-(0.99010456033717779485729)],_
+[1.44,0.99145834819168646252760,sin(1.44),sin(1.44)-(0.99145834819168646252760)],_
+[1.45,0.99271299103758849766535,sin(1.45),sin(1.45)-(0.99271299103758849766535)],_
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+[1.49,0.99673775204314338855320,sin(1.49),sin(1.49)-(0.99673775204314338855320)],_
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+[1.51,0.99815247249754811924274,sin(1.51),sin(1.51)-(0.99815247249754811924274)],_
+[1.52,0.99871014397558300717231,sin(1.52),sin(1.52)-(0.99871014397558300717231)],_
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+[1.54,0.99952583060547905600596,sin(1.54),sin(1.54)-(0.99952583060547905600596)],_
+[1.55,0.99978376418935696389761,sin(1.55),sin(1.55)-(0.99978376418935696389761)],_
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+[1.60,0.99957360304150516434211,sin(1.60),sin(1.60)-(0.99957360304150516434211)]]
+--R 
+--R
+--R   (1)
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+--R    [1.03,0.8572989891 8860337214,0.8572989891 8860337215,0.3 E -20],
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+--R    [1.12,0.9001004421 7650499712,0.9001004421 7650499712,0.0],
+--R    [1.13,0.9044121893 7882591604,0.9044121893 7882591604,0.0],
+--R    [1.14,0.9086334961 1588326459,0.9086334961 1588326459,0.0],
+--R    [1.15,0.9127639402 6052108094,0.9127639402 6052108094,0.0],
+--R    [1.16,0.9168031087 7176692662,0.9168031087 7176692662,- 0.3 E -20],
+--R    [1.17,0.9207505977 3613563957,0.9207505977 3613563958,0.3 E -20],
+--R    [1.18,0.9246060124 0802034611,0.9246060124 0802034611,0.0],
+--R    [1.19,0.9283689672 491666926,0.9283689672 491666926,0.0],
+--R    [1.2,0.9320390859 6722634967,0.9320390859 6722634967,0.3 E -20],
+--R    [1.21,0.9356160015 5338593342,0.9356160015 5338593342,0.0],
+--R    [1.22,0.9390993563 1906758094,0.9390993563 1906758094,0.0],
+--R    [1.23,0.9424888019 3169751002,0.9424888019 3169751002,0.0],
+--R    [1.24,0.9457839994 4953898629,0.9457839994 4953898629,0.0],
+--R    [1.25,0.9489846193 5558621435,0.9489846193 5558621435,0.0],
+--R    [1.26,0.9520903415 9051576386,0.9520903415 9051576386,0.0],
+--R    [1.27,0.9551008555 8469223509,0.9551008555 8469223509,0.0],
+--R    [1.28,0.9580158602 892249637,0.9580158602 892249637,0.0],
+--R    [1.29,0.9608350642 0607265891,0.9608350642 0607265891,0.0],
+--R    [1.3,0.9635581854 171929647,0.9635581854 171929647,- 0.3 E -20],
+--R    [1.31,0.9661849516 1273402917,0.9661849516 1273402917,0.0],
+--R    [1.32,0.9687151001 1826526273,0.9687151001 1826526273,0.0],
+--R    [1.33,0.9711483779 2104456234,0.9711483779 2104456234,0.0],
+--R    [1.34,0.9734845416 9531937479,0.9734845416 9531937479,0.0],
+--R    [1.35,0.9757233578 2665906926,0.9757233578 2665906926,0.0],
+--R    [1.36,0.9778646024 3531618568,0.9778646024 3531618568,0.0],
+--R    [1.37,0.9799080613 9861422289,0.9799080613 9861422289,0.0],
+--R    [1.38,0.9818535303 7235972788,0.9818535303 7235972788,0.0],
+--R    [1.39,0.9837008148 1127654484,0.9837008148 1127654484,0.0],
+--R    [1.4,0.9854497299 8846018066,0.9854497299 8846018066,0.0],
+--R    [1.41,0.9871001010 1385034143,0.9871001010 1385034143,0.0],
+--R    [1.42,0.9886517628 5171979274,0.9886517628 5171979274,0.0],
+--R    [1.43,0.9901045603 3717779486,0.9901045603 3717779486,0.0],
+--R    [1.44,0.9914583481 9168646253,0.9914583481 9168646253,0.0],
+--R    [1.45,0.9927129910 3758849767,0.9927129910 3758849767,0.0],
+--R    [1.46,0.9938683634 1164484229,0.9938683634 1164484229,0.0],
+--R    [1.47,0.9949243497 7758089786,0.9949243497 7758089786,0.0],
+--R    [1.48,0.9958808445 3764005648,0.9958808445 3764005648,0.0],
+--R    [1.49,0.9967377520 4314338855,0.9967377520 4314338855,0.0],
+--R    [1.5,0.9974949866 0405443094,0.9974949866 0405443094,0.0],
+--R    [1.51,0.9981524724 9754811924,0.9981524724 9754811924,0.0],
+--R    [1.52,0.9987101439 7558300717,0.9987101439 7558300717,0.0],
+--R    [1.53,0.9991679452 7147601592,0.9991679452 7147601592,0.0],
+--R    [1.54,0.9995258306 0547905601,0.9995258306 0547905601,0.0],
+--R    [1.55,0.9997837641 893569639,0.9997837641 893569639,0.0],
+--R    [1.56,0.9999417202 2996629574,0.9999417202 2996629574,0.0],
+--R    [1.57,0.9999996829 3183462021,0.9999996829 3183462021,0.0],
+--R    [1.58,0.9999576464 9874005255,0.9999576464 9874005255,0.0],
+--R    [1.59,0.9998156151 3429087198,0.9998156151 3429087198,0.0],
+--R    [1.6,0.9995736030 4150516434,0.9995736030 4150516434,0.0]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of cos(x), ranging from 0.00 to 1.60. The
+second column is the reference value of sin from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp142-173
+
+<<*>>=
+--S 2 of 2
+[[0.01,0.99995000041666527778026,cos(0.01),cos(0.01)-(0.99995000041666527778026)],_
+[0.02,0.99980000666657777841270,cos(0.02),cos(0.02)-(0.99980000666657777841270)],_
+[0.03,0.99955003374898751627216,cos(0.03),cos(0.03)-(0.99955003374898751627216)],_
+[0.04,0.99920010666097794031457,cos(0.04),cos(0.04)-(0.99920010666097794031457)],_
+[0.05,0.99875026039496624656287,cos(0.05),cos(0.05)-(0.99875026039496624656287)],_
+[0.06,0.99820053993520416554766,cos(0.06),cos(0.06)-(0.99820053993520416554766)],_
+[0.07,0.99755100025327957462091,cos(0.07),cos(0.07)-(0.99755100025327957462091)],_
+[0.08,0.99680170630261938497771,cos(0.08),cos(0.08)-(0.99680170630261938497771)],_
+[0.09,0.99595273301199425309284,cos(0.09),cos(0.09)-(0.99595273301199425309284)],_
+[0.10,0.99500416527802576609556,cos(0.10),cos(0.10)-(0.99500416527802576609556)],_
+[0.11,0.99395609795669685035784,cos(0.11),cos(0.11)-(0.99395609795669685035784)],_
+[0.12,0.99280863585386625224810,cos(0.12),cos(0.12)-(0.99280863585386625224810)],_
+[0.13,0.99156189371478803959451,cos(0.13),cos(0.13)-(0.99156189371478803959451)],_
+[0.14,0.99021599621263717189895,cos(0.14),cos(0.14)-(0.99021599621263717189895)],_
+[0.15,0.98877107793604228673498,cos(0.15),cos(0.15)-(0.98877107793604228673498)],_
+[0.16,0.98722728337562694904095,cos(0.16),cos(0.16)-(0.98722728337562694904095)],_
+[0.17,0.98558476690956070917193,cos(0.17),cos(0.17)-(0.98558476690956070917193)],_
+[0.18,0.98384369278812141459272,cos(0.18),cos(0.18)-(0.98384369278812141459272)],_
+[0.19,0.98200423511727031896788,cos(0.19),cos(0.19)-(0.98200423511727031896788)],_
+[0.20,0.98006657784124163112420,cos(0.20),cos(0.20)-(0.98006657784124163112420)],_
+[0.21,0.97803091472414824491614,cos(0.21),cos(0.21)-(0.97803091472414824491614)],_
+[0.22,0.97589744933060548940602,cos(0.22),cos(0.22)-(0.97589744933060548940602)],_
+[0.23,0.97366639500537483696773,cos(0.23),cos(0.23)-(0.97366639500537483696773)],_
+[0.24,0.97133797485202960492618,cos(0.24),cos(0.24)-(0.97133797485202960492618)],_
+[0.25,0.96891242171064478414459,cos(0.25),cos(0.25)-(0.96891242171064478414459)],_
+[0.26,0.96638997813451322555822,cos(0.26),cos(0.26)-(0.96638997813451322555822)],_
+[0.27,0.96377089636589051301623,cos(0.27),cos(0.27)-(0.96377089636589051301623)],_
+[0.28,0.96105543831077094792459,cos(0.28),cos(0.28)-(0.96105543831077094792459)],_
+[0.29,0.95824387551269716807013,cos(0.29),cos(0.29)-(0.95824387551269716807013)],_
+[0.30,0.95533648912560601964231,cos(0.30),cos(0.30)-(0.95533648912560601964231)],_
+[0.31,0.95233356988571339784281,cos(0.31),cos(0.31)-(0.95233356988571339784281)],_
+[0.32,0.94923541808244086757531,cos(0.32),cos(0.32)-(0.94923541808244086757531)],_
+[0.33,0.94604234352838697152941,cos(0.33),cos(0.33)-(0.94604234352838697152941)],_
+[0.34,0.94275466552834622850264,cos(0.34),cos(0.34)-(0.94275466552834622850264)],_
+[0.35,0.93937271284737892003503,cos(0.35),cos(0.35)-(0.93937271284737892003503)],_
+[0.36,0.93589682367793485835091,cos(0.36),cos(0.36)-(0.93589682367793485835091)],_
+[0.37,0.93232734560603442320381,cos(0.37),cos(0.37)-(0.93232734560603442320381)],_
+[0.38,0.92866463557651024949253,cos(0.38),cos(0.38)-(0.92866463557651024949253)],_
+[0.39,0.92490905985731304145068,cos(0.39),cos(0.39)-(0.92490905985731304145068)],_
+[0.40,0.92106099400288508279853,cos(0.40),cos(0.40)-(0.92106099400288508279853)],_
+[0.41,0.91712082281660510547564,cos(0.41),cos(0.41)-(0.91712082281660510547564)],_
+[0.42,0.91308894031230827243609,cos(0.42),cos(0.42)-(0.91308894031230827243609)],_
+[0.43,0.90896574967488512247591,cos(0.43),cos(0.43)-(0.90896574967488512247591)],_
+[0.44,0.90475166321996341716554,cos(0.44),cos(0.44)-(0.90475166321996341716554)],_
+[0.45,0.90044710235267692166884,cos(0.45),cos(0.45)-(0.90044710235267692166884)],_
+[0.46,0.89605249752552524253639,cos(0.46),cos(0.46)-(0.89605249752552524253639)],_
+[0.47,0.89156828819532893645402,cos(0.47),cos(0.47)-(0.89156828819532893645402)],_
+[0.48,0.88699492277928419439995,cos(0.48),cos(0.48)-(0.88699492277928419439995)],_
+[0.49,0.88233285861012149570547,cos(0.49),cos(0.49)-(0.88233285861012149570547)],_
+[0.50,0.87758256189037271611628,cos(0.50),cos(0.50)-(0.87758256189037271611628)],_
+[0.51,0.87274450764575126310581,cos(0.51),cos(0.51)-(0.87274450764575126310581)],_
+[0.52,0.86781917967764990038785,cos(0.52),cos(0.52)-(0.86781917967764990038785)],_
+[0.53,0.86280707051476101180670,cos(0.53),cos(0.53)-(0.86280707051476101180670)],_
+[0.54,0.85770868136382414253797,cos(0.54),cos(0.54)-(0.85770868136382414253797)],_
+[0.55,0.85252452205950574280498,cos(0.55),cos(0.55)-(0.85252452205950574280498)],_
+[0.56,0.84725511101341612609452,cos(0.56),cos(0.56)-(0.84725511101341612609452)],_
+[0.57,0.84190097516226874013376,cos(0.57),cos(0.57)-(0.84190097516226874013376)],_
+[0.58,0.83646264991518693465789,cos(0.58),cos(0.58)-(0.83646264991518693465789)],_
+[0.59,0.83094067910016349524800,cos(0.59),cos(0.59)-(0.83094067910016349524800)],_
+[0.60,0.82533561490967829724095,cos(0.60),cos(0.60)-(0.82533561490967829724095)],_
+[0.61,0.81964801784547951790075,cos(0.61),cos(0.61)-(0.81964801784547951790075)],_
+[0.62,0.81387845666253392868400,cos(0.62),cos(0.62)-(0.81387845666253392868400)],_
+[0.63,0.80802750831215187252371,cos(0.63),cos(0.63)-(0.80802750831215187252371)],_
+[0.64,0.80209575788429261358611,cos(0.64),cos(0.64)-(0.80209575788429261358611)],_
+[0.65,0.79608379854905582891760,cos(0.65),cos(0.65)-(0.79608379854905582891760)],_
+[0.66,0.78999223149736509278382,cos(0.66),cos(0.66)-(0.78999223149736509278382)],_
+[0.67,0.78382166588084928530294,cos(0.67),cos(0.67)-(0.78382166588084928530294)],_
+[0.68,0.77757271875092793718239,cos(0.68),cos(0.68)-(0.77757271875092793718239)],_
+[0.69,0.77124601499710660197354,cos(0.69),cos(0.69)-(0.77124601499710660197354)],_
+[0.70,0.76484218728448842625586,cos(0.70),cos(0.70)-(0.76484218728448842625586)],_
+[0.71,0.75836187599050816654146,cos(0.71),cos(0.71)-(0.75836187599050816654146)],_
+[0.72,0.75180572914089497944549,cos(0.72),cos(0.72)-(0.75180572914089497944549)],_
+[0.73,0.74517440234487038879013,cos(0.73),cos(0.73)-(0.74517440234487038879013)],_
+[0.74,0.73846855872958790979142,cos(0.74),cos(0.74)-(0.73846855872958790979142)],_
+[0.75,0.73168886887382088631184,cos(0.75),cos(0.75)-(0.73168886887382088631184)],_
+[0.76,0.72483601074090517233969,cos(0.76),cos(0.76)-(0.72483601074090517233969)],_
+[0.77,0.71791066961094336337129,cos(0.77),cos(0.77)-(0.71791066961094336337129)],_
+[0.78,0.71091353801227735721626,cos(0.78),cos(0.78)-(0.71091353801227735721626)],_
+[0.79,0.70384531565223609691278,cos(0.79),cos(0.79)-(0.70384531565223609691278)],_
+[0.80,0.69670670934716542092075,cos(0.80),cos(0.80)-(0.69670670934716542092075)],_
+[0.81,0.68949843295174701754964,cos(0.81),cos(0.81)-(0.68949843295174701754964)],_
+[0.82,0.68222120728761355166656,cos(0.82),cos(0.82)-(0.68222120728761355166656)],_
+[0.83,0.67487576007126710211246,cos(0.83),cos(0.83)-(0.67487576007126710211246)],_
+[0.84,0.66746282584130811792267,cos(0.84),cos(0.84)-(0.66746282584130811792267)],_
+[0.85,0.65998314588498217039542,cos(0.85),cos(0.85)-(0.65998314588498217039542)],_
+[0.86,0.65243746816405184627203,cos(0.86),cos(0.86)-(0.65243746816405184627203)],_
+[0.87,0.64482654724000119477766,cos(0.87),cos(0.87)-(0.64482654724000119477766)],_
+[0.88,0.63715114419858020801550,cos(0.88),cos(0.88)-(0.63715114419858020801550)],_
+[0.89,0.62941202657369688020355,cos(0.89),cos(0.89)-(0.62941202657369688020355)],_
+[0.90,0.62160996827066445648472,cos(0.90),cos(0.90)-(0.62160996827066445648472)],_
+[0.91,0.61374574948881154652118,cos(0.91),cos(0.91)-(0.61374574948881154652118)],_
+[0.92,0.60582015664346284179741,cos(0.92),cos(0.92)-(0.60582015664346284179741)],_
+[0.93,0.59783398228729823849491,cos(0.93),cos(0.93)-(0.59783398228729823849491)],_
+[0.94,0.58978802503109822996099,cos(0.94),cos(0.94)-(0.58978802503109822996099)],_
+[0.95,0.58168308946388349416618,cos(0.95),cos(0.95)-(0.58168308946388349416618)],_
+[0.96,0.57351998607245666212505,cos(0.96),cos(0.96)-(0.57351998607245666212505)],_
+[0.97,0.56529953116035431303653,cos(0.97),cos(0.97)-(0.56529953116035431303653)],_
+[0.98,0.55702254676621730087666,cos(0.98),cos(0.98)-(0.55702254676621730087666)],_
+[0.99,0.54868986058158757534313,cos(0.99),cos(0.99)-(0.54868986058158757534313)],_
+[1.00,0.54030230586813971740094,cos(1.00),cos(1.00)-(0.54030230586813971740094)],_
+[1.01,0.53186072137435546620673,cos(1.01),cos(1.01)-(0.53186072137435546620673)],_
+[1.02,0.52336595125164956988961,cos(1.02),cos(1.02)-(0.52336595125164956988961)],_
+[1.03,0.51481884496995534753350,cos(1.03),cos(1.03)-(0.51481884496995534753350)],_
+[1.04,0.50622025723277840373447,cos(1.04),cos(1.04)-(0.50622025723277840373447)],_
+[1.05,0.49757104789172699029085,cos(1.05),cos(1.05)-(0.49757104789172699029085)],_
+[1.06,0.48887208186052756191864,cos(1.06),cos(1.06)-(0.48887208186052756191864)],_
+[1.07,0.48012422902853412436509,cos(1.07),cos(1.07)-(0.48012422902853412436509)],_
+[1.08,0.47132836417374002391353,cos(1.08),cos(1.08)-(0.47132836417374002391353)],_
+[1.09,0.46248536687530087702790,cos(1.09),cos(1.09)-(0.46248536687530087702790)],_
+[1.10,0.45359612142557738777137,cos(1.10),cos(1.10)-(0.45359612142557738777137)],_
+[1.11,0.44466151674170684864374,cos(1.11),cos(1.11)-(0.44466151674170684864374)],_
+[1.12,0.43568244627671216761399,cos(1.12),cos(1.12)-(0.43568244627671216761399)],_
+[1.13,0.42665980793015731037122,cos(1.13),cos(1.13)-(0.42665980793015731037122)],_
+[1.14,0.41759450395835809217519,cos(1.14),cos(1.14)-(0.41759450395835809217519)],_
+[1.15,0.40848744088415729815258,cos(1.15),cos(1.15)-(0.40848744088415729815258)],_
+[1.16,0.39933952940627315445164,cos(1.16),cos(1.16)-(0.39933952940627315445164)],_
+[1.17,0.39015168430823021533266,cos(1.17),cos(1.17)-(0.39015168430823021533266)],_
+[1.18,0.38092482436688177302960,cos(1.18),cos(1.18)-(0.38092482436688177302960)],_
+[1.19,0.37165987226053293806568,cos(1.19),cos(1.19)-(0.37165987226053293806568)],_
+[1.20,0.36235775447667357763837,cos(1.20),cos(1.20)-(0.36235775447667357763837)],_
+[1.21,0.35301940121933033870301,cos(1.21),cos(1.21)-(0.35301940121933033870301)],_
+[1.22,0.34364574631604702047552,cos(1.22),cos(1.22)-(0.34364574631604702047552)],_
+[1.23,0.33423772712450259823955,cos(1.23),cos(1.23)-(0.33423772712450259823955)],_
+[1.24,0.32479628443877623657769,cos(1.24),cos(1.24)-(0.32479628443877623657769)],_
+[1.25,0.31532236239526866544754,cos(1.25),cos(1.25)-(0.31532236239526866544754)],_
+[1.26,0.30581690837828932688634,cos(1.26),cos(1.26)-(0.30581690837828932688634)],_
+[1.27,0.29628087292531873355114,cos(1.27),cos(1.27)-(0.29628087292531873355114)],_
+[1.28,0.28671520963195551277939,cos(1.28),cos(1.28)-(0.28671520963195551277939)],_
+[1.29,0.27712087505655764138661,cos(1.29),cos(1.29)-(0.27712087505655764138661)],_
+[1.30,0.26749882862458740699798,cos(1.30),cos(1.30)-(0.26749882862458740699798)],_
+[1.31,0.25785003253266966133818,cos(1.31),cos(1.31)-(0.25785003253266966133818)],_
+[1.32,0.24817545165237295957398,cos(1.32),cos(1.32)-(0.24817545165237295957398)],_
+[1.33,0.23847605343372320751578,cos(1.33),cos(1.33)-(0.23847605343372320751578)],_
+[1.34,0.22875280780845946523264,cos(1.34),cos(1.34)-(0.22875280780845946523264)],_
+[1.35,0.21900668709304158142002,cos(1.35),cos(1.35)-(0.21900668709304158142002)],_
+[1.36,0.20923866589141935767598,cos(1.36),cos(1.36)-(0.20923866589141935767598)],_
+[1.37,0.19944972099757296568820,cos(1.37),cos(1.37)-(0.19944972099757296568820)],_
+[1.38,0.18964083129783436320915,cos(1.38),cos(1.38)-(0.18964083129783436320915)],_
+[1.39,0.17981297767299947659616,cos(1.39),cos(1.39)-(0.17981297767299947659616)],_
+[1.40,0.16996714290024093861675,cos(1.40),cos(1.40)-(0.16996714290024093861675)],_
+[1.41,0.16010431155483119016356,cos(1.41),cos(1.41)-(0.16010431155483119016356)],_
+[1.42,0.15022546991168577348698,cos(1.42),cos(1.42)-(0.15022546991168577348698)],_
+[1.43,0.14033160584673666253390,cos(1.43),cos(1.43)-(0.14033160584673666253390)],_
+[1.44,0.13042370873814549297752,cos(1.44),cos(1.44)-(0.13042370873814549297752)],_
+[1.45,0.12050276936736657053287,cos(1.45),cos(1.45)-(0.12050276936736657053287)],_
+[1.46,0.11056977982006955117465,cos(1.46),cos(1.46)-(0.11056977982006955117465)],_
+[1.47,0.10062573338693170090698,cos(1.47),cos(1.47)-(0.10062573338693170090698)],_
+[1.48,0.09067162446430965577623,cos(1.48),cos(1.48)-(0.09067162446430965577623)],_
+[1.49,0.08070844845480061486832,cos(1.49),cos(1.49)-(0.08070844845480061486832)],_
+[1.50,0.07073720166770291008819,cos(1.50),cos(1.50)-(0.07073720166770291008819)],_
+[1.51,0.06075888121938590658160,cos(1.51),cos(1.51)-(0.06075888121938590658160)],_
+[1.52,0.05077448493357919672613,cos(1.52),cos(1.52)-(0.05077448493357919672613)],_
+[1.53,0.04078501124159105868899,cos(1.53),cos(1.53)-(0.04078501124159105868899)],_
+[1.54,0.03079145908246615762248,cos(1.54),cos(1.54)-(0.03079145908246615762248)],_
+[1.55,0.02079482780309247364391,cos(1.55),cos(1.55)-(0.02079482780309247364391)],_
+[1.56,0.01079611705826744582392,cos(1.56),cos(1.56)-(0.01079611705826744582392)],_
+[1.57,0.00079632671073332548541,cos(1.57),cos(1.57)-(0.00079632671073332548541)],_
+[1.58,-0.00920354326880826480539,cos(1.58),cos(1.58)-(-0.00920354326880826480539)],_
+[1.59,-0.01920249290169256809503,cos(1.59),cos(1.59)-(-0.01920249290169256809503)],_
+[1.60,-0.02919952230128872620577,cos(1.60),cos(1.60)-(-0.02919952230128872620577)]]
+--R 
+--R
+--R   (2)
+--R   [[0.01,0.9999500004 1666527778,0.9999500004 1666527778,0.0],
+--R    [0.02,0.9998000066 6657777841,0.9998000066 6657777841,0.0],
+--R    [0.03,0.9995500337 4898751627,0.9995500337 4898751627,0.0],
+--R    [0.04,0.9992001066 6097794031,0.9992001066 6097794031,0.0],
+--R    [0.05,0.9987502603 9496624656,0.9987502603 9496624656,0.0],
+--R    [0.06,0.9982005399 3520416555,0.9982005399 3520416555,0.0],
+--R    [0.07,0.9975510002 5327957462,0.9975510002 5327957462,0.0],
+--R    [0.08,0.9968017063 0261938498,0.9968017063 0261938498,0.0],
+--R    [0.09,0.9959527330 1199425309,0.9959527330 1199425309,0.0],
+--R    [0.1,0.9950041652 7802576609,0.9950041652 7802576609,0.0],
+--R    [0.11,0.9939560979 5669685036,0.9939560979 5669685036,0.0],
+--R    [0.12,0.9928086358 5386625225,0.9928086358 5386625225,0.0],
+--R    [0.13,0.9915618937 1478803959,0.9915618937 1478803959,0.0],
+--R    [0.14,0.9902159962 126371719,0.9902159962 126371719,0.0],
+--R    [0.15,0.9887710779 3604228674,0.9887710779 3604228674,0.0],
+--R    [0.16,0.9872272833 7562694904,0.9872272833 7562694904,0.0],
+--R    [0.17,0.9855847669 0956070917,0.9855847669 0956070917,0.0],
+--R    [0.18,0.9838436927 8812141459,0.9838436927 8812141459,0.0],
+--R    [0.19,0.9820042351 1727031897,0.9820042351 1727031897,0.0],
+--R    [0.2,0.9800665778 4124163112,0.9800665778 4124163112,0.0],
+--R    [0.21,0.9780309147 2414824492,0.9780309147 2414824492,0.0],
+--R    [0.22,0.9758974493 3060548941,0.9758974493 3060548941,0.0],
+--R    [0.23,0.9736663950 0537483697,0.9736663950 0537483697,0.0],
+--R    [0.24,0.9713379748 5202960493,0.9713379748 5202960493,0.0],
+--R    [0.25,0.9689124217 1064478414,0.9689124217 1064478414,0.0],
+--R    [0.26,0.9663899781 3451322556,0.9663899781 3451322556,0.0],
+--R    [0.27,0.9637708963 6589051302,0.9637708963 6589051302,0.0],
+--R    [0.28,0.9610554383 1077094792,0.9610554383 1077094792,0.0],
+--R    [0.29,0.9582438755 1269716807,0.9582438755 1269716807,0.0],
+--R    [0.3,0.9553364891 2560601964,0.9553364891 2560601964,0.0],
+--R    [0.31,0.9523335698 8571339784,0.9523335698 8571339784,0.0],
+--R    [0.32,0.9492354180 8244086758,0.9492354180 8244086758,0.0],
+--R    [0.33,0.9460423435 2838697153,0.9460423435 2838697153,0.0],
+--R    [0.34,0.9427546655 283462285,0.9427546655 283462285,0.0],
+--R    [0.35,0.9393727128 4737892004,0.9393727128 4737892004,0.0],
+--R    [0.36,0.9358968236 7793485835,0.9358968236 7793485835,0.0],
+--R    [0.37,0.9323273456 060344232,0.9323273456 060344232,0.0],
+--R    [0.38,0.9286646355 7651024949,0.9286646355 7651024949,0.0],
+--R    [0.39,0.9249090598 5731304145,0.9249090598 5731304145,0.0],
+--R    [0.4,0.9210609940 028850828,0.9210609940 028850828,0.0],
+--R    [0.41,0.9171208228 1660510548,0.9171208228 1660510548,0.0],
+--R    [0.42,0.9130889403 1230827244,0.9130889403 1230827244,0.0],
+--R    [0.43,0.9089657496 7488512248,0.9089657496 7488512248,0.0],
+--R    [0.44,0.9047516632 1996341716,0.9047516632 1996341717,0.3 E -20],
+--R    [0.45,0.9004471023 5267692167,0.9004471023 5267692167,0.0],
+--R    [0.46,0.8960524975 2552524254,0.8960524975 2552524254,0.0],
+--R    [0.47,0.8915682881 9532893645,0.8915682881 9532893645,0.0],
+--R    [0.48,0.8869949227 792841944,0.8869949227 792841944,0.0],
+--R    [0.49,0.8823328586 101214957,0.8823328586 101214957,0.0],
+--R    [0.5,0.8775825618 9037271612,0.8775825618 9037271612,0.0],
+--R    [0.51,0.8727445076 457512631,0.8727445076 457512631,0.0],
+--R    [0.52,0.8678191796 7764990039,0.8678191796 7764990039,0.0],
+--R    [0.53,0.8628070705 1476101181,0.8628070705 1476101181,0.0],
+--R    [0.54,0.8577086813 6382414254,0.8577086813 6382414254,0.0],
+--R    [0.55,0.8525245220 595057428,0.8525245220 595057428,0.0],
+--R    [0.56,0.8472551110 1341612609,0.8472551110 1341612609,0.0],
+--R    [0.57,0.8419009751 6226874013,0.8419009751 6226874013,0.0],
+--R    [0.58,0.8364626499 1518693466,0.8364626499 1518693466,0.0],
+--R    [0.59,0.8309406791 0016349525,0.8309406791 0016349525,0.0],
+--R    [0.6,0.8253356149 0967829724,0.8253356149 0967829724,0.0],
+--R    [0.61,0.8196480178 454795179,0.8196480178 454795179,0.0],
+--R    [0.62,0.8138784566 6253392868,0.8138784566 6253392868,0.0],
+--R    [0.63,0.8080275083 1215187252,0.8080275083 1215187253,0.3 E -20],
+--R    [0.64,0.8020957578 8429261359,0.8020957578 8429261358,- 0.3 E -20],
+--R    [0.65,0.7960837985 4905582892,0.7960837985 4905582892,0.0],
+--R    [0.66,0.7899922314 9736509279,0.7899922314 9736509279,0.0],
+--R    [0.67,0.7838216658 808492853,0.7838216658 808492853,0.0],
+--R    [0.68,0.7775727187 5092793718,0.7775727187 5092793718,0.0],
+--R    [0.69,0.7712460149 9710660197,0.7712460149 9710660197,0.0],
+--R    [0.7,0.7648421872 8448842626,0.7648421872 8448842626,0.0],
+--R    [0.71,0.7583618759 9050816654,0.7583618759 9050816654,- 0.3 E -20],
+--R    [0.72,0.7518057291 4089497945,0.7518057291 4089497945,0.0],
+--R    [0.73,0.7451744023 4487038879,0.7451744023 4487038879,0.0],
+--R    [0.74,0.7384685587 2958790979,0.7384685587 2958790979,0.3 E -20],
+--R    [0.75,0.7316888688 7382088631,0.7316888688 7382088631,0.0],
+--R    [0.76,0.7248360107 4090517234,0.7248360107 4090517234,0.0],
+--R    [0.77,0.7179106696 1094336337,0.7179106696 1094336337,0.0],
+--R    [0.78,0.7109135380 1227735722,0.7109135380 1227735722,0.0],
+--R    [0.79,0.7038453156 5223609691,0.7038453156 5223609691,0.0],
+--R    [0.8,0.6967067093 4716542092,0.6967067093 4716542092,- 0.3 E -20],
+--R    [0.81,0.6894984329 5174701755,0.6894984329 5174701755,0.0],
+--R    [0.82,0.6822212072 8761355167,0.6822212072 8761355167,0.0],
+--R    [0.83,0.6748757600 7126710211,0.6748757600 7126710211,0.3 E -20],
+--R    [0.84,0.6674628258 4130811792,0.6674628258 4130811792,0.0],
+--R    [0.85,0.6599831458 8498217039,0.6599831458 8498217039,0.0],
+--R    [0.86,0.6524374681 6405184627,0.6524374681 6405184627,0.0],
+--R    [0.87,0.6448265472 4000119478,0.6448265472 4000119478,- 0.3 E -20],
+--R    [0.88,0.6371511441 9858020802,0.6371511441 9858020802,0.0],
+--R    [0.89,0.6294120265 736968802,0.6294120265 736968802,0.0],
+--R    [0.9,0.6216099682 7066445648,0.6216099682 7066445648,0.0],
+--R    [0.91,0.6137457494 8881154652,0.6137457494 8881154652,0.0],
+--R    [0.92,0.6058201566 434628418,0.6058201566 434628418,0.0],
+--R    [0.93,0.5978339822 872982385,0.5978339822 872982385,0.0],
+--R    [0.94,0.5897880250 3109822996,0.5897880250 3109822996,- 0.3 E -20],
+--R    [0.95,0.5816830894 6388349417,0.5816830894 6388349417,0.0],
+--R    [0.96,0.5735199860 7245666213,0.5735199860 7245666212,- 0.3 E -20],
+--R    [0.97,0.5652995311 6035431304,0.5652995311 6035431304,0.0],
+--R    [0.98,0.5570225467 6621730088,0.5570225467 6621730087,- 0.3 E -20],
+--R    [0.99,0.5486898605 8158757534,0.5486898605 8158757535,0.3 E -20],
+--R    [1.0,0.5403023058 681397174,0.5403023058 681397174,0.0],
+--R    [1.01,0.5318607213 7435546621,0.5318607213 7435546621,0.0],
+--R    [1.02,0.5233659512 5164956989,0.5233659512 5164956989,- 0.3 E -20],
+--R    [1.03,0.5148188449 6995534753,0.5148188449 6995534753,0.0],
+--R    [1.04,0.5062202572 3277840374,0.5062202572 3277840374,0.0],
+--R    [1.05,0.4975710478 9172699029,0.4975710478 9172699029,0.3 E -20],
+--R    [1.06,0.4888720818 6052756192,0.4888720818 6052756192,- 0.2 E -20],
+--R    [1.07,0.4801242290 2853412436,0.4801242290 2853412437,0.2 E -20],
+--R    [1.08,0.4713283641 7374002391,0.4713283641 7374002391,0.0],
+--R    [1.09,0.4624853668 7530087703,0.4624853668 7530087702,- 0.3 E -20],
+--R    [1.1,0.4535961214 2557738777,0.4535961214 2557738777,- 0.2 E -20],
+--R    [1.11,0.4446615167 4170684864,0.4446615167 4170684864,0.0],
+--R    [1.12,0.4356824462 7671216761,0.4356824462 7671216762,0.2 E -20],
+--R    [1.13,0.4266598079 3015731037,0.4266598079 3015731037,- 0.3 E -20],
+--R    [1.14,0.4175945039 5835809217,0.4175945039 5835809217,0.0],
+--R    [1.15,0.4084874408 8415729815,0.4084874408 8415729815,0.0],
+--R    [1.16,0.3993395294 0627315445,0.3993395294 0627315445,0.3 E -20],
+--R    [1.17,0.3901516843 0823021533,0.3901516843 0823021533,- 0.2 E -20],
+--R    [1.18,0.3809248243 6688177303,0.3809248243 6688177303,0.0],
+--R    [1.19,0.3716598722 6053293807,0.3716598722 6053293807,0.2 E -20],
+--R    [1.2,0.3623577544 7667357764,0.3623577544 7667357763,- 0.3 E -20],
+--R    [1.21,0.3530194012 193303387,0.3530194012 193303387,- 0.2 E -20],
+--R    [1.22,0.3436457463 1604702048,0.3436457463 1604702048,0.0],
+--R    [1.23,0.3342377271 2450259824,0.3342377271 2450259824,0.2 E -20],
+--R    [1.24,0.3247962844 3877623658,0.3247962844 3877623657,- 0.3 E -20],
+--R    [1.25,0.3153223623 9526866545,0.3153223623 9526866545,0.0],
+--R    [1.26,0.3058169083 7828932689,0.3058169083 7828932689,0.2 E -20],
+--R    [1.27,0.2962808729 2531873355,0.2962808729 2531873355,- 0.3 E -20],
+--R    [1.28,0.2867152096 3195551278,0.2867152096 3195551278,- 0.2 E -20],
+--R    [1.29,0.2771208750 5655764139,0.2771208750 5655764139,0.0],
+--R    [1.3,0.2674988286 24587407,0.2674988286 24587407,0.2 E -20],
+--R    [1.31,0.2578500325 3266966134,0.2578500325 3266966134,- 0.3 E -20],
+--R    [1.32,0.2481754516 5237295957,0.2481754516 5237295957,0.0],
+--R    [1.33,0.2384760534 3372320752,0.2384760534 3372320752,0.8 E -21],
+--R    [1.34,0.2287528078 0845946523,0.2287528078 0845946523,- 0.3 E -20],
+--R    [1.35,0.2190066870 9304158142,0.2190066870 9304158142,- 0.2 E -20],
+--R    [1.36,0.2092386658 9141935768,0.2092386658 9141935768,0.0],
+--R    [1.37,0.1994497209 9757296569,0.1994497209 9757296569,0.2 E -20],
+--R    [1.38,0.1896408312 9783436321,0.1896408312 9783436321,- 0.3 E -20],
+--R    [1.39,0.1798129776 729994766,0.1798129776 729994766,- 0.8 E -21],
+--R    [1.4,0.1699671429 0024093862,0.1699671429 0024093862,0.8 E -21],
+--R    [1.41,0.1601043115 5483119016,0.1601043115 5483119017,0.3 E -20],
+--R    [1.42,0.1502254699 1168577349,0.1502254699 1168577349,- 0.2 E -20],
+--R    [1.43,0.1403316058 4673666253,0.1403316058 4673666253,0.0],
+--R    [1.44,0.1304237087 3814549298,0.1304237087 3814549298,0.2 E -20],
+--R    [1.45,0.1205027693 6736657053,0.1205027693 6736657053,- 0.3 E -20],
+--R    [1.46,0.1105697798 2006955117,0.1105697798 2006955117,- 0.8 E -21],
+--R    [1.47,0.1006257333 8693170091,0.1006257333 8693170091,0.8 E -21],
+--R    [1.48,0.0906716244 6430965577 6,0.0906716244 6430965577 9,0.3 E -20],
+--R    [1.49,0.0807084484 5480061486 8,0.0807084484 5480061486 6,- 0.2 E -20],
+--R    [1.5,0.0707372016 6770291008 8,0.0707372016 6770291008 8,0.0],
+--R    [1.51,0.0607588812 1938590658 2,0.0607588812 1938590658 3,0.2 E -20],
+--R    [1.52,0.0507744849 3357919672 6,0.0507744849 3357919672 3,- 0.3 E -20],
+--R    [1.53,0.0407850112 4159105868 9,0.0407850112 4159105868 8,- 0.1 E -20],
+--R    [1.54,0.0307914590 8246615762 2,0.0307914590 8246615762 3,0.8 E -21],
+--R    [1.55,0.0207948278 0309247364 4,0.0207948278 0309247364 7,0.3 E -20],
+--R    [1.56,0.0107961170 5826744582 4,0.0107961170 5826744582 2,- 0.2 E -20],
+--R    [1.57,0.0007963267 1073332548 541,0.0007963267 1073332548 514,- 0.3 E -21],
+--R    [1.58,- 0.0092035432 6880826480 54,- 0.0092035432 6880826480 38,0.2 E -20],
+--R    [1.59,- 0.0192024929 0169256809 5,- 0.0192024929 0169256809 8,- 0.3 E -20],
+--R    [1.6,- 0.0291995223 0128872620 6,- 0.0291995223 0128872620 7,- 0.1 E -20]]
+--R                                                        Type: List List Float
+--E 2
+ 
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun,``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp142-173
+\end{thebibliography}
+\end{document}
+
diff --git a/src/input/sinhcosh.input.pamphlet b/src/input/sinhcosh.input.pamphlet
new file mode 100644
index 0000000..aa23267
--- /dev/null
+++ b/src/input/sinhcosh.input.pamphlet
@@ -0,0 +1,861 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input sinhcosh.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a set of tests for the sinh and cosh functions.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of sinh, ranging from 0.00 to 2.00. The
+second column is the reference value of sinh from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+
+<<*>>=
+)spool sinhcosh.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.00,0.000000000,sinh(0.00),sinh(0.00)-0.000000000],_
+[0.01,0.010000167,sinh(0.01),sinh(0.01)-0.010000167],_
+[0.02,0.020001333,sinh(0.02),sinh(0.02)-0.020001333],_
+[0.03,0.030004500,sinh(0.03),sinh(0.03)-0.030004500],_
+[0.04,0.040010668,sinh(0.04),sinh(0.04)-0.040010668],_
+[0.05,0.050020836,sinh(0.05),sinh(0.05)-0.050020836],_
+[0.06,0.060036006,sinh(0.06),sinh(0.06)-0.060036006],_
+[0.07,0.070057181,sinh(0.07),sinh(0.07)-0.070057181],_
+[0.08,0.080085361,sinh(0.08),sinh(0.08)-0.080085361],_
+[0.09,0.090121549,sinh(0.09),sinh(0.09)-0.090121549],_
+[0.10,0.100166750,sinh(0.10),sinh(0.10)-0.100166750],_
+[0.11,0.110221968,sinh(0.11),sinh(0.11)-0.110221968],_
+[0.12,0.120288207,sinh(0.12),sinh(0.12)-0.120288207],_
+[0.13,0.130366476,sinh(0.13),sinh(0.13)-0.130366476],_
+[0.14,0.140457782,sinh(0.14),sinh(0.14)-0.140457782],_
+[0.15,0.150563133,sinh(0.15),sinh(0.15)-0.150563133],_
+[0.16,0.160683541,sinh(0.16),sinh(0.16)-0.160683541],_
+[0.17,0.170820017,sinh(0.17),sinh(0.17)-0.170820017],_
+[0.18,0.180973576,sinh(0.18),sinh(0.18)-0.180973576],_
+[0.19,0.191145232,sinh(0.19),sinh(0.19)-0.191145232],_
+[0.20,0.201336003,sinh(0.20),sinh(0.20)-0.201336003],_
+[0.21,0.211546907,sinh(0.21),sinh(0.21)-0.211546907],_
+[0.22,0.221778966,sinh(0.22),sinh(0.22)-0.221778966],_
+[0.23,0.232033204,sinh(0.23),sinh(0.23)-0.232033204],_
+[0.24,0.242310645,sinh(0.24),sinh(0.24)-0.242310645],_
+[0.25,0.252612317,sinh(0.25),sinh(0.25)-0.252612317],_
+[0.26,0.262939250,sinh(0.26),sinh(0.26)-0.262939250],_
+[0.27,0.273292478,sinh(0.27),sinh(0.27)-0.273292478],_
+[0.28,0.283673035,sinh(0.28),sinh(0.28)-0.283673035],_
+[0.29,0.294081960,sinh(0.29),sinh(0.29)-0.294081960],_
+[0.30,0.304520293,sinh(0.30),sinh(0.30)-0.304520293],_
+[0.31,0.314989079,sinh(0.31),sinh(0.31)-0.314989079],_
+[0.32,0.325489364,sinh(0.32),sinh(0.32)-0.325489364],_
+[0.33,0.336022198,sinh(0.33),sinh(0.33)-0.336022198],_
+[0.34,0.346588634,sinh(0.34),sinh(0.34)-0.346588634],_
+[0.35,0.357189729,sinh(0.35),sinh(0.35)-0.357189729],_
+[0.36,0.367826544,sinh(0.36),sinh(0.36)-0.367826544],_
+[0.37,0.378500142,sinh(0.37),sinh(0.37)-0.378500142],_
+[0.38,0.389211590,sinh(0.38),sinh(0.38)-0.389211590],_
+[0.39,0.399961960,sinh(0.39),sinh(0.39)-0.399961960],_
+[0.40,0.410752326,sinh(0.40),sinh(0.40)-0.410752326],_
+[0.41,0.421583767,sinh(0.41),sinh(0.41)-0.421583767],_
+[0.42,0.432457368,sinh(0.42),sinh(0.42)-0.432457368],_
+[0.43,0.443374214,sinh(0.43),sinh(0.43)-0.443374214],_
+[0.44,0.454335399,sinh(0.44),sinh(0.44)-0.454335399],_
+[0.45,0.465342017,sinh(0.45),sinh(0.45)-0.465342017],_
+[0.46,0.476395170,sinh(0.46),sinh(0.46)-0.476395170],_
+[0.47,0.487495962,sinh(0.47),sinh(0.47)-0.487495962],_
+[0.48,0.498645505,sinh(0.48),sinh(0.48)-0.498645505],_
+[0.49,0.509844913,sinh(0.49),sinh(0.49)-0.509844913],_
+[0.50,0.521095305,sinh(0.50),sinh(0.50)-0.521095305],_
+[0.51,0.532397808,sinh(0.51),sinh(0.51)-0.532397808],_
+[0.52,0.543753551,sinh(0.52),sinh(0.52)-0.543753551],_
+[0.53,0.555163669,sinh(0.53),sinh(0.53)-0.555163669],_
+[0.54,0.566629305,sinh(0.54),sinh(0.54)-0.566629305],_
+[0.55,0.578151604,sinh(0.55),sinh(0.55)-0.578151604],_
+[0.56,0.589731718,sinh(0.56),sinh(0.56)-0.589731718],_
+[0.57,0.601370806,sinh(0.57),sinh(0.57)-0.601370806],_
+[0.58,0.613070032,sinh(0.58),sinh(0.58)-0.613070032],_
+[0.59,0.624830565,sinh(0.59),sinh(0.59)-0.624830565],_
+[0.60,0.636653582,sinh(0.60),sinh(0.60)-0.636653582],_
+[0.61,0.648540265,sinh(0.61),sinh(0.61)-0.648540265],_
+[0.62,0.660491802,sinh(0.62),sinh(0.62)-0.660491802],_
+[0.63,0.672509389,sinh(0.63),sinh(0.63)-0.672509389],_
+[0.64,0.684594228,sinh(0.64),sinh(0.64)-0.684594228],_
+[0.65,0.696747526,sinh(0.65),sinh(0.65)-0.696747526],_
+[0.66,0.708970500,sinh(0.66),sinh(0.66)-0.708970500],_
+[0.67,0.721264371,sinh(0.67),sinh(0.67)-0.721264371],_
+[0.68,0.733630370,sinh(0.68),sinh(0.68)-0.733630370],_
+[0.69,0.746069732,sinh(0.69),sinh(0.69)-0.746069732],_
+[0.70,0.758583702,sinh(0.70),sinh(0.70)-0.758583702],_
+[0.71,0.771173531,sinh(0.71),sinh(0.71)-0.771173531],_
+[0.72,0.783840477,sinh(0.72),sinh(0.72)-0.783840477],_
+[0.73,0.796585809,sinh(0.73),sinh(0.73)-0.796585809],_
+[0.74,0.809410799,sinh(0.74),sinh(0.74)-0.809410799],_
+[0.75,0.822316732,sinh(0.75),sinh(0.75)-0.822316732],_
+[0.76,0.835304897,sinh(0.76),sinh(0.76)-0.835304897],_
+[0.77,0.848376593,sinh(0.77),sinh(0.77)-0.848376593],_
+[0.78,0.861533127,sinh(0.78),sinh(0.78)-0.861533127],_
+[0.79,0.874775815,sinh(0.79),sinh(0.79)-0.874775815],_
+[0.80,0.888105982,sinh(0.80),sinh(0.80)-0.888105982],_
+[0.81,0.901524960,sinh(0.81),sinh(0.81)-0.901524960],_
+[0.82,0.915034092,sinh(0.82),sinh(0.82)-0.915034092],_
+[0.83,0.928634727,sinh(0.83),sinh(0.83)-0.928634727],_
+[0.84,0.942328227,sinh(0.84),sinh(0.84)-0.942328227],_
+[0.85,0.956115960,sinh(0.85),sinh(0.85)-0.956115960],_
+[0.86,0.969999306,sinh(0.86),sinh(0.86)-0.969999306],_
+[0.87,0.983979652,sinh(0.87),sinh(0.87)-0.983979652],_
+[0.88,0.998058397,sinh(0.88),sinh(0.88)-0.998058397],_
+[0.89,1.012236949,sinh(0.89),sinh(0.89)-1.012236949],_
+[0.90,1.026516726,sinh(0.90),sinh(0.90)-1.026516726],_
+[0.91,1.040899155,sinh(0.91),sinh(0.91)-1.040899155],_
+[0.92,1.055385674,sinh(0.92),sinh(0.92)-1.055385674],_
+[0.93,1.069977734,sinh(0.93),sinh(0.93)-1.069977734],_
+[0.94,1.084676791,sinh(0.94),sinh(0.94)-1.084676791],_
+[0.95,1.099484318,sinh(0.95),sinh(0.95)-1.099484318],_
+[0.96,1.114401794,sinh(0.96),sinh(0.96)-1.114401794],_
+[0.97,1.129430711,sinh(0.97),sinh(0.97)-1.129430711],_
+[0.98,1.144572572,sinh(0.98),sinh(0.98)-1.144572572],_
+[0.99,1.159828891,sinh(0.99),sinh(0.99)-1.159828891],_
+[1.00,1.175201194,sinh(1.00),sinh(1.00)-1.175201194],_
+[1.01,1.190691018,sinh(1.01),sinh(1.01)-1.190691018],_
+[1.02,1.206299912,sinh(1.02),sinh(1.02)-1.206299912],_
+[1.03,1.222029437,sinh(1.03),sinh(1.03)-1.222029437],_
+[1.04,1.237881166,sinh(1.04),sinh(1.04)-1.237881166],_
+[1.05,1.253856684,sinh(1.05),sinh(1.05)-1.253856684],_
+[1.06,1.269957589,sinh(1.06),sinh(1.06)-1.269957589],_
+[1.07,1.286185491,sinh(1.07),sinh(1.07)-1.286185491],_
+[1.08,1.302542013,sinh(1.08),sinh(1.08)-1.302542013],_
+[1.09,1.319028789,sinh(1.09),sinh(1.09)-1.319028789],_
+[1.10,1.335647470,sinh(1.10),sinh(1.10)-1.335647470],_
+[1.11,1.352399717,sinh(1.11),sinh(1.11)-1.352399717],_
+[1.12,1.369287204,sinh(1.12),sinh(1.12)-1.369287204],_
+[1.13,1.386311622,sinh(1.13),sinh(1.13)-1.386311622],_
+[1.14,1.403474672,sinh(1.14),sinh(1.14)-1.403474672],_
+[1.15,1.420778070,sinh(1.15),sinh(1.15)-1.420778070],_
+[1.16,1.438223548,sinh(1.16),sinh(1.16)-1.438223548],_
+[1.17,1.455812849,sinh(1.17),sinh(1.17)-1.455812849],_
+[1.18,1.473547732,sinh(1.18),sinh(1.18)-1.473547732],_
+[1.19,1.491429972,sinh(1.19),sinh(1.19)-1.491429972],_
+[1.20,1.509461355,sinh(1.20),sinh(1.20)-1.509461355],_
+[1.21,1.527643687,sinh(1.21),sinh(1.21)-1.527643687],_
+[1.22,1.545978783,sinh(1.22),sinh(1.22)-1.545978783],_
+[1.23,1.564468479,sinh(1.23),sinh(1.23)-1.564468479],_
+[1.24,1.583114623,sinh(1.24),sinh(1.24)-1.583114623],_
+[1.25,1.601919080,sinh(1.25),sinh(1.25)-1.601919080],_
+[1.26,1.620883730,sinh(1.26),sinh(1.26)-1.620883730],_
+[1.27,1.640010470,sinh(1.27),sinh(1.27)-1.640010470],_
+[1.28,1.659301213,sinh(1.28),sinh(1.28)-1.659301213],_
+[1.29,1.678757886,sinh(1.29),sinh(1.29)-1.678757886],_
+[1.30,1.698382437,sinh(1.30),sinh(1.30)-1.698382437],_
+[1.31,1.718176828,sinh(1.31),sinh(1.31)-1.718176828],_
+[1.32,1.738143038,sinh(1.32),sinh(1.32)-1.738143038],_
+[1.33,1.758283063,sinh(1.33),sinh(1.33)-1.758283063],_
+[1.34,1.778598918,sinh(1.34),sinh(1.34)-1.778598918],_
+[1.35,1.799092635,sinh(1.35),sinh(1.35)-1.799092635],_
+[1.36,1.819766262,sinh(1.36),sinh(1.36)-1.819766262],_
+[1.37,1.840621868,sinh(1.37),sinh(1.37)-1.840621868],_
+[1.38,1.861661537,sinh(1.38),sinh(1.38)-1.861661537],_
+[1.39,1.882887374,sinh(1.39),sinh(1.39)-1.882887374],_
+[1.40,1.904301501,sinh(1.40),sinh(1.40)-1.904301501],_
+[1.41,1.925906060,sinh(1.41),sinh(1.41)-1.925906060],_
+[1.42,1.947703212,sinh(1.42),sinh(1.42)-1.947703212],_
+[1.43,1.969695135,sinh(1.43),sinh(1.43)-1.969695135],_
+[1.44,1.991884029,sinh(1.44),sinh(1.44)-1.991884029],_
+[1.45,2.014272114,sinh(1.45),sinh(1.45)-2.014272114],_
+[1.46,2.036861627,sinh(1.46),sinh(1.46)-2.036861627],_
+[1.47,2.059654828,sinh(1.47),sinh(1.47)-2.059654828],_
+[1.48,2.082653996,sinh(1.48),sinh(1.48)-2.082653996],_
+[1.49,2.105861432,sinh(1.49),sinh(1.49)-2.105861432],_
+[1.50,2.129279455,sinh(1.50),sinh(1.50)-2.129279455],_
+[1.51,2.152910408,sinh(1.51),sinh(1.51)-2.152910408],_
+[1.52,2.176756654,sinh(1.52),sinh(1.52)-2.176756654],_
+[1.53,2.200820577,sinh(1.53),sinh(1.53)-2.200820577],_
+[1.54,2.225104585,sinh(1.54),sinh(1.54)-2.225104585],_
+[1.55,2.249611104,sinh(1.55),sinh(1.55)-2.249611104],_
+[1.56,2.274342587,sinh(1.56),sinh(1.56)-2.274342587],_
+[1.57,2.299301506,sinh(1.57),sinh(1.57)-2.299301506],_
+[1.58,2.324490357,sinh(1.58),sinh(1.58)-2.324490357],_
+[1.59,2.349911658,sinh(1.59),sinh(1.59)-2.349911658],_
+[1.60,2.375567953,sinh(1.60),sinh(1.60)-2.375567953],_
+[1.61,2.401461807,sinh(1.61),sinh(1.61)-2.401461807],_
+[1.62,2.427595809,sinh(1.62),sinh(1.62)-2.427595809],_
+[1.63,2.453972572,sinh(1.63),sinh(1.63)-2.453972572],_
+[1.64,2.480594735,sinh(1.64),sinh(1.64)-2.480594735],_
+[1.65,2.507464959,sinh(1.65),sinh(1.65)-2.507464959],_
+[1.66,2.534585932,sinh(1.66),sinh(1.66)-2.534585932],_
+[1.67,2.561960366,sinh(1.67),sinh(1.67)-2.561960366],_
+[1.68,2.589590998,sinh(1.68),sinh(1.68)-2.589590998],_
+[1.69,2.617480591,sinh(1.69),sinh(1.69)-2.617480591],_
+[1.70,2.645631934,sinh(1.70),sinh(1.70)-2.645631934],_
+[1.71,2.674047843,sinh(1.71),sinh(1.71)-2.674047843],_
+[1.72,2.702731158,sinh(1.72),sinh(1.72)-2.702731158],_
+[1.73,2.731684749,sinh(1.73),sinh(1.73)-2.731684749],_
+[1.74,2.760911511,sinh(1.74),sinh(1.74)-2.760911511],_
+[1.75,2.790414366,sinh(1.75),sinh(1.75)-2.790414366],_
+[1.76,2.820196265,sinh(1.76),sinh(1.76)-2.820196265],_
+[1.77,2.850260186,sinh(1.77),sinh(1.77)-2.850260186],_
+[1.78,2.880609136,sinh(1.78),sinh(1.78)-2.880609136],_
+[1.79,2.911246148,sinh(1.79),sinh(1.79)-2.911246148],_
+[1.80,2.942174288,sinh(1.80),sinh(1.80)-2.942174288],_
+[1.81,2.973396648,sinh(1.81),sinh(1.81)-2.973396648],_
+[1.82,3.004916349,sinh(1.82),sinh(1.82)-3.004916349],_
+[1.83,3.036736545,sinh(1.83),sinh(1.83)-3.036736545],_
+[1.84,3.068860417,sinh(1.84),sinh(1.84)-3.068860417],_
+[1.85,3.101291178,sinh(1.85),sinh(1.85)-3.101291178],_
+[1.86,3.134032071,sinh(1.86),sinh(1.86)-3.134032071],_
+[1.87,3.167086369,sinh(1.87),sinh(1.87)-3.167086369],_
+[1.88,3.200457378,sinh(1.88),sinh(1.88)-3.200457378],_
+[1.89,3.234148436,sinh(1.89),sinh(1.89)-3.234148436],_
+[1.90,3.268162912,sinh(1.90),sinh(1.90)-3.268162912],_
+[1.91,3.302504206,sinh(1.91),sinh(1.91)-3.302504206],_
+[1.92,3.337175754,sinh(1.92),sinh(1.92)-3.337175754],_
+[1.93,3.372181022,sinh(1.93),sinh(1.93)-3.372181022],_
+[1.94,3.407523510,sinh(1.94),sinh(1.94)-3.407523510],_
+[1.95,3.443206754,sinh(1.95),sinh(1.95)-3.443206754],_
+[1.96,3.479234322,sinh(1.96),sinh(1.96)-3.479234322],_
+[1.97,3.515609816,sinh(1.97),sinh(1.97)-3.515609816],_
+[1.98,3.552336874,sinh(1.98),sinh(1.98)-3.552336874],_
+[1.99,3.589419168,sinh(1.99),sinh(1.99)-3.589419168],_
+[2.00,3.626860408,sinh(2.00),sinh(2.00)-3.626860408]]
+--R 
+--R
+--R   (1)
+--R   [[0.0,0.0,0.0,0.0],
+--R    [0.01,0.010000167,0.0100001666 6750000198 4,- 0.3324999980 16 E -9],
+--R    [0.02,0.020001333,0.0200013333 6000025397,0.3600002539 7 E -9],
+--R    [0.03,0.0300045,0.0300045002 0250433934,0.2025043393 4 E -9],
+--R    [0.04,0.040010668,0.0400106675 2003250865 9,- 0.4799674913 41 E -9],
+--R    [0.05,0.050020836,0.0500208359 3765501530 3,- 0.6234498469 7 E -10],
+--R    [0.06,0.060036006,0.0600360064 8055545634 4,0.4805554563 44 E -9],
+--R    [0.07,0.070057181,0.0700571806 7413412509 7,- 0.3258658749 E -9],
+--R    [0.08,0.080085361,0.0800853606 4416138576 3,- 0.3558386142 4 E -9],
+--R    [0.09,0.090121549,0.0901215492 1699108556 3,0.2169910855 6 E -9],
+--R    [0.1,0.10016675,0.1001667500 1984402582,0.1984402582 E -10],
+--R    [0.11,0.110221968,0.1102219675 8117152041,- 0.4188284795 9 E -9],
+--R    [0.12,0.120288207,0.1202882074 3110907798,0.4311090779 8 E -9],
+--R    [0.13,0.130366476,0.1303664762 0203025344,0.2020302534 4 E -9],
+--R    [0.14,0.140457782,0.1404577817 2921072423,- 0.2707892757 7 E -9],
+--R    [0.15,0.150563133,0.1505631331 5161265769,0.1516126576 9 E -9],
+--R    [0.16,0.160683541,0.1606835410 1279944828,0.127994483 E -10],
+--R    [0.17,0.170820017,0.1708200173 61990916,0.361990916 E -9],
+--R    [0.18,0.180973576,0.1809735758 5526907178,- 0.1447309282 2 E -9],
+--R    [0.19,0.191145232,0.1911452318 5694457039,- 0.1430554296 1 E -9],
+--R    [0.2,0.201336003,0.2013360025 4109398763,- 0.4589060123 7 E -9],
+--R    [0.21,0.211546907,0.2115469069 9327807565,- 0.672192435 E -11],
+--R    [0.22,0.221778966,0.2217789663 1245116827,0.3124511682 7 E -9],
+--R    [0.23,0.232033204,0.2320332037 1307192732,- 0.2869280726 8 E -9],
+--R    [0.24,0.242310645,0.2423106446 274256412,- 0.3725743588 E -9],
+--R    [0.25,0.252612317,0.2526123168 0816830791,- 0.1918316921 E -9],
+--R    [0.26,0.26293925,0.2629392504 3110275717,0.4311027571 7 E -9],
+--R    [0.27,0.273292478,0.2732924781 9819708912,0.1981970891 E -9],
+--R    [0.28,0.283673035,0.2836730354 4085573175,0.4408557317 5 E -9],
+--R    [0.29,0.29408196,0.2940819602 2345344414,0.2234534441 E -9],
+--R    [0.3,0.304520293,0.3045202934 4714261896,0.4471426189 6 E -9],
+--R    [0.31,0.314989079,0.3149890789 5394426513,- 0.4605573487 E -10],
+--R    [0.32,0.325489364,0.3254893636 3113307984,- 0.3688669201 6 E -9],
+--R    [0.33,0.336022198,0.3360221975 1592704834,- 0.4840729516 6 E -9],
+--R    [0.34,0.346588634,0.3465886339 0049204088,- 0.9950795912 E -10],
+--R    [0.35,0.357189729,0.3571897294 3727190696,0.4372719069 6 E -9],
+--R    [0.36,0.367826544,0.3678265442 4465460028,0.2446546002 8 E -9],
+--R    [0.37,0.378500142,0.3785001420 1298490101,0.12984901 E -10],
+--R    [0.38,0.38921159,0.3892115901 1093433664,0.1109343366 E -9],
+--R    [0.39,0.39996196,0.3999619596 922389386,- 0.3077610614 E -9],
+--R    [0.4,0.410752326,0.4107523258 0281550854,- 0.1971844915 E -9],
+--R    [0.41,0.421583767,0.4215837674 8826710585,0.4882671058 5 E -9],
+--R    [0.42,0.432457368,0.4324573679 0178850717,- 0.9821149283 E -10],
+--R    [0.43,0.443374214,0.4433742144 1248242854,0.4124824285 4 E -9],
+--R    [0.44,0.454335399,0.4543353987 1409734176,- 0.2859026582 4 E -9],
+--R    [0.45,0.465342017,0.4653420169 3419775902,- 0.6580224098 E -10],
+--R    [0.46,0.47639517,0.4763951697 4377790279,- 0.2562220972 1 E -9],
+--R    [0.47,0.487495962,0.4874959624 6732972245,0.4673297224 5 E -9],
+--R    [0.48,0.498645505,0.4986455051 9337626463,0.1933762646 E -9],
+--R    [0.49,0.509844913,0.5098449128 8548145057,- 0.1145185494 E -9],
+--R    [0.5,0.521095305,0.5210953054 9374736162,0.4937473616 2 E -9],
+--R    [0.51,0.532397808,0.5323978080 6681018282,0.6681018282 E -10],
+--R    [0.52,0.543753551,0.5437535508 6434600397,- 0.135653996 E -9],
+--R    [0.53,0.555163669,0.5551636694 700977292,0.4700977292 E -9],
+--R    [0.54,0.566629305,0.5666293049 0543439756,- 0.9456560244 E -10],
+--R    [0.55,0.578151604,0.5781516037 4345427075,- 0.2565457293 E -9],
+--R    [0.56,0.589731718,0.5897317182 2364309847,0.2236430985 E -9],
+--R    [0.57,0.601370806,0.6013708063 6709902716,0.3670990272 E -9],
+--R    [0.58,0.613070032,0.6130700320 9233567487,0.9233567487 E -10],
+--R    [0.59,0.624830565,0.6248305653 3167495254,0.3316749525 E -9],
+--R    [0.6,0.636653582,0.6366535821 4824127113,0.1482412711 E -9],
+--R    [0.61,0.648540265,0.6485402648 5356883411,- 0.1464311659 E -9],
+--R    [0.62,0.660491802,0.6604918021 2583377619,0.1258337762 E -9],
+--R    [0.63,0.672509389,0.6725093891 2872297146,0.1287229715 E -9],
+--R    [0.64,0.684594228,0.6845942276 3095139805,- 0.369048602 E -9],
+--R    [0.65,0.696747526,0.6967475261 2644001112,0.1264400111 E -9],
+--R    [0.66,0.7089705,0.7089704999 5516614206,- 0.448338579 E -10],
+--R    [0.67,0.721264371,0.7212643714 2469850893,0.4246985089 E -9],
+--R    [0.68,0.73363037,0.7336303699 3242899202,- 0.6757100798 E -10],
+--R    [0.69,0.746069732,0.7460697320 8851339749,0.8851339749 E -10],
+--R    [0.7,0.758583702,0.7585837018 3953350346,- 0.1604664965 E -9],
+--R    [0.71,0.771173531,0.7711735305 9289275483,- 0.4071072452 E -9],
+--R    [0.72,0.783840477,0.7838404773 4195804645,0.3419580464 E -9],
+--R    [0.73,0.796585809,0.7965858087 91960109,- 0.208039891 E -9],
+--R    [0.74,0.809410799,0.8094107994 8666508762,0.4866650876 2 E -9],
+--R    [0.75,0.822316732,0.8223167319 358299807,- 0.641700193 E -10],
+--R    [0.76,0.835304897,0.8353048967 4345468428,- 0.2565453157 E -9],
+--R    [0.77,0.848376593,0.8483765927 3684346756,- 0.2631565324 E -9],
+--R    [0.78,0.861533127,0.8615331270 9648878576,0.9648878576 E -10],
+--R    [0.79,0.874775815,0.8747758154 8679041856,0.4867904185 6 E -9],
+--R    [0.8,0.888105982,0.8881059821 8762300658,0.1876230066 E -9],
+--R    [0.81,0.90152496,0.9015249602 2676514235,0.2267651423 E -9],
+--R    [0.82,0.915034092,0.9150340915 1320325907,- 0.4867967409 3 E -9],
+--R    [0.83,0.928634727,0.9286347269 7132364747,- 0.286763525 E -10],
+--R    [0.84,0.942328227,0.9423282266 7600602026,- 0.3239939797 E -9],
+--R    [0.85,0.95611596,0.9561159599 8863213347,- 0.113678665 E -10],
+--R    [0.86,0.969999306,0.9699993056 9402306576,- 0.3059769342 E -9],
+--R    [0.87,0.983979652,0.9839796521 3831884955,0.1383188496 E -9],
+--R    [0.88,0.998058397,0.9980583973 6781424195,0.367814242 E -9],
+--R    [0.89,1.012236949,1.0122369492 687645193,0.2687645193 E -9],
+--R    [0.9,1.026516726,1.0265167257 08175276,- 0.291824724 E -9],
+--R    [0.91,1.040899155,1.0408991546 755903063,- 0.3244096937 E -9],
+--R    [0.92,1.055385674,1.0553856744 258917494,0.4258917494 E -9],
+--R    [0.93,1.069977734,1.0699777336 231267752,- 0.3768732248 E -9],
+--R    [0.94,1.084676791,1.0846767914 853751972,0.4853751972 E -9],
+--R    [0.95,1.099484318,1.0994843179 30672496,- 0.69327504 E -10],
+--R    [0.96,1.114401794,1.1144017937 240028478,- 0.2759971522 E -9],
+--R    [0.97,1.129430711,1.1294307106 253768569,- 0.3746231431 E -9],
+--R    [0.98,1.144572572,1.1445725715 390087998,- 0.4609912002 E -9],
+--R    [0.99,1.159828891,1.1598288906 636082993,- 0.3363917007 E -9],
+--R    [1.0,1.175201194,1.1752011936 438014569,- 0.3561985431 E -9],
+--R    [1.01,1.190691018,1.1906910177 226965871,- 0.2773034129 E -9],
+--R    [1.02,1.206299912,1.2062999118 956098093,- 0.104390191 E -9],
+--R    [1.03,1.222029437,1.2220294370 649658698,0.649658698 E -10],
+--R    [1.04,1.237881166,1.2378811661 963896857,0.1963896857 E -9],
+--R    [1.05,1.253856684,1.2538566844 760042176,0.4760042176 E -9],
+--R    [1.06,1.269957589,1.2699575894 689504033,0.4689504033 E -9],
+--R    [1.07,1.286185491,1.2861854912 791450029,0.2791450029 E -9],
+--R    [1.08,1.302542013,1.3025420127 102923325,- 0.2897076676 E -9],
+--R    [1.09,1.319028789,1.3190287894 281659867,0.4281659867 E -9],
+--R    [1.1,1.33564747,1.3356474701 241767794,0.1241767794 E -9],
+--R    [1.11,1.352399717,1.3523997166 802432583,- 0.3197567417 E -9],
+--R    [1.12,1.369287204,1.3692872043 349812821,0.3349812821 E -9],
+--R    [1.13,1.386311622,1.3863116218 51229278,- 0.148770722 E -9],
+--R    [1.14,1.403474672,1.4034746716 849259325,- 0.3150740675 E -9],
+--R    [1.15,1.42077807,1.4207780701 553572045,0.1553572045 E -9],
+--R    [1.16,1.438223548,1.4382235476 16789684,- 0.383210316 E -9],
+--R    [1.17,1.455812849,1.4558128486 315074606,- 0.3684925394 E -9],
+--R    [1.18,1.473547732,1.4735477321 442698057,0.1442698057 E -9],
+--R    [1.19,1.491429972,1.4914299716 582071144,- 0.3417928856 E -9],
+--R    [1.2,1.509461355,1.5094613554 121726964,0.4121726964 E -9],
+--R    [1.21,1.527643687,1.5276436865 595681515,- 0.4404318485 E -9],
+--R    [1.22,1.545978783,1.5459787833 486602124,0.3486602124 E -9],
+--R    [1.23,1.564468479,1.5644684793 044070864,0.3044070864 E -9],
+--R    [1.24,1.583114623,1.5831146234 118124797,0.4118124797 E -9],
+--R    [1.25,1.60191908,1.6019190803 008256379,0.3008256379 E -9],
+--R    [1.26,1.62088373,1.6208837304 328058954,0.4328058954 E -9],
+--R    [1.27,1.64001047,1.6400104702 88570378,0.288570378 E -9],
+--R    [1.28,1.659301213,1.6593012125 580436651,- 0.4419563349 E -9],
+--R    [1.29,1.678757886,1.6787578863 315283762,0.3315283762 E -9],
+--R    [1.3,1.698382437,1.6983824372 926158087,0.2926158087 E -9],
+--R    [1.31,1.718176828,1.7181768279 127559182,- 0.872440818 E -10],
+--R    [1.32,1.738143038,1.7381430376 475060993,- 0.3524939007 E -9],
+--R    [1.33,1.758283063,1.7582830631 344783905,0.1344783905 E -9],
+--R    [1.34,1.778598918,1.7785989183 930048997,0.3930048997 E -9],
+--R    [1.35,1.799092635,1.7990926350 255414153,0.255414153 E -10],
+--R    [1.36,1.819766262,1.8197662624 20829345,0.420829345 E -9],
+--R    [1.37,1.840621868,1.8406218679 588362979,- 0.411637021 E -10],
+--R    [1.38,1.861661537,1.8616615372 174958039,0.2174958039 E -9],
+--R    [1.39,1.882887374,1.8828873741 812668452,0.1812668451 E -9],
+--R    [1.4,1.904301501,1.9043015014 515340551,0.4515340551 E -9],
+--R    [1.41,1.92590606,1.9259060604 588696261,0.4588696261 E -9],
+--R    [1.42,1.947703212,1.9477032116 771781509,- 0.3228218491 E -9],
+--R    [1.43,1.969695135,1.9696951348 397458135,- 0.1602541865 E -9],
+--R    [1.44,1.991884029,1.9918840291 572155345,0.1572155345 E -9],
+--R    [1.45,2.014272114,2.0142721135 375098676,- 0.4624901324 E -9],
+--R    [1.46,2.036861627,2.0368616268 077236416,- 0.192276358 E -9],
+--R    [1.47,2.059654828,2.0596548279 380085349,- 0.619914651 E -10],
+--R    [1.48,2.082653996,2.0826539962 674719736,0.2674719736 E -9],
+--R    [1.49,2.105861432,2.1058614317 321129415,- 0.2678870585 E -9],
+--R    [1.5,2.129279455,2.1292794550 948174968,0.948174968 E -10],
+--R    [1.51,2.152910408,2.1529104081 774369945,0.177436994 E -9],
+--R    [1.52,2.176756654,2.1767566540 94972223,0.94972223 E -10],
+--R    [1.53,2.200820577,2.2008205774 918868736,0.4918868736 E -9],
+--R    [1.54,2.225104585,2.2251045847 805739743,- 0.219426026 E -9],
+--R    [1.55,2.249611104,2.2496111043 819991339,0.3819991339 E -9],
+--R    [1.56,2.274342587,2.2743425869 685446626,- 0.314553374 E -10],
+--R    [1.57,2.299301506,2.2993015057 090788526,- 0.2909211474 E -9],
+--R    [1.58,2.324490357,2.3244903565 162749255,- 0.4837250745 E -9],
+--R    [1.59,2.349911658,2.3499116582 962043799,0.2962043799 E -9],
+--R    [1.6,2.375567953,2.3755679532 002296976,0.200229698 E -9],
+--R    [1.61,2.401461807,2.4014618068 79221598,- 0.120778402 E -9],
+--R    [1.62,2.427595809,2.4275958087 401262638,- 0.2598737362 E -9],
+--R    [1.63,2.453972572,2.4539725722 049081927,0.204908193 E -9],
+--R    [1.64,2.480594735,2.4805947349 718945727,- 0.281054273 E -10],
+--R    [1.65,2.507464959,2.5074649592 795473117,0.2795473117 E -9],
+--R    [1.66,2.534585932,2.5345859321 726891034,0.172689103 E -9],
+--R    [1.67,2.561960366,2.5619603657 712101481,- 0.228789852 E -9],
+--R    [1.68,2.589590998,2.5895909975 412824018,- 0.4587175982 E -9],
+--R    [1.69,2.617480591,2.6174805905 69108475,- 0.430891525 E -9],
+--R    [1.7,2.645631934,2.6456319338 372325553,- 0.162767445 E -9],
+--R    [1.71,2.674047843,2.6740478425 034409861,- 0.4965590139 E -9],
+--R    [1.72,2.702731158,2.7027311581 822803909,0.182280391 E -9],
+--R    [1.73,2.731684749,2.7316847492 292214966,0.229221497 E -9],
+--R    [1.74,2.760911511,2.7609115110 274970701,0.274970701 E -10],
+--R    [1.75,2.790414366,2.7904143662 776426551,0.2776426551 E -9],
+--R    [1.76,2.820196265,2.8201962652 897690607,0.2897690607 E -9],
+--R    [1.77,2.850260186,2.8502601862 785958316,0.2785958316 E -9],
+--R    [1.78,2.880609136,2.8806091356 612752013,- 0.3387247987 E -9],
+--R    [1.79,2.911246148,2.9112461483 580363133,0.3580363133 E -9],
+--R    [1.8,2.942174288,2.9421742880 956797727,0.956797727 E -10],
+--R    [1.81,2.973396648,2.9733966477 139528796,- 0.2860471204 E -9],
+--R    [1.82,3.004916349,3.0049163494 748361809,0.4748361809 E -9],
+--R    [1.83,3.036736545,3.0367365453 747722708,0.3747722708 E -9],
+--R    [1.84,3.068860417,3.0688604174 598680611,0.4598680611 E -9],
+--R    [1.85,3.101291178,3.1012911781 441020441,0.144102044 E -9],
+--R    [1.86,3.134032071,3.1340320705 305683671,- 0.4694316329 E -9],
+--R    [1.87,3.167086369,3.1670863687 357898447,- 0.2642101554 E -9],
+--R    [1.88,3.200457378,3.2004573782 171323393,0.217132339 E -9],
+--R    [1.89,3.234148436,3.2341484361 033532526,0.103353253 E -9],
+--R    [1.9,3.268162912,3.2681629115 283171817,- 0.4716828183 E -9],
+--R    [1.91,3.302504206,3.3025042059 679121137,- 0.320878863 E -10],
+--R    [1.92,3.337175754,3.3371757535 801998489,- 0.4198001511 E -9],
+--R    [1.93,3.372181022,3.3721810215 488346686,- 0.4511653313 E -9],
+--R    [1.94,3.40752351,3.4075235104 297845903,0.4297845903 E -9],
+--R    [1.95,3.443206754,3.4432067545 013898812,0.5013898812 E -9],
+--R    [1.96,3.479234322,3.4792343221 177938377,0.117793838 E -9],
+--R    [1.97,3.515609816,3.5156098160 657811731,0.657811731 E -10],
+--R    [1.98,3.552336874,3.5523368739 25059699,- 0.74940301 E -10],
+--R    [1.99,3.589419168,3.5894191684 320213268,0.4320213268 E -9],
+--R    [2.0,3.626860408,3.6268604078 470187677,- 0.152981232 E -9]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of cosh, ranging from 0.00 to 2.00. The
+second column is the reference value of sinh from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+
+<<*>>=
+--S 2 of 2
+[[0.00,1.000000000,cosh(0.00),cosh(0.00)-1.000000000],_
+[0.01,1.000050000,cosh(0.01),cosh(0.01)-1.000050000],_
+[0.02,1.000200007,cosh(0.02),cosh(0.02)-1.000200007],_
+[0.03,1.000450034,cosh(0.03),cosh(0.03)-1.000450034],_
+[0.04,1.000800107,cosh(0.04),cosh(0.04)-1.000800107],_
+[0.05,1.001250260,cosh(0.05),cosh(0.05)-1.001250260],_
+[0.06,1.001800540,cosh(0.06),cosh(0.06)-1.001800540],_
+[0.07,1.002451001,cosh(0.07),cosh(0.07)-1.002451001],_
+[0.08,1.003201707,cosh(0.08),cosh(0.08)-1.003201707],_
+[0.09,1.004052734,cosh(0.09),cosh(0.09)-1.004052734],_
+[0.10,1.005004168,cosh(0.10),cosh(0.10)-1.005004168],_
+[0.11,1.006056103,cosh(0.11),cosh(0.11)-1.006056103],_
+[0.12,1.007208644,cosh(0.12),cosh(0.12)-1.007208644],_
+[0.13,1.008461907,cosh(0.13),cosh(0.13)-1.008461907],_
+[0.14,1.009816017,cosh(0.14),cosh(0.14)-1.009816017],_
+[0.15,1.011271110,cosh(0.15),cosh(0.15)-1.011271110],_
+[0.16,1.012827330,cosh(0.16),cosh(0.16)-1.012827330],_
+[0.17,1.014484834,cosh(0.17),cosh(0.17)-1.014484834],_
+[0.18,1.016243787,cosh(0.18),cosh(0.18)-1.016243787],_
+[0.19,1.018104366,cosh(0.19),cosh(0.19)-1.018104366],_
+[0.20,1.020066756,cosh(0.20),cosh(0.20)-1.020066756],_
+[0.21,1.022131153,cosh(0.21),cosh(0.21)-1.022131153],_
+[0.22,1.024297764,cosh(0.22),cosh(0.22)-1.024297764],_
+[0.23,1.026566806,cosh(0.23),cosh(0.23)-1.026566806],_
+[0.24,1.028938506,cosh(0.24),cosh(0.24)-1.028938506],_
+[0.25,1.031413100,cosh(0.25),cosh(0.25)-1.031413100],_
+[0.26,1.033990836,cosh(0.26),cosh(0.26)-1.033990836],_
+[0.27,1.036671973,cosh(0.27),cosh(0.27)-1.036671973],_
+[0.28,1.039456777,cosh(0.28),cosh(0.28)-1.039456777],_
+[0.29,1.042345528,cosh(0.29),cosh(0.29)-1.042345528],_
+[0.30,1.045338514,cosh(0.30),cosh(0.30)-1.045338514],_
+[0.31,1.048436035,cosh(0.31),cosh(0.31)-1.048436035],_
+[0.32,1.051638401,cosh(0.32),cosh(0.32)-1.051638401],_
+[0.33,1.054945931,cosh(0.33),cosh(0.33)-1.054945931],_
+[0.34,1.058358957,cosh(0.34),cosh(0.34)-1.058358957],_
+[0.35,1.061877819,cosh(0.35),cosh(0.35)-1.061877819],_
+[0.36,1.065502870,cosh(0.36),cosh(0.36)-1.065502870],_
+[0.37,1.069234473,cosh(0.37),cosh(0.37)-1.069234473],_
+[0.38,1.073072999,cosh(0.38),cosh(0.38)-1.073072999],_
+[0.39,1.077018834,cosh(0.39),cosh(0.39)-1.077018834],_
+[0.40,1.081072372,cosh(0.40),cosh(0.40)-1.081072372],_
+[0.41,1.085234018,cosh(0.41),cosh(0.41)-1.085234018],_
+[0.42,1.089504188,cosh(0.42),cosh(0.42)-1.089504188],_
+[0.43,1.093883309,cosh(0.43),cosh(0.43)-1.093883309],_
+[0.44,1.098371820,cosh(0.44),cosh(0.44)-1.098371820],_
+[0.45,1.102970169,cosh(0.45),cosh(0.45)-1.102970169],_
+[0.46,1.107678815,cosh(0.46),cosh(0.46)-1.107678815],_
+[0.47,1.112498231,cosh(0.47),cosh(0.47)-1.112498231],_
+[0.48,1.117428897,cosh(0.48),cosh(0.48)-1.117428897],_
+[0.49,1.122471307,cosh(0.49),cosh(0.49)-1.122471307],_
+[0.50,1.127625965,cosh(0.50),cosh(0.50)-1.127625965],_
+[0.51,1.132893387,cosh(0.51),cosh(0.51)-1.132893387],_
+[0.52,1.138274099,cosh(0.52),cosh(0.52)-1.138274099],_
+[0.53,1.143768639,cosh(0.53),cosh(0.53)-1.143768639],_
+[0.54,1.149377557,cosh(0.54),cosh(0.54)-1.149377557],_
+[0.55,1.155101414,cosh(0.55),cosh(0.55)-1.155101414],_
+[0.56,1.160940782,cosh(0.56),cosh(0.56)-1.160940782],_
+[0.57,1.166896245,cosh(0.57),cosh(0.57)-1.166896245],_
+[0.58,1.172968399,cosh(0.58),cosh(0.58)-1.172968399],_
+[0.59,1.179157850,cosh(0.59),cosh(0.59)-1.179157850],_
+[0.60,1.185465218,cosh(0.60),cosh(0.60)-1.185465218],_
+[0.61,1.191891134,cosh(0.61),cosh(0.61)-1.191891134],_
+[0.62,1.198436240,cosh(0.62),cosh(0.62)-1.198436240],_
+[0.63,1.205101190,cosh(0.63),cosh(0.63)-1.205101190],_
+[0.64,1.211886652,cosh(0.64),cosh(0.64)-1.211886652],_
+[0.65,1.218793303,cosh(0.65),cosh(0.65)-1.218793303],_
+[0.66,1.225821834,cosh(0.66),cosh(0.66)-1.225821834],_
+[0.67,1.232972949,cosh(0.67),cosh(0.67)-1.232972949],_
+[0.68,1.240247362,cosh(0.68),cosh(0.68)-1.240247362],_
+[0.69,1.247645801,cosh(0.69),cosh(0.69)-1.247645801],_
+[0.70,1.255169006,cosh(0.70),cosh(0.70)-1.255169006],_
+[0.71,1.262817728,cosh(0.71),cosh(0.71)-1.262817728],_
+[0.72,1.270592733,cosh(0.72),cosh(0.72)-1.270592733],_
+[0.73,1.278494799,cosh(0.73),cosh(0.73)-1.278494799],_
+[0.74,1.286524715,cosh(0.74),cosh(0.74)-1.286524715],_
+[0.75,1.294683285,cosh(0.75),cosh(0.75)-1.294683285],_
+[0.76,1.302971324,cosh(0.76),cosh(0.76)-1.302971324],_
+[0.77,1.311389661,cosh(0.77),cosh(0.77)-1.311389661],_
+[0.78,1.319939138,cosh(0.78),cosh(0.78)-1.319939138],_
+[0.79,1.328620611,cosh(0.79),cosh(0.79)-1.328620611],_
+[0.80,1.337434946,cosh(0.80),cosh(0.80)-1.337434946],_
+[0.81,1.346383026,cosh(0.81),cosh(0.81)-1.346383026],_
+[0.82,1.355465746,cosh(0.82),cosh(0.82)-1.355465746],_
+[0.83,1.364684013,cosh(0.83),cosh(0.83)-1.364684013],_
+[0.84,1.374038750,cosh(0.84),cosh(0.84)-1.374038750],_
+[0.85,1.383530892,cosh(0.85),cosh(0.85)-1.383530892],_
+[0.86,1.393161388,cosh(0.86),cosh(0.86)-1.393161388],_
+[0.87,1.402931201,cosh(0.87),cosh(0.87)-1.402931201],_
+[0.88,1.412841309,cosh(0.88),cosh(0.88)-1.412841309],_
+[0.89,1.422892702,cosh(0.89),cosh(0.89)-1.422892702],_
+[0.90,1.433086385,cosh(0.90),cosh(0.90)-1.433086385],_
+[0.91,1.443423379,cosh(0.91),cosh(0.91)-1.443423379],_
+[0.92,1.453904716,cosh(0.92),cosh(0.92)-1.453904716],_
+[0.93,1.464531444,cosh(0.93),cosh(0.93)-1.464531444],_
+[0.94,1.475304627,cosh(0.94),cosh(0.94)-1.475304627],_
+[0.95,1.486225341,cosh(0.95),cosh(0.95)-1.486225341],_
+[0.96,1.497294680,cosh(0.96),cosh(0.96)-1.497294680],_
+[0.97,1.508513749,cosh(0.97),cosh(0.97)-1.508513749],_
+[0.98,1.519883670,cosh(0.98),cosh(0.98)-1.519883670],_
+[0.99,1.531405582,cosh(0.99),cosh(0.99)-1.531405582],_
+[1.00,1.543080635,cosh(1.00),cosh(1.00)-1.543080635],_
+[1.01,1.554909997,cosh(1.01),cosh(1.01)-1.554909997],_
+[1.02,1.566894852,cosh(1.02),cosh(1.02)-1.566894852],_
+[1.03,1.579036398,cosh(1.03),cosh(1.03)-1.579036398],_
+[1.04,1.591335848,cosh(1.04),cosh(1.04)-1.591335848],_
+[1.05,1.603794434,cosh(1.05),cosh(1.05)-1.603794434],_
+[1.06,1.616413400,cosh(1.06),cosh(1.06)-1.616413400],_
+[1.07,1.629194009,cosh(1.07),cosh(1.07)-1.629194009],_
+[1.08,1.642137538,cosh(1.08),cosh(1.08)-1.642137538],_
+[1.09,1.655245283,cosh(1.09),cosh(1.09)-1.655245283],_
+[1.10,1.668518554,cosh(1.10),cosh(1.10)-1.668518554],_
+[1.11,1.681958678,cosh(1.11),cosh(1.11)-1.681958678],_
+[1.12,1.695566999,cosh(1.12),cosh(1.12)-1.695566999],_
+[1.13,1.709344878,cosh(1.13),cosh(1.13)-1.709344878],_
+[1.14,1.723293694,cosh(1.14),cosh(1.14)-1.723293694],_
+[1.15,1.737414840,cosh(1.15),cosh(1.15)-1.737414840],_
+[1.16,1.751709728,cosh(1.16),cosh(1.16)-1.751709728],_
+[1.17,1.766179790,cosh(1.17),cosh(1.17)-1.766179790],_
+[1.18,1.780826471,cosh(1.18),cosh(1.18)-1.780826471],_
+[1.19,1.795651236,cosh(1.19),cosh(1.19)-1.795651236],_
+[1.20,1.810655567,cosh(1.20),cosh(1.20)-1.810655567],_
+[1.21,1.825840966,cosh(1.21),cosh(1.21)-1.825840966],_
+[1.22,1.841208950,cosh(1.22),cosh(1.22)-1.841208950],_
+[1.23,1.856761057,cosh(1.23),cosh(1.23)-1.856761057],_
+[1.24,1.872498841,cosh(1.24),cosh(1.24)-1.872498841],_
+[1.25,1.888423877,cosh(1.25),cosh(1.25)-1.888423877],_
+[1.26,1.904537757,cosh(1.26),cosh(1.26)-1.904537757],_
+[1.27,1.920842092,cosh(1.27),cosh(1.27)-1.920842092],_
+[1.28,1.937338513,cosh(1.28),cosh(1.28)-1.937338513],_
+[1.29,1.954028669,cosh(1.29),cosh(1.29)-1.954028669],_
+[1.30,1.970914230,cosh(1.30),cosh(1.30)-1.970914230],_
+[1.31,1.987996884,cosh(1.31),cosh(1.31)-1.987996884],_
+[1.32,2.005278340,cosh(1.32),cosh(1.32)-2.005278340],_
+[1.33,2.022760324,cosh(1.33),cosh(1.33)-2.022760324],_
+[1.34,2.040444587,cosh(1.34),cosh(1.34)-2.040444587],_
+[1.35,2.058332896,cosh(1.35),cosh(1.35)-2.058332896],_
+[1.36,2.076427039,cosh(1.36),cosh(1.36)-2.076427039],_
+[1.37,2.094728828,cosh(1.37),cosh(1.37)-2.094728828],_
+[1.38,2.113240090,cosh(1.38),cosh(1.38)-2.113240090],_
+[1.39,2.131962679,cosh(1.39),cosh(1.39)-2.131962679],_
+[1.40,2.150898465,cosh(1.40),cosh(1.40)-2.150898465],_
+[1.41,2.170049344,cosh(1.41),cosh(1.41)-2.170049344],_
+[1.42,2.189417229,cosh(1.42),cosh(1.42)-2.189417229],_
+[1.43,2.209004057,cosh(1.43),cosh(1.43)-2.209004057],_
+[1.44,2.228811788,cosh(1.44),cosh(1.44)-2.228811788],_
+[1.45,2.248842402,cosh(1.45),cosh(1.45)-2.248842402],_
+[1.46,2.269097902,cosh(1.46),cosh(1.46)-2.269097902],_
+[1.47,2.289580313,cosh(1.47),cosh(1.47)-2.289580313],_
+[1.48,2.310291685,cosh(1.48),cosh(1.48)-2.310291685],_
+[1.49,2.331234087,cosh(1.49),cosh(1.49)-2.331234087],_
+[1.50,2.352409615,cosh(1.50),cosh(1.50)-2.352409615],_
+[1.51,2.373820386,cosh(1.51),cosh(1.51)-2.373820386],_
+[1.52,2.395468541,cosh(1.52),cosh(1.52)-2.395468541],_
+[1.53,2.417356245,cosh(1.53),cosh(1.53)-2.417356245],_
+[1.54,2.439485686,cosh(1.54),cosh(1.54)-2.439485686],_
+[1.55,2.461859078,cosh(1.55),cosh(1.55)-2.461859078],_
+[1.56,2.484478658,cosh(1.56),cosh(1.56)-2.484478658],_
+[1.57,2.507346688,cosh(1.57),cosh(1.57)-2.507346688],_
+[1.58,2.530465455,cosh(1.58),cosh(1.58)-2.530465455],_
+[1.59,2.553837270,cosh(1.59),cosh(1.59)-2.553837270],_
+[1.60,2.577464471,cosh(1.60),cosh(1.60)-2.577464471],_
+[1.61,2.601349421,cosh(1.61),cosh(1.61)-2.601349421],_
+[1.62,2.625494508,cosh(1.62),cosh(1.62)-2.625494508],_
+[1.63,2.649902146,cosh(1.63),cosh(1.63)-2.649902146],_
+[1.64,2.674574777,cosh(1.64),cosh(1.64)-2.674574777],_
+[1.65,2.699514868,cosh(1.65),cosh(1.65)-2.699514868],_
+[1.66,2.724724912,cosh(1.66),cosh(1.66)-2.724724912],_
+[1.67,2.750207431,cosh(1.67),cosh(1.67)-2.750207431],_
+[1.68,2.775964974,cosh(1.68),cosh(1.68)-2.775964974],_
+[1.69,2.802000115,cosh(1.69),cosh(1.69)-2.802000115],_
+[1.70,2.828315458,cosh(1.70),cosh(1.70)-2.828315458],_
+[1.71,2.854913635,cosh(1.71),cosh(1.71)-2.854913635],_
+[1.72,2.881797306,cosh(1.72),cosh(1.72)-2.881797306],_
+[1.73,2.908969159,cosh(1.73),cosh(1.73)-2.908969159],_
+[1.74,2.936431912,cosh(1.74),cosh(1.74)-2.936431912],_
+[1.75,2.964188310,cosh(1.75),cosh(1.75)-2.964188310],_
+[1.76,2.992241129,cosh(1.76),cosh(1.76)-2.992241129],_
+[1.77,3.020593175,cosh(1.77),cosh(1.77)-3.020593175],_
+[1.78,3.049247283,cosh(1.78),cosh(1.78)-3.049247283],_
+[1.79,3.078206318,cosh(1.79),cosh(1.79)-3.078206318],_
+[1.80,3.107473176,cosh(1.80),cosh(1.80)-3.107473176],_
+[1.81,3.137050785,cosh(1.81),cosh(1.81)-3.137050785],_
+[1.82,3.166942100,cosh(1.82),cosh(1.82)-3.166942100],_
+[1.83,3.197150113,cosh(1.83),cosh(1.83)-3.197150113],_
+[1.84,3.227677844,cosh(1.84),cosh(1.84)-3.227677844],_
+[1.85,3.258528344,cosh(1.85),cosh(1.85)-3.258528344],_
+[1.86,3.289704701,cosh(1.86),cosh(1.86)-3.289704701],_
+[1.87,3.321210031,cosh(1.87),cosh(1.87)-3.321210031],_
+[1.88,3.353047484,cosh(1.88),cosh(1.88)-3.353047484],_
+[1.89,3.385220245,cosh(1.89),cosh(1.89)-3.385220245],_
+[1.90,3.417731531,cosh(1.90),cosh(1.90)-3.417731531],_
+[1.91,3.450584593,cosh(1.91),cosh(1.91)-3.450584593],_
+[1.92,3.483782716,cosh(1.92),cosh(1.92)-3.483782716],_
+[1.93,3.517329220,cosh(1.93),cosh(1.93)-3.517329220],_
+[1.94,3.551227460,cosh(1.94),cosh(1.94)-3.551227460],_
+[1.95,3.585480826,cosh(1.95),cosh(1.95)-3.585480826],_
+[1.96,3.620092743,cosh(1.96),cosh(1.96)-3.620092743],_
+[1.97,3.655066672,cosh(1.97),cosh(1.97)-3.655066672],_
+[1.98,3.690406111,cosh(1.98),cosh(1.98)-3.690406111],_
+[1.99,3.726114594,cosh(1.99),cosh(1.99)-3.726114594],_
+[2.00,3.762195691,cosh(2.00),cosh(2.00)-3.762195691]]
+--R 
+--R
+--R   (2)
+--R   [[0.0,1.0,1.0,0.0], [0.01,1.00005,1.0000500004 166680556,0.4166680556 E -9],
+--R    [0.02,1.000200007,1.0002000066 667555562,- 0.3332444438 E -9],
+--R    [0.03,1.000450034,1.0004500337 510125163,- 0.2489874837 E -9],
+--R    [0.04,1.000800107,1.0008001066 723557181,- 0.3276442819 E -9],
+--R    [0.05,1.00125026,1.0012502604 383690244,0.4383690244 E -9],
+--R    [0.06,1.00180054,1.0018005400 648041659,0.648041659 E -10],
+--R    [0.07,1.002451001,1.0024510005 80082354,- 0.419917646 E -9],
+--R    [0.08,1.003201707,1.0032017070 307971687,0.307971687 E -10],
+--R    [0.09,1.004052734,1.0040527344 882192723,0.4882192723 E -9],
+--R    [0.1,1.005004168,1.0050041680 55803599,0.55803599 E -10],
+--R    [0.11,1.006056103,1.0060561028 776997711,- 0.1223002289 E -9],
+--R    [0.12,1.007208644,1.0072086441 482665935,0.1482665935 E -9],
+--R    [0.13,1.008461907,1.0084619071 225915772,0.1225915772 E -9],
+--R    [0.14,1.009816017,1.0098160171 280165439,0.1280165439 E -9],
+--R    [0.15,1.01127111,1.0112711095 766704649,- 0.4233295351 E -9],
+--R    [0.16,1.01282733,1.0128273299 790107867,- 0.209892133 E -10],
+--R    [0.17,1.014484834,1.0144848339 58374598,- 0.41625402 E -10],
+--R    [0.18,1.016243787,1.0162437872 665410931,0.2665410931 E -9],
+--R    [0.19,1.018104366,1.0181043658 003068879,- 0.1996931121 E -9],
+--R    [0.2,1.020066756,1.0200667556 190758463,- 0.3809241537 E -9],
+--R    [0.21,1.022131153,1.0221311529 634651755,- 0.365348245 E -10],
+--R    [0.22,1.024297764,1.0242977642 749296513,0.2749296512 E -9],
+--R    [0.23,1.026566806,1.0265668062 164059355,0.2164059355 E -9],
+--R    [0.24,1.028938506,1.0289385056 939790504,- 0.3060209496 E -9],
+--R    [0.25,1.0314131,1.0314130998 795731762,- 0.1204268238 E -9],
+--R    [0.26,1.033990836,1.0339908362 346690408,0.2346690408 E -9],
+--R    [0.27,1.036671973,1.0366719725 350502748,- 0.4649497252 E -9],
+--R    [0.28,1.039456777,1.0394567768 965812039,- 0.103418796 E -9],
+--R    [0.29,1.042345528,1.0423455278 020186592,- 0.1979813408 E -9],
+--R    [0.3,1.045338514,1.0453385141 28860485,0.128860485 E -9],
+--R    [0.31,1.048436035,1.0484360351 78233529,0.178233529 E -9],
+--R    [0.32,1.051638401,1.0516384007 048240047,- 0.2951759953 E -9],
+--R    [0.33,1.054945931,1.0549459309 478532179,- 0.521467821 E -10],
+--R    [0.34,1.058358957,1.0583589566 63101756,- 0.336898244 E -9],
+--R    [0.35,1.061877819,1.0618778191 559853413,0.1559853413 E -9],
+--R    [0.36,1.06550287,1.0655028703 156856575,0.3156856575 E -9],
+--R    [0.37,1.069234473,1.0692344726 503395606,- 0.3496604394 E -9],
+--R    [0.38,1.073072999,1.0730729993 232901949,0.3232901949 E -9],
+--R    [0.39,1.077018834,1.0770188341 904036385,0.1904036385 E -9],
+--R    [0.4,1.081072372,1.0810723718 384548093,- 0.1615451907 E -9],
+--R    [0.41,1.085234018,1.0852340176 245864718,- 0.3754135282 E -9],
+--R    [0.42,1.089504188,1.0895041877 168452888,- 0.2831547112 E -9],
+--R    [0.43,1.093883309,1.0938833091 357989732,0.1357989732 E -9],
+--R    [0.44,1.09837182,1.0983718197 972387003,- 0.2027612997 E -9],
+--R    [0.45,1.102970169,1.1029701685 559710522,- 0.4440289478 E -9],
+--R    [0.46,1.107678815,1.1076788152 507038721,0.2507038721 E -9],
+--R    [0.47,1.112498231,1.1124982307 500305186,- 0.2499694814 E -9],
+--R    [0.48,1.117428897,1.1174288969 995171175,- 0.4828825 E -12],
+--R    [0.49,1.122471307,1.1224713070 698975196,0.698975196 E -10],
+--R    [0.5,1.127625965,1.1276259652 063807852,0.2063807852 E -9],
+--R    [0.51,1.132893387,1.1328933868 790761256,- 0.1209238744 E -9],
+--R    [0.52,1.138274099,1.1382740988 345403429,- 0.1654596571 E -9],
+--R    [0.53,1.143768639,1.1437686391 484529252,0.1484529252 E -9],
+--R    [0.54,1.149377557,1.1493775572 794240625,0.2794240625 E -9],
+--R    [0.55,1.155101414,1.1551014141 239409661,0.1239409661 E -9],
+--R    [0.56,1.160940782,1.1609407820 724579841,0.724579841 E -10],
+--R    [0.57,1.166896245,1.1668962450 666361245,0.666361245 E -10],
+--R    [0.58,1.172968399,1.1729683986 577377074,- 0.3422622926 E -9],
+--R    [0.59,1.17915785,1.1791578500 661819879,0.661819879 E -10],
+--R    [0.6,1.185465218,1.1854652182 422677038,0.2422677038 E -9],
+--R    [0.61,1.191891134,1.1918911339 280686212,- 0.719313788 E -10],
+--R    [0.62,1.19843624,1.1984362397 20508268,- 0.279491732 E -9],
+--R    [0.63,1.20510119,1.2051011901 35620161,0.135620161 E -9],
+--R    [0.64,1.211886652,1.2118866516 739999553,- 0.3260000447 E -9],
+--R    [0.65,1.218793303,1.2187933028 87456059,- 0.112543941 E -9],
+--R    [0.66,1.225821834,1.2258218344 468653796,0.4468653796 E -9],
+--R    [0.67,1.232972949,1.2329729492 112409872,0.2112409872 E -9],
+--R    [0.68,1.240247362,1.2402473622 980186015,0.2980186015 E -9],
+--R    [0.69,1.247645801,1.2476458011 545689314,0.1545689314 E -9],
+--R    [0.7,1.255169006,1.2551690056 309430182,- 0.3690569818 E -9],
+--R    [0.71,1.262817728,1.2628177280 538578572,0.538578572 E -10],
+--R    [0.72,1.270592733,1.2705927333 019296965,0.3019296965 E -9],
+--R    [0.73,1.278494799,1.2784947988 82162536,- 0.117837464 E -9],
+--R    [0.74,1.286524715,1.2865247150 076994755,0.769947551 E -11],
+--R    [0.75,1.294683285,1.2946832846 768446878,- 0.3231553122 E -9],
+--R    [0.76,1.302971324,1.3029713237 533639182,- 0.2466360818 E -9],
+--R    [0.77,1.311389661,1.3113896610 480715408,0.480715408 E -10],
+--R    [0.78,1.319939138,1.3199391384 017123309,0.4017123309 E -9],
+--R    [0.79,1.328620611,1.3286206107 691462407,- 0.2308537593 E -9],
+--R    [0.8,1.337434946,1.3374349463 04844598,0.304844598 E -9],
+--R    [0.81,1.346383026,1.3463830264 497062768,0.4497062768 E -9],
+--R    [0.82,1.355465746,1.3554657460 192025216,0.192025216 E -10],
+--R    [0.83,1.364684013,1.3646840132 928592402,0.2928592402 E -9],
+--R    [0.84,1.37403875,1.3740387501 050857132,0.105085713 E -9],
+--R    [0.85,1.383530892,1.3835308919 373588034,- 0.626411966 E -10],
+--R    [0.86,1.393161388,1.3931613880 117718825,0.117718825 E -10],
+--R    [0.87,1.402931201,1.4029312013 859578321,0.3859578321 E -9],
+--R    [0.88,1.412841309,1.4128413090 493956089,0.493956089 E -10],
+--R    [0.89,1.422892702,1.4228927020 211100075,0.211100075 E -10],
+--R    [0.9,1.433086385,1.4330863854 487743878,0.4487743878 E -9],
+--R    [0.91,1.443423379,1.4434233787 09226281,- 0.290773719 E -9],
+--R    [0.92,1.453904716,1.4539047155 104059219,- 0.4895940781 E -9],
+--R    [0.93,1.464531444,1.4645314439 947279049,- 0.52720951 E -11],
+--R    [0.94,1.475304627,1.4753046268 438962989,- 0.1561037011 E -9],
+--R    [0.95,1.486225341,1.4862253413 85173703,0.385173703 E -9],
+--R    [0.96,1.49729468,1.4972946796 991148706,- 0.3008851294 E -9],
+--R    [0.97,1.508513749,1.5085137487 287756753,- 0.2712243247 E -9],
+--R    [0.98,1.51988367,1.5198836703 904083385,0.3904083385 E -9],
+--R    [0.99,1.531405582,1.5314055816 856539898,- 0.3143460102 E -9],
+--R    [1.0,1.543080635,1.5430806348 152437785,- 0.1847562215 E -9],
+--R    [1.01,1.554909997,1.5549099972 942199069,0.2942199069 E -9],
+--R    [1.02,1.566894852,1.5668948520 686881075,0.686881075 E -10],
+--R    [1.03,1.579036398,1.5790363976 341132396,- 0.3658867604 E -9],
+--R    [1.04,1.591335848,1.5913358481 551698338,0.1551698338 E -9],
+--R    [1.05,1.603794434,1.6037944335 871595723,- 0.4128404277 E -9],
+--R    [1.06,1.6164134,1.6164133997 99007843,- 0.200992157 E -9],
+--R    [1.07,1.629194009,1.6291940086 97851671,- 0.302148329 E -9],
+--R    [1.08,1.642137538,1.6421375383 552314837,0.3552314837 E -9],
+--R    [1.09,1.655245283,1.6552452831 348993296,0.1348993296 E -9],
+--R    [1.1,1.668518554,1.6685185538 222563327,- 0.1777436673 E -9],
+--R    [1.11,1.681958678,1.6819586777 554323243,- 0.2445676757 E -9],
+--R    [1.12,1.695566999,1.6955669989 580207628,- 0.419792372 E -10],
+--R    [1.13,1.709344878,1.7093448782 734822124,0.2734822124 E -9],
+--R    [1.14,1.723293694,1.7232936935 012298237,- 0.4987701763 E -9],
+--R    [1.15,1.73741484,1.7374148395 344104227,- 0.4655895773 E -9],
+--R    [1.16,1.751709728,1.7517097284 993949886,0.4993949886 E -9],
+--R    [1.17,1.76617979,1.7661797898 96992467,- 0.103007533 E -9],
+--R    [1.18,1.780826471,1.7808264707 454010422,- 0.2545989578 E -9],
+--R    [1.19,1.795651236,1.7956512357 249111632,- 0.2750888368 E -9],
+--R    [1.2,1.810655567,1.8106555673 243747931,0.3243747931 E -9],
+--R    [1.21,1.825840966,1.8258409659 894555295,- 0.105444705 E -10],
+--R    [1.22,1.84120895,1.8412089502 726744215,0.2726744215 E -9],
+--R    [1.23,1.856761057,1.8567610569 852664874,- 0.147335126 E -10],
+--R    [1.24,1.872498841,1.8724988413 508631184,0.3508631184 E -9],
+--R    [1.25,1.888423877,1.8884238771 610157382,0.1610157382 E -9],
+--R    [1.26,1.904537757,1.9045377569 325762696,- 0.674237304 E -10],
+--R    [1.27,1.920842092,1.9208420920 669501464,0.669501464 E -10],
+--R    [1.28,1.937338513,1.9373385130 112377973,0.112377973 E -10],
+--R    [1.29,1.954028669,1.9540286694 212807143,0.4212807143 E -9],
+--R    [1.3,1.97091423,1.9709142303 266284118,0.3266284118 E -9],
+--R    [1.31,1.987996884,1.9879968842 974427722,0.2974427722 E -9],
+--R    [1.32,2.00527834,2.0052783396 133564693,- 0.3866435307 E -9],
+--R    [1.33,2.022760324,2.0227603244 343023553,0.4343023553 E -9],
+--R    [1.34,2.040444587,2.0404445869 73330894,- 0.26669106 E -10],
+--R    [1.35,2.058332896,2.0583328956 714329229,- 0.3285670771 E -9],
+--R    [1.36,2.076427039,2.0764270393 743852257,0.3743852257 E -9],
+--R    [1.37,2.094728828,2.0947288275 11636601,- 0.488363399 E -9],
+--R    [1.38,2.11324009,2.1132400902 77252315,0.277252315 E -9],
+--R    [1.39,2.131962679,2.1319626788 129350329,- 0.187064967 E -9],
+--R    [1.4,2.150898465,2.1508984653 931405321,0.3931405321 E -9],
+--R    [1.41,2.170049344,2.1700493436 123067079,- 0.3876932921 E -9],
+--R    [1.42,2.189417229,2.1894172285 742145953,- 0.4257854047 E -9],
+--R    [1.43,2.209004057,2.2090040570 835003431,0.83500343 E -10],
+--R    [1.44,2.228811788,2.2288117878 393372912,- 0.160662709 E -9],
+--R    [1.45,2.248842402,2.2488424016 313075208,- 0.3686924792 E -9],
+--R    [1.46,2.269097902,2.2690979015 374824626,- 0.4625175374 E -9],
+--R    [1.47,2.289580313,2.2895803131 247323736,0.124732374 E -9],
+--R    [1.48,2.310291685,2.3102916846 512847122,- 0.3487152878 E -9],
+--R    [1.49,2.331234087,2.3312340872 715516672,0.2715516672 E -9],
+--R    [1.5,2.352409615,2.3524096152 432473258,0.2432473258 E -9],
+--R    [1.51,2.373820386,2.3738203861 368151896,0.13681519 E -9],
+--R    [1.52,2.395468541,2.3954685410 47186984,0.47186984 E -10],
+--R    [1.53,2.417356245,2.4173562448 078939354,- 0.192106065 E -9],
+--R    [1.54,2.439485686,2.4394856862 075519285,0.207551928 E -9],
+--R    [1.55,2.461859078,2.4618590782 087421916,0.208742192 E -9],
+--R    [1.56,2.484478658,2.4844786581 693093993,0.169309399 E -9],
+--R    [1.57,2.507346688,2.5073466880 66099321,0.66099321 E -10],
+--R    [1.58,2.530465455,2.5304654547 21158391,- 0.278841609 E -9],
+--R    [1.59,2.55383727,2.5538372700 304178181,0.304178181 E -10],
+--R    [1.6,2.577464471,2.5774644711 948851061,0.194885106 E -9],
+--R    [1.61,2.601349421,2.6013494209 543661015,- 0.456338985 E -10],
+--R    [1.62,2.625494508,2.6254945078 237409436,- 0.176259056 E -9],
+--R    [1.63,2.649902146,2.6499021463 318175428,0.3318175428 E -9],
+--R    [1.64,2.674574777,2.6745747772 627864732,0.2627864732 E -9],
+--R    [1.65,2.699514868,2.6995148679 00301426,- 0.99698574 E -10],
+--R    [1.66,2.724724912,2.7247249122 742096308,0.2742096308 E -9],
+--R    [1.67,2.750207431,2.7502074314 099569189,0.4099569189 E -9],
+--R    [1.68,2.775964974,2.7759649735 806923683,- 0.4193076317 E -9],
+--R    [1.69,2.802000115,2.8020001145 620977426,- 0.4379022574 E -9],
+--R    [1.7,2.828315458,2.8283154578 899672055,- 0.110032794 E -9],
+--R    [1.71,2.854913635,2.8549136351 205630698,0.12056307 E -9],
+--R    [1.72,2.881797306,2.8817973060 937736167,0.937736167 E -10],
+--R    [1.73,2.908969159,2.9089691591 990993011,0.199099301 E -9],
+--R    [1.74,2.936431912,2.9364319116 444939418,- 0.3555060582 E -9],
+--R    [1.75,2.96418831,2.9641883097 280877818,- 0.2719122182 E -9],
+--R    [1.76,2.992241129,2.9922411291 128195891,0.112819589 E -9],
+--R    [1.77,3.020593175,3.0205931751 040052646,0.104005265 E -9],
+--R    [1.78,3.049247283,3.0492472829 298707103,- 0.701292897 E -10],
+--R    [1.79,3.078206318,3.0782063180 250770157,0.250770157 E -10],
+--R    [1.8,3.107473176,3.1074731763 17266311,0.317266311 E -9],
+--R    [1.81,3.137050785,3.1370507845 166569452,- 0.4833430548 E -9],
+--R    [1.82,3.1669421,3.1669421004 087169461,0.4087169461 E -9],
+--R    [1.83,3.197150113,3.1971501131 499450329,0.149945033 E -9],
+--R    [1.84,3.227677844,3.2276778435 667887561,- 0.4332112439 E -9],
+--R    [1.85,3.258528344,3.2585283444 577296603,0.4577296603 E -9],
+--R    [1.86,3.289704701,3.2897047008 98565676,- 0.101434324 E -9],
+--R    [1.87,3.321210031,3.3212100305 509212704,- 0.4490787296 E -9],
+--R    [1.88,3.353047484,3.3530474839 74016208,- 0.25983792 E -10],
+--R    [1.89,3.385220245,3.3852202449 397240979,- 0.602759021 E -10],
+--R    [1.9,3.417731531,3.4177315307 509522343,- 0.2490477656 E -9],
+--R    [1.91,3.450584593,3.4505845925 633745687,- 0.4366254312 E -9],
+--R    [1.92,3.483782716,3.4837827157 105499861,- 0.289450014 E -9],
+--R    [1.93,3.51732922,3.5173292200 324583986,0.324583986 E -10],
+--R    [1.94,3.55122746,3.5512274602 074875108,0.207487511 E -9],
+--R    [1.95,3.585480826,3.5854808260 879034531,0.879034531 E -10],
+--R    [1.96,3.620092743,3.6200927430 388388338,0.388388338 E -10],
+--R    [1.97,3.655066672,3.6550666722 808321068,0.2808321068 E -9],
+--R    [1.98,3.690406111,3.6904061112 359525095,0.2359525095 E -9],
+--R    [1.99,3.726114594,3.7261145938 775451847,- 0.122454815 E -9],
+--R    [2.0,3.762195691,3.7621956910 836314596,0.836314596 E -10]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf Digital Library of Mathematical Functions},
+http://dlmf.nist.gov
+\bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+\end{thebibliography}
+\end{document}
diff --git a/src/input/tancot.input.pamphlet b/src/input/tancot.input.pamphlet
new file mode 100644
index 0000000..8ef163c
--- /dev/null
+++ b/src/input/tancot.input.pamphlet
@@ -0,0 +1,696 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input tancot.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's tan and cot functions with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of tan(x), ranging from 0.01 to 1.60. The
+second column is the reference value of tan from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+<<*>>=
+)spool tancot.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.01,0.010000333,tan(0.01),tan(0.01)-(0.010000333)],_
+[0.02,0.020002667,tan(0.02),tan(0.02)-(0.020002667)],_
+[0.03,0.030009003,tan(0.03),tan(0.03)-(0.030009003)],_
+[0.04,0.040021347,tan(0.04),tan(0.04)-(0.040021347)],_
+[0.05,0.050041708,tan(0.05),tan(0.05)-(0.050041708)],_
+[0.06,0.060072104,tan(0.06),tan(0.06)-(0.060072104)],_
+[0.07,0.070114558,tan(0.07),tan(0.07)-(0.070114558)],_
+[0.08,0.080171105,tan(0.08),tan(0.08)-(0.080171105)],_
+[0.09,0.090243790,tan(0.09),tan(0.09)-(0.090243790)],_
+[0.10,0.10033467,tan(0.10),tan(0.10)-(0.10033467)],_
+[0.11,0.11044582,tan(0.11),tan(0.11)-(0.11044582)],_
+[0.12,0.12057934,tan(0.12),tan(0.12)-(0.12057934)],_
+[0.13,0.13073732,tan(0.13),tan(0.13)-(0.13073732)],_
+[0.14,0.14092189,tan(0.14),tan(0.14)-(0.14092189)],_
+[0.15,0.15113522,tan(0.15),tan(0.15)-(0.15113522)],_
+[0.16,0.16137946,tan(0.16),tan(0.16)-(0.16137946)],_
+[0.17,0.17165682,tan(0.17),tan(0.17)-(0.17165682)],_
+[0.18,0.18196953,tan(0.18),tan(0.18)-(0.18196953)],_
+[0.19,0.19231984,tan(0.19),tan(0.19)-(0.19231984)],_
+[0.20,0.20271004,tan(0.20),tan(0.20)-(0.20271004)],_
+[0.21,0.21314244,tan(0.21),tan(0.21)-(0.21314244)],_
+[0.22,0.22361942,tan(0.22),tan(0.22)-(0.22361942)],_
+[0.23,0.23414336,tan(0.23),tan(0.23)-(0.23414336)],_
+[0.24,0.24471670,tan(0.24),tan(0.24)-(0.24471670)],_
+[0.25,0.25534192,tan(0.25),tan(0.25)-(0.25534192)],_
+[0.26,0.26602154,tan(0.26),tan(0.26)-(0.26602154)],_
+[0.27,0.27675814,tan(0.27),tan(0.27)-(0.27675814)],_
+[0.28,0.28755433,tan(0.28),tan(0.28)-(0.28755433)],_
+[0.29,0.29841279,tan(0.29),tan(0.29)-(0.29841279)],_
+[0.30,0.30933625,tan(0.30),tan(0.30)-(0.30933625)],_
+[0.31,0.32032751,tan(0.31),tan(0.31)-(0.32032751)],_
+[0.32,0.33138941,tan(0.32),tan(0.32)-(0.33138941)],_
+[0.33,0.34252487,tan(0.33),tan(0.33)-(0.34252487)],_
+[0.34,0.35373688,tan(0.34),tan(0.34)-(0.35373688)],_
+[0.35,0.36502849,tan(0.35),tan(0.35)-(0.36502849)],_
+[0.36,0.37640285,tan(0.36),tan(0.36)-(0.37640285)],_
+[0.37,0.38786316,tan(0.37),tan(0.37)-(0.38786316)],_
+[0.38,0.39941272,tan(0.38),tan(0.38)-(0.39941272)],_
+[0.39,0.41105492,tan(0.39),tan(0.39)-(0.41105492)],_
+[0.40,0.42279322,tan(0.40),tan(0.40)-(0.42279322)],_
+[0.41,0.43463120,tan(0.41),tan(0.41)-(0.43463120)],_
+[0.42,0.44657255,tan(0.42),tan(0.42)-(0.44657255)],_
+[0.43,0.45862102,tan(0.43),tan(0.43)-(0.45862102)],_
+[0.44,0.47078053,tan(0.44),tan(0.44)-(0.47078053)],_
+[0.45,0.48305507,tan(0.45),tan(0.45)-(0.48305507)],_
+[0.46,0.49544877,tan(0.46),tan(0.46)-(0.49544877)],_
+[0.47,0.50796590,tan(0.47),tan(0.47)-(0.50796590)],_
+[0.48,0.52061084,tan(0.48),tan(0.48)-(0.52061084)],_
+[0.49,0.53338815,tan(0.49),tan(0.49)-(0.53338815)],_
+[0.50,0.54630249,tan(0.50),tan(0.50)-(0.54630249)],_
+[0.51,0.55935872,tan(0.51),tan(0.51)-(0.55935872)],_
+[0.52,0.57256183,tan(0.52),tan(0.52)-(0.57256183)],_
+[0.53,0.58591701,tan(0.53),tan(0.53)-(0.58591701)],_
+[0.54,0.59942962,tan(0.54),tan(0.54)-(0.59942962)],_
+[0.55,0.61310521,tan(0.55),tan(0.55)-(0.61310521)],_
+[0.56,0.62694954,tan(0.56),tan(0.56)-(0.62694954)],_
+[0.57,0.64096855,tan(0.57),tan(0.57)-(0.64096855)],_
+[0.58,0.65516845,tan(0.58),tan(0.58)-(0.65516845)],_
+[0.59,0.66955565,tan(0.59),tan(0.59)-(0.66955565)],_
+[0.60,0.68413681,tan(0.60),tan(0.60)-(0.68413681)],_
+[0.61,0.69891886,tan(0.61),tan(0.61)-(0.69891886)],_
+[0.62,0.71390901,tan(0.62),tan(0.62)-(0.71390901)],_
+[0.63,0.72911473,tan(0.63),tan(0.63)-(0.72911473)],_
+[0.64,0.74454382,tan(0.64),tan(0.64)-(0.74454382)],_
+[0.65,0.76020440,tan(0.65),tan(0.65)-(0.76020440)],_
+[0.66,0.77610491,tan(0.66),tan(0.66)-(0.77610491)],_
+[0.67,0.79225417,tan(0.67),tan(0.67)-(0.79225417)],_
+[0.68,0.80866138,tan(0.68),tan(0.68)-(0.80866138)],_
+[0.69,0.82533611,tan(0.69),tan(0.69)-(0.82533611)],_
+[0.70,0.84228838,tan(0.70),tan(0.70)-(0.84228838)],_
+[0.71,0.85952867,tan(0.71),tan(0.71)-(0.85952867)],_
+[0.72,0.87706790,tan(0.72),tan(0.72)-(0.87706790)],_
+[0.73,0.89491753,tan(0.73),tan(0.73)-(0.89491753)],_
+[0.74,0.91308953,tan(0.74),tan(0.74)-(0.91308953)],_
+[0.75,0.93159646,tan(0.75),tan(0.75)-(0.93159646)],_
+[0.76,0.95045146,tan(0.76),tan(0.76)-(0.95045146)],_
+[0.77,0.96966833,tan(0.77),tan(0.77)-(0.96966833)],_
+[0.78,0.98926154,tan(0.78),tan(0.78)-(0.98926154)],_
+[0.79,1.00924629,tan(0.79),tan(0.79)-(1.00924629)],_
+[0.80,1.02963857,tan(0.80),tan(0.80)-(1.02963857)],_
+[0.81,1.05045514,tan(0.81),tan(0.81)-(1.05045514)],_
+[0.82,1.07171372,tan(0.82),tan(0.82)-(1.07171372)],_
+[0.83,1.09343292,tan(0.83),tan(0.83)-(1.09343292)],_
+[0.84,1.11563235,tan(0.84),tan(0.84)-(1.11563235)],_
+[0.85,1.13833271,tan(0.85),tan(0.85)-(1.13833271)],_
+[0.86,1.16155586,tan(0.86),tan(0.86)-(1.16155586)],_
+[0.87,1.18532486,tan(0.87),tan(0.87)-(1.18532486)],_
+[0.88,1.20966412,tan(0.88),tan(0.88)-(1.20966412)],_
+[0.89,1.23459946,tan(0.89),tan(0.89)-(1.23459946)],_
+[0.90,1.26015822,tan(0.90),tan(0.90)-(1.26015822)],_
+[0.91,1.28636938,tan(0.91),tan(0.91)-(1.28636938)],_
+[0.92,1.31326370,tan(0.92),tan(0.92)-(1.31326370)],_
+[0.93,1.34087383,tan(0.93),tan(0.93)-(1.34087383)],_
+[0.94,1.36923448,tan(0.94),tan(0.94)-(1.36923448)],_
+[0.95,1.39838259,tan(0.95),tan(0.95)-(1.39838259)],_
+[0.96,1.42835749,tan(0.96),tan(0.96)-(1.42835749)],_
+[0.97,1.45920113,tan(0.97),tan(0.97)-(1.45920113)],_
+[0.98,1.49095827,tan(0.98),tan(0.98)-(1.49095827)],_
+[0.99,1.52367674,tan(0.99),tan(0.99)-(1.52367674)],_
+[1.00,1.55740772,tan(1.00),tan(1.00)-(1.55740772)],_
+[1.01,1.5922060,tan(1.01),tan(1.01)-(1.5922060)],_
+[1.02,1.6281304,tan(1.02),tan(1.02)-(1.6281304)],_
+[1.03,1.6652440,tan(1.03),tan(1.03)-(1.6652440)],_
+[1.04,1.7036146,tan(1.04),tan(1.04)-(1.7036146)],_
+[1.05,1.7433153,tan(1.05),tan(1.05)-(1.7433153)],_
+[1.06,1.7844248,tan(1.06),tan(1.06)-(1.7844248)],_
+[1.07,1.8270282,tan(1.07),tan(1.07)-(1.8270282)],_
+[1.08,1.8712173,tan(1.08),tan(1.08)-(1.8712173)],_
+[1.09,1.9170918,tan(1.09),tan(1.09)-(1.9170918)],_
+[1.10,1.9647597,tan(1.10),tan(1.10)-(1.9647597)],_
+[1.11,2.0143382,tan(1.11),tan(1.11)-(2.0143382)],_
+[1.12,2.0659553,tan(1.12),tan(1.12)-(2.0659553)],_
+[1.13,2.1197501,tan(1.13),tan(1.13)-(2.1197501)],_
+[1.14,2.1758751,tan(1.14),tan(1.14)-(2.1758751)],_
+[1.15,2.2344969,tan(1.15),tan(1.15)-(2.2344969)],_
+[1.16,2.2957985,tan(1.16),tan(1.16)-(2.2957985)],_
+[1.17,2.3599811,tan(1.17),tan(1.17)-(2.3599811)],_
+[1.18,2.4272664,tan(1.18),tan(1.18)-(2.4272664)],_
+[1.19,2.4978994,tan(1.19),tan(1.19)-(2.4978994)],_
+[1.20,2.5721516,tan(1.20),tan(1.20)-(2.5721516)],_
+[1.21,2.6503246,tan(1.21),tan(1.21)-(2.6503246)],_
+[1.22,2.7327542,tan(1.22),tan(1.22)-(2.7327542)],_
+[1.23,2.8198157,tan(1.23),tan(1.23)-(2.8198157)],_
+[1.24,2.9119299,tan(1.24),tan(1.24)-(2.9119299)],_
+[1.25,3.0095697,tan(1.25),tan(1.25)-(3.0095697)],_
+[1.26,3.1132691,tan(1.26),tan(1.26)-(3.1132691)],_
+[1.27,3.2236332,tan(1.27),tan(1.27)-(3.2236332)],_
+[1.28,3.3413500,tan(1.28),tan(1.28)-(3.3413500)],_
+[1.29,3.4672057,tan(1.29),tan(1.29)-(3.4672057)],_
+[1.30,3.6021024,tan(1.30),tan(1.30)-(3.6021024)],_
+[1.31,3.7470810,tan(1.31),tan(1.31)-(3.7470810)],_
+[1.32,3.9033478,tan(1.32),tan(1.32)-(3.9033478)],_
+[1.33,4.0723098,tan(1.33),tan(1.33)-(4.0723098)],_
+[1.34,4.2556179,tan(1.34),tan(1.34)-(4.2556179)],_
+[1.35,4.4552218,tan(1.35),tan(1.35)-(4.4552218)],_
+[1.36,4.6734412,tan(1.36),tan(1.36)-(4.6734412)],_
+[1.37,4.9130581,tan(1.37),tan(1.37)-(4.9130581)],_
+[1.38,5.1774374,tan(1.38),tan(1.38)-(5.1774374)],_
+[1.39,5.4706886,tan(1.39),tan(1.39)-(5.4706886)],_
+[1.40,5.7978837,tan(1.40),tan(1.40)-(5.7978837)],_
+[1.41,6.1653561,tan(1.41),tan(1.41)-(6.1653561)],_
+[1.42,6.5811195,tan(1.42),tan(1.42)-(6.5811195)],_
+[1.43,7.0554638,tan(1.43),tan(1.43)-(7.0554638)],_
+[1.44,7.6018261,tan(1.44),tan(1.44)-(7.6018261)],_
+[1.45,8.2380928,tan(1.45),tan(1.45)-(8.2380928)],_
+[1.46,8.9886076,tan(1.46),tan(1.46)-(8.9886076)],_
+[1.47,9.8873749,tan(1.47),tan(1.47)-(9.8873749)],_
+[1.48,10.9833793,tan(1.48),tan(1.48)-(10.9833793)],_
+[1.49,12.3498564,tan(1.49),tan(1.49)-(12.3498564)],_
+[1.50,14.1014199,tan(1.50),tan(1.50)-(14.1014199)],_
+[1.51,16.4280917,tan(1.51),tan(1.51)-(16.4280917)],_
+[1.52,19.6695278,tan(1.52),tan(1.52)-(19.6695278)],_
+[1.53,24.4984104,tan(1.53),tan(1.53)-(24.4984104)],_
+[1.54,32.4611389,tan(1.54),tan(1.54)-(32.4611389)],_
+[1.55,48.0784825,tan(1.55),tan(1.55)-(48.0784825)],_
+[1.56,92.6204963,tan(1.56),tan(1.56)-(92.6204963)],_
+[1.57,1255.7655915,tan(1.57),tan(1.57)-(1255.7655915)],_
+[1.58,-108.6492036,tan(1.58),tan(1.58)-(-108.6492036)],_
+[1.59,-52.0669696,tan(1.59),tan(1.59)-(-52.0669696)],_
+[1.60,-34.2325327,tan(1.60),tan(1.60)-(-34.2325327)]]
+--R 
+--R
+--R   (1)
+--R   [[0.01,0.010000333,0.0100003333 4666720637 1,0.3466672063 71 E -9],
+--R    [0.02,0.020002667,0.0200026670 9340242389 7,0.9340242389 7 E -10],
+--R    [0.03,0.030009003,0.0300090032 4118071632 9,0.2411807163 29 E -9],
+--R    [0.04,0.040021347,0.0400213469 9551456207 2,- 0.4485437928 E -11],
+--R    [0.05,0.050041708,0.0500417083 7553878891 2,0.3755387889 12 E -9],
+--R    [0.06,0.060072104,0.0600721038 3129728751 1,- 0.1687027124 9 E -9],
+--R    [0.07,0.070114558,0.0701145578 7200271322 9,- 0.1279972867 7 E -9],
+--R    [0.08,0.080171105,0.0801711047 0807255711 8,- 0.2919274428 8 E -9],
+--R    [0.09,0.09024379,0.0902437899 0978545046 6,- 0.9021454953 4 E -10],
+--R    [0.1,0.10033467,0.1003346720 8545054506,0.2085450545 06 E -8],
+--R    [0.11,0.11044582,0.1104458245 820405045,0.4582040504 5 E -8],
+--R    [0.12,0.12057934,0.1205793372 1130531183,- 0.2788694688 17 E -8],
+--R    [0.13,0.13073732,0.1307373180 0446004867,- 0.1995539951 33 E -8],
+--R    [0.14,0.14092189,0.1409218949 9862537921,0.4998625379 21 E -8],
+--R    [0.15,0.15113522,0.1511352180 5829507125,- 0.1941704928 75 E -8],
+--R    [0.16,0.16137946,0.1613794607 3521095024,0.7352109502 4 E -9],
+--R    [0.17,0.17165682,0.1716568221 7014270414,0.2170142704 14 E -8],
+--R    [0.18,0.18196953,0.1819695290 4019848684,- 0.9598015131 64 E -9],
+--R    [0.19,0.19231984,0.1923198375 554329145,- 0.2444567085 5 E -8],
+--R    [0.2,0.20271004,0.2027100355 0867248332,- 0.4491327516 68 E -8],
+--R    [0.21,0.21314244,0.2131424443 8264539723,0.4382645397 23 E -8],
+--R    [0.22,0.22361942,0.2236194215 1868409245,0.1518684092 45 E -8],
+--R    [0.23,0.23414336,0.2341433623 5146527061,0.2351465270 61 E -8],
+--R    [0.24,0.2447167,0.2447167027 1446497862,0.2714464978 62 E -8],
+--R    [0.25,0.25534192,0.2553419212 2103626651,0.1221036266 5 E -8],
+--R    [0.26,0.26602154,0.2660215417 2626537908,0.1726265379 1 E -8],
+--R    [0.27,0.27675814,0.2767581358 7503056579,- 0.4124969434 21 E -8],
+--R    [0.28,0.28755433,0.2875543257 41976815,- 0.4258023185 E -8],
+--R    [0.29,0.29841279,0.2984127865 694316513,- 0.3430568348 7 E -8],
+--R    [0.3,0.30933625,0.3093362496 0962323304,- 0.3903767669 6 E -9],
+--R    [0.31,0.32032751,0.3203275050 7792416023,- 0.4922075839 77 E -8],
+--R    [0.32,0.33138941,0.3313894052 2423462352,- 0.4775765376 48 E -8],
+--R    [0.33,0.34252487,0.3425248675 3003894803,- 0.2469961051 97 E -8],
+--R    [0.34,0.35373688,0.3537368780 3912256577,- 0.1960877434 23 E -8],
+--R    [0.35,0.36502849,0.3650284948 3042455832,0.4830424558 32 E -8],
+--R    [0.36,0.37640285,0.3764028516 4202695764,0.1642026957 6 E -8],
+--R    [0.37,0.38786316,0.3878631616 5584905222,0.1655849052 2 E -8],
+--R    [0.38,0.39941272,0.3994127214 5322637827,0.1453226378 3 E -8],
+--R    [0.39,0.41105492,0.4110549151 5221356343,- 0.4847786436 57 E -8],
+--R    [0.4,0.42279322,0.4227932187 3816176198,- 0.1261838238 E -8],
+--R    [0.41,0.4346312,0.4346312045 9988949299,0.4599889492 99 E -8],
+--R    [0.42,0.44657255,0.4465725462 8459510803,- 0.3715404891 96 E -8],
+--R    [0.43,0.45862102,0.4586210234 8555518632,0.3485555186 32 E -8],
+--R    [0.44,0.47078053,0.4707805272 7762171492,- 0.2722378285 08 E -8],
+--R    [0.45,0.48305507,0.4830550656 1657837051,- 0.4383421629 49 E -8],
+--R    [0.46,0.49544877,0.4954487691 1954962242,- 0.8804503775 8 E -9],
+--R    [0.47,0.5079659,0.5079658971 4488348004,- 0.285511652 E -8],
+--R    [0.48,0.52061084,0.5206108441 9125804964,0.4191258049 65 E -8],
+--R    [0.49,0.53338815,0.5333881466 3720305695,- 0.3362796943 1 E -8],
+--R    [0.5,0.54630249,0.5463024898 4379051326,- 0.1562094867 E -9],
+--R    [0.51,0.55935872,0.5593587156 4494521344,- 0.4355054786 56 E -8],
+--R    [0.52,0.57256183,0.5725618302 5166841478,0.2516684148 E -9],
+--R    [0.53,0.58591701,0.5859170125 9847085812,0.2598470858 1 E -8],
+--R    [0.54,0.59942962,0.5994296231 6248975451,0.3162489754 5 E -8],
+--R    [0.55,0.61310521,0.6131052132 8813564222,0.3288135642 2 E -8],
+--R    [0.56,0.62694954,0.6269495350 5269815933,- 0.4947301840 67 E -8],
+--R    [0.57,0.64096855,0.6409685517 1115591313,0.1711155913 1 E -8],
+--R    [0.58,0.65516845,0.6551684487 6150824025,- 0.1238491759 7 E -8],
+--R    [0.59,0.66955565,0.6695556456 753018611,- 0.4324698138 9 E -8],
+--R    [0.6,0.68413681,0.6841368083 4169231707,- 0.1658307682 9 E -8],
+--R    [0.61,0.69891886,0.6989188622 7739105048,0.2277391050 5 E -8],
+--R    [0.62,0.71390901,0.7139090066 5924020594,- 0.3340759794 1 E -8],
+--R    [0.63,0.72911473,0.7291147292 4096908976,- 0.7590309102 4 E -9],
+--R    [0.64,0.74454382,0.7445438222 2096388599,0.2220963886 E -8],
+--R    [0.65,0.7602044,0.7602043991 3367625635,- 0.8663237436 5 E -9],
+--R    [0.66,0.77610491,0.7761049128 4366351779,0.2843663517 8 E -8],
+--R    [0.67,0.79225417,0.7922541747 2825678628,0.4728256786 28 E -8],
+--R    [0.68,0.80866138,0.8086613751 4256524544,- 0.4857434754 56 E -8],
+--R    [0.69,0.82533611,0.8253361052 6902491172,- 0.4730975088 28 E -8],
+--R    [0.7,0.84228838,0.8422883804 6307944813,0.4630794481 E -9],
+--R    [0.71,0.85952867,0.8595286652 1694081593,- 0.4783059184 07 E -8],
+--R    [0.72,0.8770679,0.8770678998 7483414069,- 0.1251658593 E -9],
+--R    [0.73,0.89491753,0.8949175292 4581448181,- 0.7541855181 9 E -9],
+--R    [0.74,0.91308953,0.9130895332 7430087206,0.3274300872 1 E -8],
+--R    [0.75,0.93159646,0.9315964599 4407246117,- 0.559275388 E -10],
+--R    [0.76,0.95045146,0.9504514606 0880299797,0.6088029979 7 E -9],
+--R    [0.77,0.96966833,0.9696683279 6148947799,- 0.2038510522 E -8],
+--R    [0.78,0.98926154,0.9892615368 7660491155,- 0.3123395088 4 E -8],
+--R    [0.79,1.00924629,1.0092462883 827548811,- 0.1617245118 9 E -8],
+--R    [0.8,1.02963857,1.0296385570 503640128,- 0.1294963598 73 E -7],
+--R    [0.81,1.05045514,1.0504551421 088292806,0.2108829280 6 E -8],
+--R    [0.82,1.07171372,1.0717137226 410736441,0.2641073644 1 E -8],
+--R    [0.83,1.09343292,1.0934329172 409999188,- 0.2759000081 2 E -8],
+--R    [0.84,1.11563235,1.1156323485 615378951,- 0.1438462104 9 E -8],
+--R    [0.85,1.13833271,1.1383327132 284394134,0.3228439413 4 E -8],
+--R    [0.86,1.16155586,1.1615558576 484476046,- 0.2351552395 4 E -8],
+--R    [0.87,1.18532486,1.1853248603 008053505,0.3008053505 E -9],
+--R    [0.88,1.20966412,1.2096641211 692683367,0.1169268336 7 E -8],
+--R    [0.89,1.23459946,1.2345994590 490045825,- 0.9509954175 3 E -9],
+--R    [0.9,1.26015822,1.2601582175 503391371,- 0.2449660862 9 E -8],
+--R    [0.91,1.28636938,1.2863693807 208075758,0.7208075758 E -9],
+--R    [0.92,1.3132637,1.3132636993 202478365,- 0.6797521635 E -9],
+--R    [0.93,1.34087383,1.3408738289 128343042,- 0.1087165695 8 E -8],
+--R    [0.94,1.36923448,1.3692344810 875628038,0.1087562803 8 E -8],
+--R    [0.95,1.39838259,1.3983825892 876991461,- 0.7123008539 E -9],
+--R    [0.96,1.42835749,1.4283574909 236105601,0.9236105601 E -9],
+--R    [0.97,1.45920113,1.4592011276 663536858,- 0.2333646314 2 E -8],
+--R    [0.98,1.49095827,1.4909582660 763114779,- 0.3923688522 1 E -8],
+--R    [0.99,1.52367674,1.5236767410 179022725,0.1017902272 5 E -8],
+--R    [1.0,1.55740772,1.5574077246 549022305,0.4654902230 5 E -8],
+--R    [1.01,1.592206,1.5922060242 195703744,0.2421957037 44 E -7],
+--R    [1.02,1.6281304,1.6281304122 125526001,0.1221255260 01 E -7],
+--R    [1.03,1.665244,1.6652439932 315124346,- 0.6768487565 4 E -8],
+--R    [1.04,1.7036146,1.7036146122 591331094,0.1225913310 94 E -7],
+--R    [1.05,1.7433153,1.7433153099 831702625,0.9983170262 47 E -8],
+--R    [1.06,1.7844248,1.7844248315 940126524,0.3159401265 24 E -7],
+--R    [1.07,1.8270282,1.8270281965 348367381,- 0.3465163261 9 E -8],
+--R    [1.08,1.8712173,1.8712173378 97878195,0.3789787819 5 E -7],
+--R    [1.09,1.9170918,1.9170918216 068594024,0.2160685940 24 E -7],
+--R    [1.1,1.9647597,1.9647596572 486519509,- 0.4275134804 91 E -7],
+--R    [1.11,2.0143382,2.0143382144 768273135,0.1447682731 3 E -7],
+--R    [1.12,2.0659553,2.0659552613 80510241,- 0.3861948975 9 E -7],
+--R    [1.13,2.1197501,2.1197501441 871810139,0.4418718101 39 E -7],
+--R    [1.14,2.1758751,2.1758751312 648761686,0.3126487616 86 E -7],
+--R    [1.15,2.2344969,2.2344969487 553259802,0.4875532598 02 E -7],
+--R    [1.16,2.2957985,2.2957985404 922076011,0.4049220760 11 E -7],
+--R    [1.17,2.3599811,2.3599810913 765482032,- 0.8623451796 7 E -8],
+--R    [1.18,2.4272664,2.4272663614 002235775,- 0.3859977642 25 E -7],
+--R    [1.19,2.4978994,2.4978993874 226530062,- 0.1257734699 4 E -7],
+--R    [1.2,2.5721516,2.5721516221 263189354,0.2212631893 54 E -7],
+--R    [1.21,2.6503246,2.6503245949 706014665,- 0.5029398533 5 E -8],
+--R    [1.22,2.7327542,2.7327541993 067149987,- 0.6932850013 E -9],
+--R    [1.23,2.8198157,2.8198157342 681519748,0.3426815197 48 E -7],
+--R    [1.24,2.9119299,2.9119298611 552260267,- 0.3884477397 34 E -7],
+--R    [1.25,3.0095697,3.0095696738 628312882,- 0.2613716871 18 E -7],
+--R    [1.26,3.1132691,3.1132691342 651312093,0.3426513120 93 E -7],
+--R    [1.27,3.2236332,3.2236331902 040711359,- 0.9795928864 E -8],
+--R    [1.28,3.34135,3.3413499811 153736983,- 0.1888462630 17 E -7],
+--R    [1.29,3.4672057,3.4672056517 213857853,- 0.4827861421 47 E -7],
+--R    [1.3,3.6021024,3.6021024479 679781512,0.4796797815 12 E -7],
+--R    [1.31,3.747081,3.7470809761 884290733,- 0.2381157092 67 E -7],
+--R    [1.32,3.9033478,3.9033477874 966235656,- 0.1250337643 4 E -7],
+--R    [1.33,4.0723098,4.0723098354 650698554,0.3546506985 54 E -7],
+--R    [1.34,4.2556179,4.2556178917 394653427,- 0.8260534657 3 E -8],
+--R    [1.35,4.4552218,4.4552217595 627031753,- 0.4043729682 47 E -7],
+--R    [1.36,4.6734412,4.6734412029 885596449,0.2988559645 E -8],
+--R    [1.37,4.9130581,4.9130580704 624720101,- 0.2953752799 E -7],
+--R    [1.38,5.1774374,5.1774373886 304102949,- 0.1136958970 5 E -7],
+--R    [1.39,5.4706886,5.4706886429 532054937,0.4295320549 37 E -7],
+--R    [1.4,5.7978837,5.7978837154 828896437,0.1548288964 4 E -7],
+--R    [1.41,6.1653561,6.1653561445 520255476,0.4455202554 75 E -7],
+--R    [1.42,6.5811195,6.5811194561 942543239,- 0.4380574567 61 E -7],
+--R    [1.43,7.0554638,7.0554637664 342109722,- 0.3356578902 78 E -7],
+--R    [1.44,7.6018261,7.6018260620 257232407,- 0.3797427675 94 E -7],
+--R    [1.45,8.2380928,8.2380927529 656070833,- 0.4703439291 7 E -7],
+--R    [1.46,8.9886076,8.9886076017 241695008,0.1724169501 E -8],
+--R    [1.47,9.8873749,9.8873748919 855531724,- 0.8014446827 5 E -8],
+--R    [1.48,10.9833793,10.9833793143 26067301,0.1432606730 1 E -7],
+--R    [1.49,12.3498564,12.3498564416 25802114,0.4162580211 4 E -7],
+--R    [1.5,14.1014199,14.1014199471 71719388,0.4717171938 8 E -7],
+--R    [1.51,16.4280917,16.4280917038 85335336,0.3885335336 E -8],
+--R    [1.52,19.6695278,19.6695278205 58866232,0.2055886623 2 E -7],
+--R    [1.53,24.4984104,24.4984104418 38034593,0.4183803459 3 E -7],
+--R    [1.54,32.4611389,32.4611389128 56765176,0.1285676518 E -7],
+--R    [1.55,48.0784825,48.0784824792 18968279,- 0.2078103172 E -7],
+--R    [1.56,92.6204963,92.6204963167 04102469,0.1670410247 E -7],
+--R    [1.57,1255.7655915,1255.7655915006 916051,0.6916051 E -9],
+--R    [1.58,- 108.6492036,- 108.6492036048 4393447,- 0.484393447 E -8],
+--R    [1.59,- 52.0669696,- 52.0669696509 12563554,- 0.5091256355 4 E -7],
+--R    [1.6,- 34.2325327,- 34.2325327355 57417056,- 0.3555741705 7 E -7]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of tan(x), ranging from 0.01 to 1.60. The
+second column is the reference value of tan from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+<<*>>=
+
+--S 2 of 2
+[[0.01,99.9966666,cot(0.01),cot(0.01)-(99.9966666)],_
+[0.02,49.9933332,cot(0.02),cot(0.02)-(49.9933332)],_
+[0.03,33.3233327,cot(0.03),cot(0.03)-(33.3233327)],_
+[0.04,24.9866652,cot(0.04),cot(0.04)-(24.9866652)],_
+[0.05,19.9833306,cot(0.05),cot(0.05)-(19.9833306)],_
+[0.06,16.6466619,cot(0.06),cot(0.06)-(16.6466619)],_
+[0.07,14.2623733,cot(0.07),cot(0.07)-(14.2623733)],_
+[0.08,12.4733219,cot(0.08),cot(0.08)-(12.4733219)],_
+[0.09,11.0810949,cot(0.09),cot(0.09)-(11.0810949)],_
+[0.10,9.9666444,cot(0.10),cot(0.10)-(9.9666444)],_
+[0.11,9.0542128,cot(0.11),cot(0.11)-(9.0542128)],_
+[0.12,8.2932949,cot(0.12),cot(0.12)-(8.2932949)],_
+[0.13,7.6489255,cot(0.13),cot(0.13)-(7.6489255)],_
+[0.14,7.0961294,cot(0.14),cot(0.14)-(7.0961294)],_
+[0.15,6.6165915,cot(0.15),cot(0.15)-(6.6165915)],_
+[0.16,6.1965754,cot(0.16),cot(0.16)-(6.1965754)],_
+[0.17,5.8255768,cot(0.17),cot(0.17)-(5.8255768)],_
+[0.18,5.4954256,cot(0.18),cot(0.18)-(5.4954256)],_
+[0.19,5.1996716,cot(0.19),cot(0.19)-(5.1996716)],_
+[0.20,4.9331549,cot(0.20),cot(0.20)-(4.9331549)],_
+[0.21,4.6916981,cot(0.21),cot(0.21)-(4.6916981)],_
+[0.22,4.4718835,cot(0.22),cot(0.22)-(4.4718835)],_
+[0.23,4.2708877,cot(0.23),cot(0.23)-(4.2708877)],_
+[0.24,4.0863578,cot(0.24),cot(0.24)-(4.0863578)],_
+[0.25,3.9163174,cot(0.25),cot(0.25)-(3.9163174)],_
+[0.26,3.7590941,cot(0.26),cot(0.26)-(3.7590941)],_
+[0.27,3.6132632,cot(0.27),cot(0.27)-(3.6132632)],_
+[0.28,3.4776037,cot(0.28),cot(0.28)-(3.4776037)],_
+[0.29,3.3510628,cot(0.29),cot(0.29)-(3.3510628)],_
+[0.30,3.2327281,cot(0.30),cot(0.30)-(3.2327281)],_
+[0.31,3.1218050,cot(0.31),cot(0.31)-(3.1218050)],_
+[0.32,3.0175980,cot(0.32),cot(0.32)-(3.0175980)],_
+[0.33,2.9194961,cot(0.33),cot(0.33)-(2.9194961)],_
+[0.34,2.8269600,cot(0.34),cot(0.34)-(2.8269600)],_
+[0.35,2.7395122,cot(0.35),cot(0.35)-(2.7395122)],_
+[0.36,2.6567280,cot(0.36),cot(0.36)-(2.6567280)],_
+[0.37,2.5782289,cot(0.37),cot(0.37)-(2.5782289)],_
+[0.38,2.5036759,cot(0.38),cot(0.38)-(2.5036759)],_
+[0.39,2.4327650,cot(0.39),cot(0.39)-(2.4327650)],_
+[0.40,2.3652224,cot(0.40),cot(0.40)-(2.3652224)],_
+[0.41,2.3008012,cot(0.41),cot(0.41)-(2.3008012)],_
+[0.42,2.2392778,cot(0.42),cot(0.42)-(2.2392778)],_
+[0.43,2.1804495,cot(0.43),cot(0.43)-(2.1804495)],_
+[0.44,2.1241320,cot(0.44),cot(0.44)-(2.1241320)],_
+[0.45,2.0701574,cot(0.45),cot(0.45)-(2.0701574)],_
+[0.46,2.0183722,cot(0.46),cot(0.46)-(2.0183722)],_
+[0.47,1.9686361,cot(0.47),cot(0.47)-(1.9686361)],_
+[0.48,1.9208205,cot(0.48),cot(0.48)-(1.9208205)],_
+[0.49,1.8748073,cot(0.49),cot(0.49)-(1.8748073)],_
+[0.50,1.83048772,cot(0.50),cot(0.50)-(1.83048772)],_
+[0.51,1.78776154,cot(0.51),cot(0.51)-(1.78776154)],_
+[0.52,1.74653626,cot(0.52),cot(0.52)-(1.74653626)],_
+[0.53,1.70672634,cot(0.53),cot(0.53)-(1.70672634)],_
+[0.54,1.66825255,cot(0.54),cot(0.54)-(1.66825255)],_
+[0.55,1.63104142,cot(0.55),cot(0.55)-(1.63104142)],_
+[0.56,1.59502471,cot(0.56),cot(0.56)-(1.59502471)],_
+[0.57,1.56013894,cot(0.57),cot(0.57)-(1.56013894)],_
+[0.58,1.52632503,cot(0.58),cot(0.58)-(1.52632503)],_
+[0.59,1.49352784,cot(0.59),cot(0.59)-(1.49352784)],_
+[0.60,1.46169595,cot(0.60),cot(0.60)-(1.46169595)],_
+[0.61,1.43078125,cot(0.61),cot(0.61)-(1.43078125)],_
+[0.62,1.40073873,cot(0.62),cot(0.62)-(1.40073873)],_
+[0.63,1.37152626,cot(0.63),cot(0.63)-(1.37152626)],_
+[0.64,1.34310429,cot(0.64),cot(0.64)-(1.34310429)],_
+[0.65,1.31543569,cot(0.65),cot(0.65)-(1.31543569)],_
+[0.66,1.28848559,cot(0.66),cot(0.66)-(1.28848559)],_
+[0.67,1.26222118,cot(0.67),cot(0.67)-(1.26222118)],_
+[0.68,1.23661155,cot(0.68),cot(0.68)-(1.23661155)],_
+[0.69,1.21162759,cot(0.69),cot(0.69)-(1.21162759)],_
+[0.70,1.18724183,cot(0.70),cot(0.70)-(1.18724183)],_
+[0.71,1.16342833,cot(0.71),cot(0.71)-(1.16342833)],_
+[0.72,1.14016258,cot(0.72),cot(0.72)-(1.14016258)],_
+[0.73,1.11742140,cot(0.73),cot(0.73)-(1.11742140)],_
+[0.74,1.09518285,cot(0.74),cot(0.74)-(1.09518285)],_
+[0.75,1.07342615,cot(0.75),cot(0.75)-(1.07342615)],_
+[0.76,1.05213158,cot(0.76),cot(0.76)-(1.05213158)],_
+[0.77,1.03128046,cot(0.77),cot(0.77)-(1.03128046)],_
+[0.78,1.01085503,cot(0.78),cot(0.78)-(1.01085503)],_
+[0.79,0.99083842,cot(0.79),cot(0.79)-(0.99083842)],_
+[0.80,0.97121460,cot(0.80),cot(0.80)-(0.97121460)],_
+[0.81,0.95196830,cot(0.81),cot(0.81)-(0.95196830)],_
+[0.82,0.93308500,cot(0.82),cot(0.82)-(0.93308500)],_
+[0.83,0.91455085,cot(0.83),cot(0.83)-(0.91455085)],_
+[0.84,0.89635264,cot(0.84),cot(0.84)-(0.89635264)],_
+[0.85,0.87847778,cot(0.85),cot(0.85)-(0.87847778)],_
+[0.86,0.86091426,cot(0.86),cot(0.86)-(0.86091426)],_
+[0.87,0.84365058,cot(0.87),cot(0.87)-(0.84365058)],_
+[0.88,0.82667575,cot(0.88),cot(0.88)-(0.82667575)],_
+[0.89,0.80997930,cot(0.89),cot(0.89)-(0.80997930)],_
+[0.90,0.79355115,cot(0.90),cot(0.90)-(0.79355115)],_
+[0.91,0.77738169,cot(0.91),cot(0.91)-(0.77738169)],_
+[0.92,0.76146169,cot(0.92),cot(0.92)-(0.76146169)],_
+[0.93,0.74578232,cot(0.93),cot(0.93)-(0.74578232)],_
+[0.94,0.73033510,cot(0.94),cot(0.94)-(0.73033510)],_
+[0.95,0.71511188,cot(0.95),cot(0.95)-(0.71511188)],_
+[0.96,0.70010485,cot(0.96),cot(0.96)-(0.70010485)],_
+[0.97,0.68530649,cot(0.97),cot(0.97)-(0.68530649)],_
+[0.98,0.67070959,cot(0.98),cot(0.98)-(0.67070959)],_
+[0.99,0.65630719,cot(0.99),cot(0.99)-(0.65630719)],_
+[1.00,0.64209262,cot(1.00),cot(1.00)-(0.64209262)],_
+[1.01,0.62805942,cot(1.01),cot(1.01)-(0.62805942)],_
+[1.02,0.61420141,cot(1.02),cot(1.02)-(0.61420141)],_
+[1.03,0.60051260,cot(1.03),cot(1.03)-(0.60051260)],_
+[1.04,0.58698722,cot(1.04),cot(1.04)-(0.58698722)],_
+[1.05,0.57361970,cot(1.05),cot(1.05)-(0.57361970)],_
+[1.06,0.56040467,cot(1.06),cot(1.06)-(0.56040467)],_
+[1.07,0.54733693,cot(1.07),cot(1.07)-(0.54733693)],_
+[1.08,0.53441147,cot(1.08),cot(1.08)-(0.53441147)],_
+[1.09,0.52162342,cot(1.09),cot(1.09)-(0.52162342)],_
+[1.10,0.50896811,cot(1.10),cot(1.10)-(0.50896811)],_
+[1.11,0.49644096,cot(1.11),cot(1.11)-(0.49644096)],_
+[1.12,0.48403759,cot(1.12),cot(1.12)-(0.48403759)],_
+[1.13,0.47175371,cot(1.13),cot(1.13)-(0.47175371)],_
+[1.14,0.45958520,cot(1.14),cot(1.14)-(0.45958520)],_
+[1.15,0.44752802,cot(1.15),cot(1.15)-(0.44752802)],_
+[1.16,0.43557829,cot(1.16),cot(1.16)-(0.43557829)],_
+[1.17,0.42373221,cot(1.17),cot(1.17)-(0.42373221)],_
+[1.18,0.41198610,cot(1.18),cot(1.18)-(0.41198610)],_
+[1.19,0.40033638,cot(1.19),cot(1.19)-(0.40033638)],_
+[1.20,0.38877957,cot(1.20),cot(1.20)-(0.38877957)],_
+[1.21,0.37731227,cot(1.21),cot(1.21)-(0.37731227)],_
+[1.22,0.36593119,cot(1.22),cot(1.22)-(0.36593119)],_
+[1.23,0.35463310,cot(1.23),cot(1.23)-(0.35463310)],_
+[1.24,0.34341486,cot(1.24),cot(1.24)-(0.34341486)],_
+[1.25,0.33227342,cot(1.25),cot(1.25)-(0.33227342)],_
+[1.26,0.32120577,cot(1.26),cot(1.26)-(0.32120577)],_
+[1.27,0.31020899,cot(1.27),cot(1.27)-(0.31020899)],_
+[1.28,0.29928023,cot(1.28),cot(1.28)-(0.29928023)],_
+[1.29,0.28841670,cot(1.29),cot(1.29)-(0.28841670)],_
+[1.30,0.27761565,cot(1.30),cot(1.30)-(0.27761565)],_
+[1.31,0.26687440,cot(1.31),cot(1.31)-(0.26687440)],_
+[1.32,0.25619034,cot(1.32),cot(1.32)-(0.25619034)],_
+[1.33,0.24556088,cot(1.33),cot(1.33)-(0.24556088)],_
+[1.34,0.23498350,cot(1.34),cot(1.34)-(0.23498350)],_
+[1.35,0.22445572,cot(1.35),cot(1.35)-(0.22445572)],_
+[1.36,0.21397509,cot(1.36),cot(1.36)-(0.21397509)],_
+[1.37,0.20353922,cot(1.37),cot(1.37)-(0.20353922)],_
+[1.38,0.19314574,cot(1.38),cot(1.38)-(0.19314574)],_
+[1.39,0.18279234,cot(1.39),cot(1.39)-(0.18279234)],_
+[1.40,0.17247673,cot(1.40),cot(1.40)-(0.17247673)],_
+[1.41,0.16219663,cot(1.41),cot(1.41)-(0.16219663)],_
+[1.42,0.15194983,cot(1.42),cot(1.42)-(0.15194983)],_
+[1.43,0.14173413,cot(1.43),cot(1.43)-(0.14173413)],_
+[1.44,0.13154734,cot(1.44),cot(1.44)-(0.13154734)],_
+[1.45,0.12138732,cot(1.45),cot(1.45)-(0.12138732)],_
+[1.46,0.11125194,cot(1.46),cot(1.46)-(0.11125194)],_
+[1.47,0.10113908,cot(1.47),cot(1.47)-(0.10113908)],_
+[1.48,0.091046660,cot(1.48),cot(1.48)-(0.091046660)],_
+[1.49,0.080972601,cot(1.49),cot(1.49)-(0.080972601)],_
+[1.50,0.070914844,cot(1.50),cot(1.50)-(0.070914844)],_
+[1.51,0.060871343,cot(1.51),cot(1.51)-(0.060871343)],_
+[1.52,0.050840061,cot(1.52),cot(1.52)-(0.050840061)],_
+[1.53,0.040818975,cot(1.53),cot(1.53)-(0.040818975)],_
+[1.54,0.030806066,cot(1.54),cot(1.54)-(0.030806066)],_
+[1.55,0.020799325,cot(1.55),cot(1.55)-(0.020799325)],_
+[1.56,0.010796746,cot(1.56),cot(1.56)-(0.010796746)],_
+[1.57,0.000796327,cot(1.57),cot(1.57)-(0.000796327)],_
+[1.58,-0.009203933,cot(1.58),cot(1.58)-(-0.009203933)],_
+[1.59,-0.019206034,cot(1.59),cot(1.59)-(-0.019206034)],_
+[1.60,-0.029211978,cot(1.60),cot(1.60)-(-0.029211978)]]
+--R 
+--R
+--R   (2)
+--R   [[0.01,99.9966666,99.9966666444 44232802,0.444442328 E -7],
+--R    [0.02,49.9933332,49.9933331555 48782798,- 0.4445121720 2 E -7],
+--R    [0.03,33.3233327,33.3233327332 81900133,0.3328190013 3 E -7],
+--R    [0.04,24.9866652,24.9866652442 27690187,0.4422769018 7 E -7],
+--R    [0.05,19.9833306,19.9833305548 94014508,- 0.4510598549 1 E -7],
+--R    [0.06,16.6466619,16.6466618650 20359708,- 0.3497964029 2 E -7],
+--R    [0.07,14.2623733,14.2623733265 9994931,0.2659994931 E -7],
+--R    [0.08,12.4733219,12.4733219486 16087812,0.4861608781 2 E -7],
+--R    [0.09,11.0810949,11.0810948986 03837279,- 0.1396162721 E -8],
+--R    [0.1,9.9666444,9.9666444232 592378598,0.2325923786 E -7],
+--R    [0.11,9.0542128,9.0542128123 384855803,0.1233848558 E -7],
+--R    [0.12,8.2932949,8.2932948805 945312103,- 0.1940546879 E -7],
+--R    [0.13,7.6489255,7.6489254580 38579468,- 0.4196142053 2 E -7],
+--R    [0.14,7.0961294,7.0961293843 639732986,- 0.1563602670 1 E -7],
+--R    [0.15,6.6165915,6.6165915055 899500987,0.5589950098 7 E -8],
+--R    [0.16,6.1965754,6.1965754219 540076277,0.2195400762 8 E -7],
+--R    [0.17,5.8255768,5.8255767953 622059416,- 0.4637794058 4 E -8],
+--R    [0.18,5.4954256,5.4954255543 470259144,- 0.4565297408 56 E -7],
+--R    [0.19,5.1996716,5.1996716132 404544488,0.1324045444 9 E -7],
+--R    [0.2,4.9331549,4.9331548755 868936573,- 0.2441310634 3 E -7],
+--R    [0.21,4.6916981,4.6916980937 15878191,- 0.6284121809 E -8],
+--R    [0.22,4.4718835,4.4718834938 782225379,- 0.6121777462 1 E -8],
+--R    [0.23,4.2708877,4.2708876730 783907862,- 0.2692160921 4 E -7],
+--R    [0.24,4.0863578,4.0863577716 916129493,- 0.2830838705 1 E -7],
+--R    [0.25,3.9163174,3.9163173646 45940105,- 0.3535405989 5 E -7],
+--R    [0.26,3.7590941,3.7590940700 170597097,- 0.2998294029 03 E -7],
+--R    [0.27,3.6132632,3.6132632445 954451122,0.4459544511 22 E -7],
+--R    [0.28,3.4776037,3.4776037446 826739597,0.4468267395 97 E -7],
+--R    [0.29,3.3510628,3.3510628398 201367735,0.3982013677 35 E -7],
+--R    [0.3,3.2327281,3.2327281437 658275137,0.4376582751 37 E -7],
+--R    [0.31,3.121805,3.1218049781 792418003,- 0.2182075819 97 E -7],
+--R    [0.32,3.017598,3.0175979806 093983387,- 0.1939060166 13 E -7],
+--R    [0.33,2.9194961,2.9194960564 791734711,- 0.4352082652 89 E -7],
+--R    [0.34,2.82696,2.8269599865 960316187,- 0.1340396838 1 E -7],
+--R    [0.35,2.7395122,2.7395121590 837832657,- 0.4091621673 43 E -7],
+--R    [0.36,2.656728,2.6567280126 534137138,0.1265341371 4 E -7],
+--R    [0.37,2.5782289,2.5782288674 460399215,- 0.3255396007 85 E -7],
+--R    [0.38,2.5036759,2.5036758878 425107672,- 0.1215748923 3 E -7],
+--R    [0.39,2.432765,2.4327649740 660567805,- 0.2593394321 95 E -7],
+--R    [0.4,2.3652224,2.3652224200 39110587,0.2003911058 7 E -7],
+--R    [0.41,2.3008012,2.3008012066 70318891,0.6670318891 E -8],
+--R    [0.42,2.2392778,2.2392778246 666163969,0.2466661639 69 E -7],
+--R    [0.43,2.1804495,2.1804495406 685083798,0.4066850837 98 E -7],
+--R    [0.44,2.124132,2.1241320361 797692227,0.3617976922 27 E -7],
+--R    [0.45,2.0701574,2.0701573613 012126215,- 0.3869878737 85 E -7],
+--R    [0.46,2.0183722,2.0183721553 634627578,- 0.4463653724 22 E -7],
+--R    [0.47,1.9686361,1.9686360947 077066311,- 0.5292293368 9 E -8],
+--R    [0.48,1.9208205,1.9208205344 885739853,0.3448857398 53 E -7],
+--R    [0.49,1.8748073,1.8748073167 815901303,0.1678159013 03 E -7],
+--R    [0.5,1.83048772,1.8304877217 124519193,0.1712451919 3 E -8],
+--R    [0.51,1.78776154,1.7877615419 775693505,0.1977569350 5 E -8],
+--R    [0.52,1.74653626,1.7465362641 453971653,0.4145397165 3 E -8],
+--R    [0.53,1.70672634,1.7067263426 353184999,0.2635318499 9 E -8],
+--R    [0.54,1.66825255,1.6682525543 602072732,0.4360207273 2 E -8],
+--R    [0.55,1.63104142,1.6310414237 662644566,0.3766264456 6 E -8],
+--R    [0.56,1.59502471,1.5950247094 703485723,- 0.5296514276 E -9],
+--R    [0.57,1.56013894,1.5601389449 30073432,0.4930073431 9 E -8],
+--R    [0.58,1.52632503,1.5263250266 253525611,- 0.3374647438 9 E -8],
+--R    [0.59,1.49352784,1.4935278441 14312362,0.4114312362 E -8],
+--R    [0.6,1.46169595,1.4616959470 781021403,- 0.2921897859 7 E -8],
+--R    [0.61,1.43078125,1.4307812451 098423602,- 0.4890157639 8 E -8],
+--R    [0.62,1.40073873,1.4007387365 5065882,0.655065882 E -8],
+--R    [0.63,1.37152626,1.3715262631 452128637,0.3145212863 7 E -8],
+--R    [0.64,1.34310429,1.3431042876 925818576,- 0.2307418142 4 E -8],
+--R    [0.65,1.31543569,1.3154356922 15926629,0.2215926629 E -8],
+--R    [0.66,1.28848559,1.2884855944 745672615,0.4474567261 5 E -8],
+--R    [0.67,1.26222118,1.2622211809 019498574,0.9019498574 E -9],
+--R    [0.68,1.23661155,1.2366115542 784545265,0.4278454526 5 E -8],
+--R    [0.69,1.21162759,1.2116275946 440535197,0.4644053519 7 E -8],
+--R    [0.7,1.18724183,1.1872418321 266793537,0.2126679353 7 E -8],
+--R    [0.71,1.16342833,1.1634283305 113563708,0.5113563708 E -9],
+--R    [0.72,1.14016258,1.1401625805 056933781,0.5056933781 E -9],
+--R    [0.73,1.1174214,1.1174214017 717845341,0.1771784534 1 E -8],
+--R    [0.74,1.09518285,1.0951828528 950954346,0.2895095434 6 E -8],
+--R    [0.75,1.07342615,1.0734261485 493773587,- 0.1450622641 3 E -8],
+--R    [0.76,1.05213158,1.0521315831 946421944,0.3194642194 4 E -8],
+--R    [0.77,1.03128046,1.0312804607 141042446,0.7141042446 E -9],
+--R    [0.78,1.01085503,1.0108550294 569216086,- 0.5430783914 E -9],
+--R    [0.79,0.99083842,0.9908384222 0755510722,0.2207555107 2 E -8],
+--R    [0.8,0.9712146,0.9712146006 5047441252,0.6504744125 2 E -9],
+--R    [0.81,0.9519683,0.9519683039 4152897039,0.3941528970 39 E -8],
+--R    [0.82,0.933085,0.9330850010 3521469901,0.1035214699 E -8],
+--R    [0.83,0.91455085,0.9145508464 5086948692,- 0.3549130513 1 E -8],
+--R    [0.84,0.89635264,0.8963526391 9101064443,- 0.8089893555 7 E -9],
+--R    [0.85,0.87847778,0.8784777845 5201177212,0.4552011772 12 E -8],
+--R    [0.86,0.86091426,0.8609142585 9147659674,- 0.1408523403 3 E -8],
+--R    [0.87,0.84365058,0.8436505750 3832779908,- 0.4961672200 92 E -8],
+--R    [0.88,0.82667575,0.8266757544 5107375391,0.4451073753 91 E -8],
+--R    [0.89,0.8099793,0.8099792954 4719434615,- 0.4552805653 85 E -8],
+--R    [0.9,0.79355115,0.7935511478 423171255,- 0.2157682874 5 E -8],
+--R    [0.91,0.77738169,0.7773816875 5202909939,- 0.2447970900 6 E -8],
+--R    [0.92,0.76146169,0.7614616931 2195656768,0.3121956567 7 E -8],
+--R    [0.93,0.74578232,0.7457823237 6329468591,0.3763294685 91 E -8],
+--R    [0.94,0.7303351,0.7303350987 8141157551,- 0.1218588424 5 E -8],
+--R    [0.95,0.71511188,0.7151118782 9460519926,- 0.1705394800 7 E -8],
+--R    [0.96,0.70010485,0.7001048451 4865796784,- 0.4851342032 16 E -8],
+--R    [0.97,0.68530649,0.6853064879 4060552823,- 0.2059394471 8 E -8],
+--R    [0.98,0.67070959,0.6707095850 721935367,- 0.4927806463 3 E -8],
+--R    [0.99,0.65630719,0.6563071897 5991155454,- 0.2400884455 E -9],
+--R    [1.0,0.64209262,0.6420926159 3433070301,- 0.4065669296 99 E -8],
+--R    [1.01,0.62805942,0.6280594249 6678856739,0.4966788567 39 E -8],
+--R    [1.02,0.61420141,0.6142014131 6631206854,0.3166312068 5 E -8],
+--R    [1.03,0.6005126,0.6005125999 9409219665,- 0.590780335 E -11],
+--R    [1.04,0.58698722,0.5869872169 4686437526,- 0.3053135624 7 E -8],
+--R    [1.05,0.5736197,0.5736196970 6424127541,- 0.2935758724 6 E -8],
+--R    [1.06,0.56040467,0.5604046650 184238209,- 0.4981576179 09 E -8],
+--R    [1.07,0.54733693,0.5473369277 478102494,- 0.2252189750 6 E -8],
+--R    [1.08,0.53441147,0.5344114655 9885875935,- 0.4401141240 65 E -8],
+--R    [1.09,0.52162342,0.5216234239 4316016472,0.3943160164 72 E -8],
+--R    [1.1,0.50896811,0.5089681052 3906440719,- 0.4760935592 81 E -8],
+--R    [1.11,0.49644096,0.4964409615 0939793968,0.1509397939 7 E -8],
+--R    [1.12,0.48403759,0.4840375872 0882520651,- 0.2791174793 49 E -8],
+--R    [1.13,0.47175371,0.4717537124 562623272,0.2456262327 2 E -8],
+--R    [1.14,0.4595852,0.4595851966 0945875941,- 0.3390541240 59 E -8],
+--R    [1.15,0.44752802,0.4475280221 6043593272,0.2160435932 72 E -8],
+--R    [1.16,0.43557829,0.4355782889 3192216044,- 0.1068077839 6 E -8],
+--R    [1.17,0.42373221,0.4237322085 562610121,- 0.1443738987 9 E -8],
+--R    [1.18,0.4119861,0.4119860992 1950525045,- 0.7804947495 5 E -9],
+--R    [1.19,0.40033638,0.4003363806 5454900775,0.6545490077 5 E -9],
+--R    [1.2,0.38877957,0.3887795693 6820491163,- 0.6317950883 7 E -9],
+--R    [1.21,0.37731227,0.3773122740 8810747104,0.4088107471 05 E -8],
+--R    [1.22,0.36593119,0.3659311914 1622565487,0.1416225654 9 E -8],
+--R    [1.23,0.3546331,0.3546331016 7660211852,0.1676602118 5 E -8],
+--R    [1.24,0.34341486,0.3434148649 4570930705,0.4945709307 05 E -8],
+--R    [1.25,0.33227342,0.3322734172 5452856774,- 0.2745471432 26 E -8],
+--R    [1.26,0.32120577,0.3212057669 5212188699,- 0.3047878113 01 E -8],
+--R    [1.27,0.31020899,0.3102089912 2108098744,0.1221080987 4 E -8],
+--R    [1.28,0.29928023,0.2992802327 358089858,0.2735808985 8 E -8],
+--R    [1.29,0.2884167,0.2884166964 5511900872,- 0.3544880991 28 E -8],
+--R    [1.3,0.27761565,0.2776156465 411251863,- 0.3458874813 7 E -8],
+--R    [1.31,0.2668744,0.2668744033 9685712292,0.3396857122 92 E -8],
+--R    [1.32,0.25619034,0.2561903408 1545187187,0.8154518718 7 E -9],
+--R    [1.33,0.24556088,0.2455608832 341699844,0.3234169984 4 E -8],
+--R    [1.34,0.2349835,0.2349835030 8684653553,0.3086846535 53 E -8],
+--R    [1.35,0.22445572,0.2244557182 4872613589,- 0.1751273864 11 E -8],
+--R    [1.36,0.21397509,0.2139750895 6794464051,- 0.4320553594 9 E -9],
+--R    [1.37,0.20353922,0.2035392184 7821123118,- 0.1521788768 82 E -8],
+--R    [1.38,0.19314574,0.1931457446 8751430396,0.4687514303 96 E -8],
+--R    [1.39,0.18279234,0.1827923439 3792453991,0.3937924539 91 E -8],
+--R    [1.4,0.17247673,0.1724767258 3179995277,- 0.4168200047 23 E -8],
+--R    [1.41,0.16219663,0.1621966317 1991176955,0.1719911769 56 E -8],
+--R    [1.42,0.15194983,0.1519498326 4720777741,0.2647207777 41 E -8],
+--R    [1.43,0.14173413,0.1417341273 5211224764,- 0.2647887752 36 E -8],
+--R    [1.44,0.13154734,0.1315473403 1542961869,0.3154296186 9 E -9],
+--R    [1.45,0.12138732,0.1213873198 5507360359,- 0.1449263964 1 E -9],
+--R    [1.46,0.11125194,0.1112519362 6298502613,- 0.3737014973 87 E -8],
+--R    [1.47,0.10113908,0.1011390799 8073116239,- 0.1926883761 E -10],
+--R    [1.48,0.09104666,0.0910466598 1039728128,- 0.1896027187 2 E -9],
+--R    [1.49,0.080972601,0.0809726011 5748799537,0.1574879953 7 E -9],
+--R    [1.5,0.070914844,0.0709148443 0265244878 9,0.3026524487 9 E -9],
+--R    [1.51,0.060871343,0.0608713426 9913372964 9,- 0.3008662703 51 E -9],
+--R    [1.52,0.050840061,0.0508400612 9291959823 6,0.2929195982 36 E -9],
+--R    [1.53,0.040818975,0.0408189748 6263901912 7,- 0.1373609808 7 E -9],
+--R    [1.54,0.030806066,0.0308060663 7630738330 6,0.3763073833 07 E -9],
+--R    [1.55,0.020799325,0.0207993253 6207296975 6,0.3620729697 56 E -9],
+--R    [1.56,0.010796746,0.0107967462 901583485,0.2901583485 E -9],
+--R    [1.57,0.000796327,0.0007963269 6322325475 679,- 0.3677674524 32 E -10],
+--R    [1.58,- 0.009203933,- 0.0092039330 8759988693 76,- 0.8759988693 75 E -10],
+--R    [1.59,- 0.019206034,- 0.0192060342 0373002777 9,- 0.2037300277 79 E -9],
+--R    [1.6,- 0.029211978,- 0.0292119781 9994480011 4,- 0.1999448001 14 E -9]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp186-188
+\end{thebibliography}
+\end{document}
+
diff --git a/src/input/tanhcoth.input.pamphlet b/src/input/tanhcoth.input.pamphlet
new file mode 100644
index 0000000..c076ef9
--- /dev/null
+++ b/src/input/tanhcoth.input.pamphlet
@@ -0,0 +1,860 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input tanhcoth.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a set of tests for the tanh and coth functions.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of tanh, ranging from 0.00 to 2.00. The
+second column is the reference value of tanh from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+
+<<*>>=
+)spool tanhcoth.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 2
+[[0.00,0.00000000,tanh(0.00),tanh(0.00)-0.00000000],_
+[0.01,0.00999967,tanh(0.01),tanh(0.01)-0.00999967],_
+[0.02,0.01999733,tanh(0.02),tanh(0.02)-0.01999733],_
+[0.03,0.02999100,tanh(0.03),tanh(0.03)-0.02999100],_
+[0.04,0.03997868,tanh(0.04),tanh(0.04)-0.03997868],_
+[0.05,0.04995838,tanh(0.05),tanh(0.05)-0.04995838],_
+[0.06,0.05992810,tanh(0.06),tanh(0.06)-0.05992810],_
+[0.07,0.06988589,tanh(0.07),tanh(0.07)-0.06988589],_
+[0.08,0.07982977,tanh(0.08),tanh(0.08)-0.07982977],_
+[0.09,0.08975779,tanh(0.09),tanh(0.09)-0.08975779],_
+[0.10,0.09966800,tanh(0.10),tanh(0.10)-0.09966800],_
+[0.11,0.10955847,tanh(0.11),tanh(0.11)-0.10955847],_
+[0.12,0.11942730,tanh(0.12),tanh(0.12)-0.11942730],_
+[0.13,0.12927258,tanh(0.13),tanh(0.13)-0.12927258],_
+[0.14,0.13909245,tanh(0.14),tanh(0.14)-0.13909245],_
+[0.15,0.14888503,tanh(0.15),tanh(0.15)-0.14888503],_
+[0.16,0.15864850,tanh(0.16),tanh(0.16)-0.15864850],_
+[0.17,0.16838105,tanh(0.17),tanh(0.17)-0.16838105],_
+[0.18,0.17808087,tanh(0.18),tanh(0.18)-0.17808087],_
+[0.19,0.18774621,tanh(0.19),tanh(0.19)-0.18774621],_
+[0.20,0.19737532,tanh(0.20),tanh(0.20)-0.19737532],_
+[0.21,0.20696650,tanh(0.21),tanh(0.21)-0.20696650],_
+[0.22,0.21651806,tanh(0.22),tanh(0.22)-0.21651806],_
+[0.23,0.22602835,tanh(0.23),tanh(0.23)-0.22602835],_
+[0.24,0.23549575,tanh(0.24),tanh(0.24)-0.23549575],_
+[0.25,0.24491866,tanh(0.25),tanh(0.25)-0.24491866],_
+[0.26,0.25429553,tanh(0.26),tanh(0.26)-0.25429553],_
+[0.27,0.26362484,tanh(0.27),tanh(0.27)-0.26362484],_
+[0.28,0.27290508,tanh(0.28),tanh(0.28)-0.27290508],_
+[0.29,0.28213481,tanh(0.29),tanh(0.29)-0.28213481],_
+[0.30,0.29131261,tanh(0.30),tanh(0.30)-0.29131261],_
+[0.31,0.30043710,tanh(0.31),tanh(0.31)-0.30043710],_
+[0.32,0.30950692,tanh(0.32),tanh(0.32)-0.30950692],_
+[0.33,0.31852078,tanh(0.33),tanh(0.33)-0.31852078],_
+[0.34,0.32747740,tanh(0.34),tanh(0.34)-0.32747740],_
+[0.35,0.33637554,tanh(0.35),tanh(0.35)-0.33637554],_
+[0.36,0.34521403,tanh(0.36),tanh(0.36)-0.34521403],_
+[0.37,0.35399171,tanh(0.37),tanh(0.37)-0.35399171],_
+[0.38,0.36270747,tanh(0.38),tanh(0.38)-0.36270747],_
+[0.39,0.37136023,tanh(0.39),tanh(0.39)-0.37136023],_
+[0.40,0.37994896,tanh(0.40),tanh(0.40)-0.37994896],_
+[0.41,0.38847268,tanh(0.41),tanh(0.41)-0.38847268],_
+[0.42,0.39693043,tanh(0.42),tanh(0.42)-0.39693043],_
+[0.43,0.40532131,tanh(0.43),tanh(0.43)-0.40532131],_
+[0.44,0.41364444,tanh(0.44),tanh(0.44)-0.41364444],_
+[0.45,0.42189901,tanh(0.45),tanh(0.45)-0.42189901],_
+[0.46,0.43008421,tanh(0.46),tanh(0.46)-0.43008421],_
+[0.47,0.43819932,tanh(0.47),tanh(0.47)-0.43819932],_
+[0.48,0.44624361,tanh(0.48),tanh(0.48)-0.44624361],_
+[0.49,0.45421643,tanh(0.49),tanh(0.49)-0.45421643],_
+[0.50,0.46211716,tanh(0.50),tanh(0.50)-0.46211716],_
+[0.51,0.46994520,tanh(0.51),tanh(0.51)-0.46994520],_
+[0.52,0.47770001,tanh(0.52),tanh(0.52)-0.47770001],_
+[0.53,0.48538109,tanh(0.53),tanh(0.53)-0.48538109],_
+[0.54,0.49298797,tanh(0.54),tanh(0.54)-0.49298797],_
+[0.55,0.50052021,tanh(0.55),tanh(0.55)-0.50052021],_
+[0.56,0.50797743,tanh(0.56),tanh(0.56)-0.50797743],_
+[0.57,0.51535928,tanh(0.57),tanh(0.57)-0.51535928],_
+[0.58,0.52266543,tanh(0.58),tanh(0.58)-0.52266543],_
+[0.59,0.52989561,tanh(0.59),tanh(0.59)-0.52989561],_
+[0.60,0.53704957,tanh(0.60),tanh(0.60)-0.53704957],_
+[0.61,0.54412710,tanh(0.61),tanh(0.61)-0.54412710],_
+[0.62,0.55112803,tanh(0.62),tanh(0.62)-0.55112803],_
+[0.63,0.55805222,tanh(0.63),tanh(0.63)-0.55805222],_
+[0.64,0.56489955,tanh(0.64),tanh(0.64)-0.56489955],_
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+[0.99,0.75736232,tanh(0.99),tanh(0.99)-0.75736232],_
+[1.00,0.76159416,tanh(1.00),tanh(1.00)-0.76159416],_
+[1.01,0.76576202,tanh(1.01),tanh(1.01)-0.76576202],_
+[1.02,0.76986654,tanh(1.02),tanh(1.02)-0.76986654],_
+[1.03,0.77390834,tanh(1.03),tanh(1.03)-0.77390834],_
+[1.04,0.77788807,tanh(1.04),tanh(1.04)-0.77788807],_
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+[1.07,0.78946122,tanh(1.07),tanh(1.07)-0.78946122],_
+[1.08,0.79319910,tanh(1.08),tanh(1.08)-0.79319910],_
+[1.09,0.79687814,tanh(1.09),tanh(1.09)-0.79687814],_
+[1.10,0.80049902,tanh(1.10),tanh(1.10)-0.80049902],_
+[1.11,0.80406239,tanh(1.11),tanh(1.11)-0.80406239],_
+[1.12,0.80756892,tanh(1.12),tanh(1.12)-0.80756892],_
+[1.13,0.81101926,tanh(1.13),tanh(1.13)-0.81101926],_
+[1.14,0.81441409,tanh(1.14),tanh(1.14)-0.81441409],_
+[1.15,0.81775408,tanh(1.15),tanh(1.15)-0.81775408],_
+[1.16,0.82103988,tanh(1.16),tanh(1.16)-0.82103988],_
+[1.17,0.82427217,tanh(1.17),tanh(1.17)-0.82427217],_
+[1.18,0.82745161,tanh(1.18),tanh(1.18)-0.82745161],_
+[1.19,0.83057887,tanh(1.19),tanh(1.19)-0.83057887],_
+[1.20,0.83365461,tanh(1.20),tanh(1.20)-0.83365461],_
+[1.21,0.83667949,tanh(1.21),tanh(1.21)-0.83667949],_
+[1.22,0.83965418,tanh(1.22),tanh(1.22)-0.83965418],_
+[1.23,0.84257933,tanh(1.23),tanh(1.23)-0.84257933],_
+[1.24,0.84545560,tanh(1.24),tanh(1.24)-0.84545560],_
+[1.25,0.84828364,tanh(1.25),tanh(1.25)-0.84828364],_
+[1.26,0.85106411,tanh(1.26),tanh(1.26)-0.85106411],_
+[1.27,0.85379765,tanh(1.27),tanh(1.27)-0.85379765],_
+[1.28,0.85648492,tanh(1.28),tanh(1.28)-0.85648492],_
+[1.29,0.85912654,tanh(1.29),tanh(1.29)-0.85912654],_
+[1.30,0.86172316,tanh(1.30),tanh(1.30)-0.86172316],_
+[1.31,0.86427541,tanh(1.31),tanh(1.31)-0.86427541],_
+[1.32,0.86678393,tanh(1.32),tanh(1.32)-0.86678393],_
+[1.33,0.86924933,tanh(1.33),tanh(1.33)-0.86924933],_
+[1.34,0.87167225,tanh(1.34),tanh(1.34)-0.87167225],_
+[1.35,0.87405329,tanh(1.35),tanh(1.35)-0.87405329],_
+[1.36,0.87639307,tanh(1.36),tanh(1.36)-0.87639307],_
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+[1.38,0.88095127,tanh(1.38),tanh(1.38)-0.88095127],_
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+[1.51,0.90693905,tanh(1.51),tanh(1.51)-0.90693905],_
+[1.52,0.90869766,tanh(1.52),tanh(1.52)-0.90869766],_
+[1.53,0.91042459,tanh(1.53),tanh(1.53)-0.91042459],_
+[1.54,0.91212037,tanh(1.54),tanh(1.54)-0.91212037],_
+[1.55,0.91378549,tanh(1.55),tanh(1.55)-0.91378549],_
+[1.56,0.91542046,tanh(1.56),tanh(1.56)-0.91542046],_
+[1.57,0.91702576,tanh(1.57),tanh(1.57)-0.91702576],_
+[1.58,0.91860189,tanh(1.58),tanh(1.58)-0.91860189],_
+[1.59,0.92014933,tanh(1.59),tanh(1.59)-0.92014933],_
+[1.60,0.92166855,tanh(1.60),tanh(1.60)-0.92166855],_
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+[1.62,0.92462422,tanh(1.62),tanh(1.62)-0.92462422],_
+[1.63,0.92606158,tanh(1.63),tanh(1.63)-0.92606158],_
+[1.64,0.92747257,tanh(1.64),tanh(1.64)-0.92747257],_
+[1.65,0.92885762,tanh(1.65),tanh(1.65)-0.92885762],_
+[1.66,0.93021718,tanh(1.66),tanh(1.66)-0.93021718],_
+[1.67,0.93155168,tanh(1.67),tanh(1.67)-0.93155168],_
+[1.68,0.93286155,tanh(1.68),tanh(1.68)-0.93286155],_
+[1.69,0.93414721,tanh(1.69),tanh(1.69)-0.93414721],_
+[1.70,0.93540907,tanh(1.70),tanh(1.70)-0.93540907],_
+[1.71,0.93664754,tanh(1.71),tanh(1.71)-0.93664754],_
+[1.72,0.93786303,tanh(1.72),tanh(1.72)-0.93786303],_
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+[1.74,0.94022664,tanh(1.74),tanh(1.74)-0.94022664],_
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+[1.76,0.94250301,tanh(1.76),tanh(1.76)-0.94250301],_
+[1.77,0.94360942,tanh(1.77),tanh(1.77)-0.94360942],_
+[1.78,0.94469516,tanh(1.78),tanh(1.78)-0.94469516],_
+[1.79,0.94576057,tanh(1.79),tanh(1.79)-0.94576057],_
+[1.80,0.94680601,tanh(1.80),tanh(1.80)-0.94680601],_
+[1.81,0.94783185,tanh(1.81),tanh(1.81)-0.94783185],_
+[1.82,0.94883842,tanh(1.82),tanh(1.82)-0.94883842],_
+[1.83,0.94982608,tanh(1.83),tanh(1.83)-0.94982608],_
+[1.84,0.95079514,tanh(1.84),tanh(1.84)-0.95079514],_
+[1.85,0.95174596,tanh(1.85),tanh(1.85)-0.95174596],_
+[1.86,0.95267884,tanh(1.86),tanh(1.86)-0.95267884],_
+[1.87,0.95359412,tanh(1.87),tanh(1.87)-0.95359412],_
+[1.88,0.95449211,tanh(1.88),tanh(1.88)-0.95449211],_
+[1.89,0.95537312,tanh(1.89),tanh(1.89)-0.95537312],_
+[1.90,0.95623746,tanh(1.90),tanh(1.90)-0.95623746],_
+[1.91,0.95708542,tanh(1.91),tanh(1.91)-0.95708542],_
+[1.92,0.95791731,tanh(1.92),tanh(1.92)-0.95791731],_
+[1.93,0.95873341,tanh(1.93),tanh(1.93)-0.95873341],_
+[1.94,0.95953401,tanh(1.94),tanh(1.94)-0.95953401],_
+[1.95,0.96031939,tanh(1.95),tanh(1.95)-0.96031939],_
+[1.96,0.96108983,tanh(1.96),tanh(1.96)-0.96108983],_
+[1.97,0.96184561,tanh(1.97),tanh(1.97)-0.96184561],_
+[1.98,0.96258698,tanh(1.98),tanh(1.98)-0.96258698],_
+[1.99,0.96331422,tanh(1.99),tanh(1.99)-0.96331422],_
+[2.00,0.96402758,tanh(2.00),tanh(2.00)-0.96402758]]
+--R 
+--R
+--R   (1)
+--R   [[0.0,0.0,0.0,0.0],
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+--R    [0.77,0.64692945,0.6469294504 4176657733,0.4417665773 E -9],
+--R    [0.78,0.65270671,0.6527067059 6198993346,- 0.4038010066 55 E -8],
+--R    [0.79,0.65840904,0.6584090359 5525105073,- 0.4044748949 27 E -8],
+--R    [0.8,0.66403677,0.6640367702 6784896369,0.2678489637 E -9],
+--R    [0.81,0.66959026,0.6695902596 1877077296,- 0.381229227 E -9],
+--R    [0.82,0.67506987,0.6750698748 386078472,0.4838607847 2 E -8],
+--R    [0.83,0.68047601,0.6804760061 1266189234,- 0.3887338107 66 E -8],
+--R    [0.84,0.68580906,0.6858090622 290945476,0.2229094547 6 E -8],
+--R    [0.85,0.69106947,0.6910694698 3293059133,- 0.1670694087 E -9],
+--R    [0.86,0.69625767,0.6962576726 8668143869,0.2686681438 7 E -8],
+--R    [0.87,0.70137413,0.7013741309 3831249855,0.9383124985 5 E -9],
+--R    [0.88,0.70641932,0.7064193203 9723523504,0.397235235 E -9],
+--R    [0.89,0.71139373,0.7113937318 1896254629,0.1818962546 3 E -8],
+--R    [0.9,0.71629787,0.7162978701 9902442081,0.1990244208 E -9],
+--R    [0.91,0.72113225,0.7211322540 7669984382,0.4076699843 82 E -8],
+--R    [0.92,0.72589742,0.7258974148 4908067729,- 0.5150919322 71 E -8],
+--R    [0.93,0.7305939,0.7305938960 9594379981,- 0.3904056200 19 E -8],
+--R    [0.94,0.73522225,0.7352222529 1586922682,0.2915869226 8 E -8],
+--R    [0.95,0.73978305,0.7397830512 7400429569,0.1274004295 7 E -8],
+--R    [0.96,0.74427687,0.7442768673 6183734344,- 0.2638162656 6 E -8],
+--R    [0.97,0.74870429,0.7487042869 6930867119,- 0.3030691328 8 E -8],
+--R    [0.98,0.75306591,0.7530659048 6955201716,- 0.5130447982 84 E -8],
+--R    [0.99,0.75736232,0.7573623242 1652628152,0.4216526281 52 E -8],
+--R    [1.0,0.76159416,0.7615941559 5576488812,- 0.4044235111 88 E -8],
+--R    [1.01,0.76576202,0.7657620182 4843895284,- 0.1751561047 2 E -8],
+--R    [1.02,0.76986654,0.7698665359 0890037199,- 0.4091099628 01 E -8],
+--R    [1.03,0.77390834,0.7739083398 5584205966,- 0.1441579403 E -9],
+--R    [1.04,0.77788807,0.7778880665 7718485812,- 0.3422815141 9 E -8],
+--R    [1.05,0.78180636,0.7818063576 087741234,- 0.2391225876 6 E -8],
+--R    [1.06,0.78566386,0.7856638590 2694365005,- 0.9730563499 5 E -9],
+--R    [1.07,0.78946122,0.7894612209 5498044002,0.9549804400 2 E -9],
+--R    [1.08,0.7931991,0.7931990970 8350083423,- 0.2916499165 8 E -8],
+--R    [1.09,0.79687814,0.7968781442 0472670169,0.4204726701 69 E -8],
+--R    [1.1,0.80049902,0.8004990217 6062970601,0.1760629706 E -8],
+--R    [1.11,0.80406239,0.8040623914 0489212927,0.1404892129 3 E -8],
+--R    [1.12,0.80756892,0.8075689165 7861430767,- 0.3421385692 3 E -8],
+--R    [1.13,0.81101926,0.8110192620 9968140547,0.2099681405 5 E -8],
+--R    [1.14,0.81441409,0.8144140937 6568599783,0.3765685997 83 E -8],
+--R    [1.15,0.81775408,0.8177540779 7028772932,- 0.2029712270 7 E -8],
+--R    [1.16,0.82103988,0.8210398813 3287713399,0.1332877134 E -8],
+--R    [1.17,0.82427217,0.8242721703 4139752183,0.3413975218 E -9],
+--R    [1.18,0.82745161,0.8274516110 0816662528,0.1008166625 3 E -8],
+--R    [1.19,0.83057887,0.8305788685 3852843014,- 0.1461471569 9 E -8],
+--R    [1.2,0.83365461,0.8336546070 1215525867,- 0.2987844741 3 E -8],
+--R    [1.21,0.83667949,0.8366794890 7681069839,- 0.9231893016 1 E -9],
+--R    [1.22,0.83965418,0.8396541756 5437534711,- 0.4345624652 89 E -8],
+--R    [1.23,0.84257933,0.8425793256 5892954289,- 0.4341070457 11 E -8],
+--R    [1.24,0.8454556,0.8454555957 2668023505,- 0.4273319764 96 E -8],
+--R    [1.25,0.84828364,0.8482836399 5751289761,- 0.424871024 E -10],
+--R    [1.26,0.85106411,0.8510641096 6794385925,- 0.3320561407 E -9],
+--R    [1.27,0.85379765,0.8537976531 55243591,0.3155243591 E -8],
+--R    [1.28,0.85648492,0.8564849154 7249732563,- 0.4527502674 37 E -8],
+--R    [1.29,0.85912654,0.8591265382 1436584793,- 0.1785634152 1 E -8],
+--R    [1.3,0.86172316,0.8617231593 1330636429,- 0.6866936357 1 E -9],
+--R    [1.31,0.86427541,0.8642754128 4601100184,0.2846011001 8 E -8],
+--R    [1.32,0.86678393,0.8667839288 4981867301,- 0.1150181327 E -8],
+--R    [1.33,0.86924933,0.8692493331 4885474185,0.3148854741 8 E -8],
+--R    [1.34,0.87167225,0.8716722471 8965211443,- 0.2810347885 6 E -8],
+--R    [1.35,0.87405329,0.8740532878 8600702097,- 0.2113992979 E -8],
+--R    [1.36,0.87639307,0.8763930674 7282283326,- 0.2527177166 7 E -8],
+--R    [1.37,0.87869219,0.8786921933 6869574227,0.3368695742 3 E -8],
+--R    [1.38,0.88095127,0.8809512680 4699698152,- 0.1953003018 5 E -8],
+--R    [1.39,0.88317089,0.8831708889 1520749667,- 0.1084792503 3 E -8],
+--R    [1.4,0.88535165,0.8853516482 0226250758,- 0.1797737492 4 E -8],
+--R    [1.41,0.88749413,0.8874941328 5366526128,0.2853665261 3 E -8],
+--R    [1.42,0.88959892,0.8895989244 3413141195,0.4434131411 95 E -8],
+--R    [1.43,0.8916666,0.8916665990 3752786391,- 0.9624721360 9 E -9],
+--R    [1.44,0.89369773,0.8936977272 0387255605,- 0.2796127443 9 E -8],
+--R    [1.45,0.89569287,0.8956928738 4316453044,0.3843164530 44 E -8],
+--R    [1.46,0.8976526,0.8976525981 6581669501,- 0.1834183305 E -8],
+--R    [1.47,0.89957745,0.8995774536 1946694194,0.3619466941 9 E -8],
+--R    [1.48,0.90146799,0.9014679878 3194670176,- 0.2168053298 2 E -8],
+--R    [1.49,0.90332474,0.9033247425 6018958154,0.2560189581 5 E -8],
+--R    [1.5,0.90514825,0.9051482536 4486643824,0.3644866438 2 E -8],
+--R    [1.51,0.90693905,0.9069390509 705370591,0.9705370591 E -9],
+--R    [1.52,0.90869766,0.9086976584 3111254663,- 0.1568887453 4 E -8],
+--R    [1.53,0.91042459,0.9104245939 004265213,0.3900426521 3 E -8],
+--R    [1.54,0.91212037,0.9121203692 077173486,- 0.7922826513 9 E -9],
+--R    [1.55,0.91378549,0.9137854901 1782775489,0.1178277549 E -9],
+--R    [1.56,0.91542046,0.9154204563 1593240882,- 0.3684067591 2 E -8],
+--R    [1.57,0.91702576,0.9170257613 9660829878,0.1396608298 8 E -8],
+--R    [1.58,0.91860189,0.9186018928 5706702341,0.2857067023 4 E -8],
+--R    [1.59,0.92014933,0.9201493320 9437242057,0.2094372420 6 E -8],
+--R    [1.6,0.92166855,0.9216685544 0647128268,0.4406471282 68 E -8],
+--R    [1.61,0.92316003,0.9231600289 9686923262,- 0.1003130767 4 E -8],
+--R    [1.62,0.92462422,0.9246242189 8278815827,- 0.1017211841 7 E -8],
+--R    [1.63,0.92606158,0.9260615814 0664591708,0.1406645917 1 E -8],
+--R    [1.64,0.92747257,0.9274725672 5070331722,- 0.2749296682 8 E -8],
+--R    [1.65,0.92885762,0.9288576214 5472765445,0.1454727654 4 E -8],
+--R    [1.66,0.93021718,0.9302171829 3652632524,0.2936526325 2 E -8],
+--R    [1.67,0.93155168,0.9315516846 1520824452,0.4615208244 51 E -8],
+--R    [1.68,0.93286155,0.9328615534 3703496255,0.3437034962 6 E -8],
+--R    [1.69,0.93414721,0.9341472104 0372749847,0.4037274985 E -9],
+--R    [1.7,0.93540907,0.9354090706 0309898052,0.6030989805 2 E -9],
+--R    [1.71,0.93664754,0.9366475432 4188720437,0.3241887204 4 E -8],
+--R    [1.72,0.93786303,0.9378630316 8066518501,0.1680665185 E -8],
+--R    [1.73,0.93905593,0.9390559334 7071168332,0.3470711683 3 E -8],
+--R    [1.74,0.94022664,0.9402266403 9272753171,0.3927275317 E -9],
+--R    [1.75,0.94137554,0.9413755384 9728736227,- 0.1502712637 7 E -8],
+--R    [1.76,0.94250301,0.9425030081 4692005344,- 0.1853079946 6 E -8],
+--R    [1.77,0.94360942,0.9436094240 5971485511,0.4059714855 11 E -8],
+--R    [1.78,0.94469516,0.9446951553 5435372611,- 0.4645646273 89 E -8],
+--R    [1.79,0.94576057,0.9457605655 9647392056,- 0.4403526079 44 E -8],
+--R    [1.8,0.94680601,0.9468060128 4626828964,0.2846268289 6 E -8],
+--R    [1.81,0.94783185,0.9478318497 0723412158,- 0.2927658784 E -9],
+--R    [1.82,0.94883842,0.9488384233 7598462509,0.3375984625 1 E -8],
+--R    [1.83,0.94982608,0.9498260756 9304036887,- 0.4306959631 13 E -8],
+--R    [1.84,0.95079514,0.9507951431 9452112204,0.3194521122 E -8],
+--R    [1.85,0.95174596,0.9517459571 6466159744,- 0.2835338402 6 E -8],
+--R    [1.86,0.95267884,0.9526788436 8907758132,0.3689077581 3 E -8],
+--R    [1.87,0.95359412,0.9535941237 0871183917,0.3708711839 2 E -8],
+--R    [1.88,0.95449211,0.9544921130 7439201893,0.3074392018 9 E -8],
+--R    [1.89,0.95537312,0.9553731226 0193552915,0.2601935529 1 E -8],
+--R    [1.9,0.95623746,0.9562374581 2773905209,- 0.1872260947 9 E -8],
+--R    [1.91,0.95708542,0.9570854205 6479296064,0.5647929606 4 E -9],
+--R    [1.92,0.95791731,0.9579173059 5906344306,- 0.4040936556 95 E -8],
+--R    [1.93,0.95873341,0.9587334055 4618760356,- 0.4453812396 44 E -8],
+--R    [1.94,0.95953401,0.9595340058 0842919816,- 0.4191570801 84 E -8],
+--R    [1.95,0.96031939,0.9603193885 3184498689,- 0.1468155013 1 E -8],
+--R    [1.96,0.96108983,0.9610898308 6361393517,0.8636139351 7 E -9],
+--R    [1.97,0.96184561,0.9618456053 6948368021,- 0.4630516319 79 E -8],
+--R    [1.98,0.96258698,0.9625869800 9129079438,0.9129079438 E -10],
+--R    [1.99,0.96331422,0.9633142186 0451342618,- 0.1395486573 8 E -8],
+--R    [2.0,0.96402758,0.9640275800 7581688395,0.7581688395 E -10]]
+--R                                                        Type: List List Float
+--E 1
+@
+In the following table there are 4 columns. The first column
+is the argument of coth, ranging from 0.01 to 2.00. The
+second column is the reference value of coth from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+
+<<*>>=
+--S 2 of 2
+[[0.01,100.0033333,coth(0.01),coth(0.01)-100.003333],_
+[0.02,50.0066665,coth(0.02),coth(0.02)-50.0066665],_
+[0.03,33.3433327,coth(0.03),coth(0.03)-33.3433327],_
+[0.04,25.0133319,coth(0.04),coth(0.04)-25.0133319],_
+[0.05,20.0166639,coth(0.05),coth(0.05)-20.0166639],_
+[0.06,16.6866619,coth(0.06),coth(0.06)-16.6866619],_
+[0.07,14.3090400,coth(0.07),coth(0.07)-14.3090400],_
+[0.08,12.5266553,coth(0.08),coth(0.08)-12.5266553],_
+[0.09,11.1410949,coth(0.09),coth(0.09)-11.1410949],_
+[0.10,10.0333111,coth(0.10),coth(0.10)-10.0333111],_
+[0.11,9.1275462,coth(0.11),coth(0.11)-9.1275462],_
+[0.12,8.3732950,coth(0.12),coth(0.12)-8.3732950],_
+[0.13,7.7355923,coth(0.13),coth(0.13)-7.7355923],_
+[0.14,7.1894629,coth(0.14),coth(0.14)-7.1894629],_
+[0.15,6.7165918,coth(0.15),coth(0.15)-6.7165918],_
+[0.16,6.3032425,coth(0.16),coth(0.16)-6.3032425],_
+[0.17,5.9389107,coth(0.17),coth(0.17)-5.9389107],_
+[0.18,5.6154264,coth(0.18),coth(0.18)-5.6154264],_
+[0.19,5.3263393,coth(0.19),coth(0.19)-5.3263393],_
+[0.20,5.0664896,coth(0.20),coth(0.20)-5.0664896],_
+[0.21,4.8316998,coth(0.21),coth(0.21)-4.8316998],_
+[0.22,4.6185523,coth(0.22),coth(0.22)-4.6185523],_
+[0.23,4.4242237,coth(0.23),coth(0.23)-4.4242237],_
+[0.24,4.2463611,coth(0.24),coth(0.24)-4.2463611],_
+[0.25,4.0829882,coth(0.25),coth(0.25)-4.0829882],_
+[0.26,3.9324324,coth(0.26),coth(0.26)-3.9324324],_
+[0.27,3.7932693,coth(0.27),coth(0.27)-3.7932693],_
+[0.28,3.6642777,coth(0.28),coth(0.28)-3.6642777],_
+[0.29,3.5444049,coth(0.29),coth(0.29)-3.5444049],_
+[0.30,3.4327384,coth(0.30),coth(0.30)-3.4327384],_
+[0.31,3.3284838,coth(0.31),coth(0.31)-3.3284838],_
+[0.32,3.2309455,coth(0.32),coth(0.32)-3.2309455],_
+[0.33,3.1395126,coth(0.33),coth(0.33)-3.1395126],_
+[0.34,3.0536459,coth(0.34),coth(0.34)-3.0536459],_
+[0.35,2.9728677,coth(0.35),coth(0.35)-2.9728677],_
+[0.36,2.8967536,coth(0.36),coth(0.36)-2.8967536],_
+[0.37,2.8249249,coth(0.37),coth(0.37)-2.8249249],_
+[0.38,2.7570428,coth(0.38),coth(0.38)-2.7570428],_
+[0.39,2.6928032,coth(0.39),coth(0.39)-2.6928032],_
+[0.40,2.6319324,coth(0.40),coth(0.40)-2.6319324],_
+[0.41,2.5741836,coth(0.41),coth(0.41)-2.5741836],_
+[0.42,2.5193332,coth(0.42),coth(0.42)-2.5193332],_
+[0.43,2.4671785,coth(0.43),coth(0.43)-2.4671785],_
+[0.44,2.4175352,coth(0.44),coth(0.44)-2.4175352],_
+[0.45,2.3702355,coth(0.45),coth(0.45)-2.3702355],_
+[0.46,2.3251260,coth(0.46),coth(0.46)-2.3251260],_
+[0.47,2.2820666,coth(0.47),coth(0.47)-2.2820666],_
+[0.48,2.2409284,coth(0.48),coth(0.48)-2.2409284],_
+[0.49,2.2015936,coth(0.49),coth(0.49)-2.2015936],_
+[0.50,2.1639534,coth(0.50),coth(0.50)-2.1639534],_
+[0.51,2.1279077,coth(0.51),coth(0.51)-2.1279077],_
+[0.52,2.0933640,coth(0.52),coth(0.52)-2.0933640],_
+[0.53,2.0602368,coth(0.53),coth(0.53)-2.0602368],_
+[0.54,2.0284471,coth(0.54),coth(0.54)-2.0284471],_
+[0.55,1.9979213,coth(0.55),coth(0.55)-1.9979213],_
+[0.56,1.9685914,coth(0.56),coth(0.56)-1.9685914],_
+[0.57,1.9403939,coth(0.57),coth(0.57)-1.9403939],_
+[0.58,1.9132698,coth(0.58),coth(0.58)-1.9132698],_
+[0.59,1.8871642,coth(0.59),coth(0.59)-1.8871642],_
+[0.60,1.8620255,coth(0.60),coth(0.60)-1.8620255],_
+[0.61,1.8378059,coth(0.61),coth(0.61)-1.8378059],_
+[0.62,1.8144604,coth(0.62),coth(0.62)-1.8144604],_
+[0.63,1.7919470,coth(0.63),coth(0.63)-1.7919470],_
+[0.64,1.7702262,coth(0.64),coth(0.64)-1.7702262],_
+[0.65,1.7492610,coth(0.65),coth(0.65)-1.7492610],_
+[0.66,1.7290167,coth(0.66),coth(0.66)-1.7290167],_
+[0.67,1.7094605,coth(0.67),coth(0.67)-1.7094605],_
+[0.68,1.6905616,coth(0.68),coth(0.68)-1.6905616],_
+[0.69,1.6722911,coth(0.69),coth(0.69)-1.6722911],_
+[0.70,1.6546216,coth(0.70),coth(0.70)-1.6546216],_
+[0.71,1.6375273,coth(0.71),coth(0.71)-1.6375273],_
+[0.72,1.6209838,coth(0.72),coth(0.72)-1.6209838],_
+[0.73,1.6049681,coth(0.73),coth(0.73)-1.6049681],_
+[0.74,1.5894583,coth(0.74),coth(0.74)-1.5894583],_
+[0.75,1.5744338,coth(0.75),coth(0.75)-1.5744338],_
+[0.76,1.5598751,coth(0.76),coth(0.76)-1.5598751],_
+[0.77,1.5457636,coth(0.77),coth(0.77)-1.5457636],_
+[0.78,1.5320817,coth(0.78),coth(0.78)-1.5320817],_
+[0.79,1.5188127,coth(0.79),coth(0.79)-1.5188127],_
+[0.80,1.5059407,coth(0.80),coth(0.80)-1.5059407],_
+[0.81,1.4934506,coth(0.81),coth(0.81)-1.4934506],_
+[0.82,1.4813281,coth(0.82),coth(0.82)-1.4813281],_
+[0.83,1.4695595,coth(0.83),coth(0.83)-1.4695595],_
+[0.84,1.4581318,coth(0.84),coth(0.84)-1.4581318],_
+[0.85,1.4470325,coth(0.85),coth(0.85)-1.4470325],_
+[0.86,1.4362499,coth(0.86),coth(0.86)-1.4362499],_
+[0.87,1.4257726,coth(0.87),coth(0.87)-1.4257726],_
+[0.88,1.4155898,coth(0.88),coth(0.88)-1.4155898],_
+[0.89,1.4056913,coth(0.89),coth(0.89)-1.4056913],_
+[0.90,1.3960673,coth(0.90),coth(0.90)-1.3960673],_
+[0.91,1.3867082,coth(0.91),coth(0.91)-1.3867082],_
+[0.92,1.3776051,coth(0.92),coth(0.92)-1.3776051],_
+[0.93,1.3687495,coth(0.93),coth(0.93)-1.3687495],_
+[0.94,1.3601329,coth(0.94),coth(0.94)-1.3601329],_
+[0.95,1.3517476,coth(0.95),coth(0.95)-1.3517476],_
+[0.96,1.3435860,coth(0.96),coth(0.96)-1.3435860],_
+[0.97,1.3356408,coth(0.97),coth(0.97)-1.3356408],_
+[0.98,1.3279050,coth(0.98),coth(0.98)-1.3279050],_
+[0.99,1.3203720,coth(0.99),coth(0.99)-1.3203720],_
+[1.00,1.3130353,coth(1.00),coth(1.00)-1.3130353],_
+[1.01,1.3058887,coth(1.01),coth(1.01)-1.3058887],_
+[1.02,1.2989264,coth(1.02),coth(1.02)-1.2989264],_
+[1.03,1.2921427,coth(1.03),coth(1.03)-1.2921427],_
+[1.04,1.2855320,coth(1.04),coth(1.04)-1.2855320],_
+[1.05,1.2790891,coth(1.05),coth(1.05)-1.2790891],_
+[1.06,1.2728090,coth(1.06),coth(1.06)-1.2728090],_
+[1.07,1.2666867,coth(1.07),coth(1.07)-1.2666867],_
+[1.08,1.2607175,coth(1.08),coth(1.08)-1.2607175],_
+[1.09,1.2548970,coth(1.09),coth(1.09)-1.2548970],_
+[1.10,1.2492208,coth(1.10),coth(1.10)-1.2492208],_
+[1.11,1.2436846,coth(1.11),coth(1.11)-1.2436846],_
+[1.12,1.2382844,coth(1.12),coth(1.12)-1.2382844],_
+[1.13,1.2330163,coth(1.13),coth(1.13)-1.2330163],_
+[1.14,1.2278766,coth(1.14),coth(1.14)-1.2278766],_
+[1.15,1.2228615,coth(1.15),coth(1.15)-1.2228615],_
+[1.16,1.2179676,coth(1.16),coth(1.16)-1.2179676],_
+[1.17,1.2131915,coth(1.17),coth(1.17)-1.2131915],_
+[1.18,1.2085299,coth(1.18),coth(1.18)-1.2085299],_
+[1.19,1.2039796,coth(1.19),coth(1.19)-1.2039796],_
+[1.20,1.1995375,coth(1.20),coth(1.20)-1.1995375],_
+[1.21,1.1952008,coth(1.21),coth(1.21)-1.1952008],_
+[1.22,1.1909665,coth(1.22),coth(1.22)-1.1909665],_
+[1.23,1.1868319,coth(1.23),coth(1.23)-1.1868319],_
+[1.24,1.1827942,coth(1.24),coth(1.24)-1.1827942],_
+[1.25,1.1788510,coth(1.25),coth(1.25)-1.1788510],_
+[1.26,1.1749996,coth(1.26),coth(1.26)-1.1749996],_
+[1.27,1.1712377,coth(1.27),coth(1.27)-1.1712377],_
+[1.28,1.1675629,coth(1.28),coth(1.28)-1.1675629],_
+[1.29,1.1639729,coth(1.29),coth(1.29)-1.1639729],_
+[1.30,1.1604655,coth(1.30),coth(1.30)-1.1604655],_
+[1.31,1.1570386,coth(1.31),coth(1.31)-1.1570386],_
+[1.32,1.1536901,coth(1.32),coth(1.32)-1.1536901],_
+[1.33,1.1504179,coth(1.33),coth(1.33)-1.1504179],_
+[1.34,1.1472202,coth(1.34),coth(1.34)-1.1472202],_
+[1.35,1.1440950,coth(1.35),coth(1.35)-1.1440950],_
+[1.36,1.1410405,coth(1.36),coth(1.36)-1.1410405],_
+[1.37,1.1380550,coth(1.37),coth(1.37)-1.1380550],_
+[1.38,1.1351366,coth(1.38),coth(1.38)-1.1351366],_
+[1.39,1.1322837,coth(1.39),coth(1.39)-1.1322837],_
+[1.40,1.1294947,coth(1.40),coth(1.40)-1.1294947],_
+[1.41,1.1267680,coth(1.41),coth(1.41)-1.1267680],_
+[1.42,1.1241021,coth(1.42),coth(1.42)-1.1241021],_
+[1.43,1.1214954,coth(1.43),coth(1.43)-1.1214954],_
+[1.44,1.1189466,coth(1.44),coth(1.44)-1.1189466],_
+[1.45,1.1164541,coth(1.45),coth(1.45)-1.1164541],_
+[1.46,1.1140167,coth(1.46),coth(1.46)-1.1140167],_
+[1.47,1.1116330,coth(1.47),coth(1.47)-1.1116330],_
+[1.48,1.1093017,coth(1.48),coth(1.48)-1.1093017],_
+[1.49,1.1070216,coth(1.49),coth(1.49)-1.1070216],_
+[1.50,1.1047914,coth(1.50),coth(1.50)-1.1047914],_
+[1.51,1.1026099,coth(1.51),coth(1.51)-1.1026099],_
+[1.52,1.1004760,coth(1.52),coth(1.52)-1.1004760],_
+[1.53,1.0983886,coth(1.53),coth(1.53)-1.0983886],_
+[1.54,1.0963465,coth(1.54),coth(1.54)-1.0963465],_
+[1.55,1.0943487,coth(1.55),coth(1.55)-1.0943487],_
+[1.56,1.0923942,coth(1.56),coth(1.56)-1.0923942],_
+[1.57,1.0904819,coth(1.57),coth(1.57)-1.0904819],_
+[1.58,1.0886109,coth(1.58),coth(1.58)-1.0886109],_
+[1.59,1.0867801,coth(1.59),coth(1.59)-1.0867801],_
+[1.60,1.0849887,coth(1.60),coth(1.60)-1.0849887],_
+[1.61,1.0832358,coth(1.61),coth(1.61)-1.0832358],_
+[1.62,1.0815204,coth(1.62),coth(1.62)-1.0815204],_
+[1.63,1.0798418,coth(1.63),coth(1.63)-1.0798418],_
+[1.64,1.0781990,coth(1.64),coth(1.64)-1.0781990],_
+[1.65,1.0765913,coth(1.65),coth(1.65)-1.0765913],_
+[1.66,1.0750178,coth(1.66),coth(1.66)-1.0750178],_
+[1.67,1.0734777,coth(1.67),coth(1.67)-1.0734777],_
+[1.68,1.0719704,coth(1.68),coth(1.68)-1.0719704],_
+[1.69,1.0704951,coth(1.69),coth(1.69)-1.0704951],_
+[1.70,1.0690510,coth(1.70),coth(1.70)-1.0690510],_
+[1.71,1.0676375,coth(1.71),coth(1.71)-1.0676375],_
+[1.72,1.0662538,coth(1.72),coth(1.72)-1.0662538],_
+[1.73,1.0648993,coth(1.73),coth(1.73)-1.0648993],_
+[1.74,1.0635734,coth(1.74),coth(1.74)-1.0635734],_
+[1.75,1.0622753,coth(1.75),coth(1.75)-1.0622753],_
+[1.76,1.0610046,coth(1.76),coth(1.76)-1.0610046],_
+[1.77,1.0597605,coth(1.77),coth(1.77)-1.0597605],_
+[1.78,1.0585425,coth(1.78),coth(1.78)-1.0585425],_
+[1.79,1.0573501,coth(1.79),coth(1.79)-1.0573501],_
+[1.80,1.0561826,coth(1.80),coth(1.80)-1.0561826],_
+[1.81,1.0550395,coth(1.81),coth(1.81)-1.0550395],_
+[1.82,1.0539202,coth(1.82),coth(1.82)-1.0539202],_
+[1.83,1.0528243,coth(1.83),coth(1.83)-1.0528243],_
+[1.84,1.0517513,coth(1.84),coth(1.84)-1.0517513],_
+[1.85,1.0507005,coth(1.85),coth(1.85)-1.0507005],_
+[1.86,1.0496717,coth(1.86),coth(1.86)-1.0496717],_
+[1.87,1.0486642,coth(1.87),coth(1.87)-1.0486642],_
+[1.88,1.0476776,coth(1.88),coth(1.88)-1.0476776],_
+[1.89,1.0467115,coth(1.89),coth(1.89)-1.0467115],_
+[1.90,1.0457653,coth(1.90),coth(1.90)-1.0457653],_
+[1.91,1.0448388,coth(1.91),coth(1.91)-1.0448388],_
+[1.92,1.0439314,coth(1.92),coth(1.92)-1.0439314],_
+[1.93,1.0430428,coth(1.93),coth(1.93)-1.0430428],_
+[1.94,1.0421725,coth(1.94),coth(1.94)-1.0421725],_
+[1.95,1.0413202,coth(1.95),coth(1.95)-1.0413202],_
+[1.96,1.0404855,coth(1.96),coth(1.96)-1.0404855],_
+[1.97,1.0396679,coth(1.97),coth(1.97)-1.0396679],_
+[1.98,1.0388672,coth(1.98),coth(1.98)-1.0388672],_
+[1.99,1.0380829,coth(1.99),coth(1.99)-1.0380829],_
+[2.00,1.0373147,coth(2.00),coth(2.00)-1.0373147]]
+--R 
+--R
+--R   (2)
+--R   [[0.01,100.0033333,100.0033333111 1132275,0.3111113227 5 E -6],
+--R    [0.02,50.0066665,50.0066664888 95661105,- 0.111043389 E -7],
+--R    [0.03,33.3433327,33.3433327333 84757277,0.3338475727 7 E -7],
+--R    [0.04,25.0133319,25.0133319113 27796018,0.1132779602 E -7],
+--R    [0.05,20.0166639,20.0166638895 50099248,- 0.1044990075 E -7],
+--R    [0.06,16.6866619,16.6866618683 11788711,- 0.3168821128 9 E -7],
+--R    [0.07,14.30904,14.3090400003 80691777,0.380691777 E -9],
+--R    [0.08,12.5266553,12.5266552958 19479794,- 0.4180520206 E -8],
+--R    [0.09,11.1410949,11.1410949235 98139558,0.2359813955 8 E -7],
+--R    [0.1,10.0333111,10.0333111322 5398961,0.3225398961 E -7],
+--R    [0.11,9.1275462,9.1275462138 41655179,0.1384165517 9 E -7],
+--R    [0.12,8.373295,8.3732949859 20466106,- 0.1407953389 4 E -7],
+--R    [0.13,7.7355923,7.7355922818 667577388,- 0.1813324226 1 E -7],
+--R    [0.14,7.1894629,7.1894629453 485603813,0.4534856038 12 E -7],
+--R    [0.15,6.7165918,6.7165918270 201652046,0.2702016520 5 E -7],
+--R    [0.16,6.3032425,6.3032425324 653055781,0.3246530557 81 E -7],
+--R    [0.17,5.9389107,5.9389107296 982815716,0.2969828157 2 E -7],
+--R    [0.18,5.6154264,5.6154263541 72649833,- 0.4582735016 7 E -7],
+--R    [0.19,5.3263393,5.3263393280 051508097,0.2800515081 E -7],
+--R    [0.2,5.0664896,5.0664895634 394727136,- 0.3656052728 64 E -7],
+--R    [0.21,4.8316998,4.8316998224 69838787,0.2246983878 7 E -7],
+--R    [0.22,4.6185523,4.6185523420 28042354,0.4202804235 4 E -7],
+--R    [0.23,4.4242237,4.4242237308 667251076,0.3086672510 75 E -7],
+--R    [0.24,4.2463611,4.2463611422 274259979,0.4222742599 79 E -7],
+--R    [0.25,4.0829882,4.0829881650 735965683,- 0.3492640343 17 E -7],
+--R    [0.26,3.9324324,3.9324324327 36408036,0.3273640803 6 E -7],
+--R    [0.27,3.7932693,3.7932693185 329284388,0.1853292843 88 E -7],
+--R    [0.28,3.6642777,3.6642776966 135092065,- 0.3386490793 5 E -8],
+--R    [0.29,3.5444049,3.5444048557 416074988,- 0.4425839250 12 E -7],
+--R    [0.3,3.4327384,3.4327384303 217415894,0.3032174158 94 E -7],
+--R    [0.31,3.3284838,3.3284837641 356108564,- 0.3586438914 36 E -7],
+--R    [0.32,3.2309455,3.2309455183 814022485,0.1838140224 85 E -7],
+--R    [0.33,3.1395126,3.1395126237 094799327,0.2370947993 27 E -7],
+--R    [0.34,3.0536459,3.0536458877 845481399,- 0.1221545186 E -7],
+--R    [0.35,2.9728677,2.9728677272 689265964,0.2726892659 64 E -7],
+--R    [0.36,2.8967536,2.8967536111 449900164,0.1114499001 6 E -7],
+--R    [0.37,2.8249249,2.8249248916 098377328,- 0.8390162267 1 E -8],
+--R    [0.38,2.7570428,2.7570427669 367181087,- 0.3306328189 13 E -7],
+--R    [0.39,2.6928032,2.6928031731 296236301,- 0.2687037636 99 E -7],
+--R    [0.4,2.6319324,2.6319324418 321883572,0.4183218835 72 E -7],
+--R    [0.41,2.5741836,2.5741835936 66919577,- 0.6333080423 E -8],
+--R    [0.42,2.5193332,2.5193331610 996456868,- 0.3890035431 32 E -7],
+--R    [0.43,2.4671785,2.4671784546 273392718,- 0.4537266072 82 E -7],
+--R    [0.44,2.4175352,2.4175352017 605355925,0.1760535592 E -8],
+--R    [0.45,2.3702355,2.3702355008 100157433,0.8100157433 E -9],
+--R    [0.46,2.325126,2.3251260415 726980245,0.4157269802 45 E -7],
+--R    [0.47,2.2820666,2.2820665531 657326625,- 0.4683426733 75 E -7],
+--R    [0.48,2.2409284,2.2409284458 829619758,0.4588296197 58 E -7],
+--R    [0.49,2.2015936,2.2015936193 562076931,0.1935620769 3 E -7],
+--R    [0.5,2.1639534,2.1639534137 386528488,0.1373865284 9 E -7],
+--R    [0.51,2.1279077,2.1279076842 79778673,- 0.1572022132 7 E -7],
+--R    [0.52,2.093364,2.0933639826 813995967,- 0.1731860040 33 E -7],
+--R    [0.53,2.0602368,2.0602368311 315780091,0.3113157800 91 E -7],
+--R    [0.54,2.0284471,2.0284470770 025658705,- 0.2299743412 95 E -7],
+--R    [0.55,1.9979213,1.9979213179 463896248,0.1794638962 48 E -7],
+--R    [0.56,1.9685914,1.9685913885 883209462,- 0.1141167905 38 E -7],
+--R    [0.57,1.9403939,1.9403939012 535294422,0.1253529442 2 E -8],
+--R    [0.58,1.9132698,1.9132698342 056208124,0.3420562081 24 E -7],
+--R    [0.59,1.8871642,1.8871641617 600075376,- 0.3823999246 24 E -7],
+--R    [0.6,1.8620255,1.8620255213 866662476,0.2138666624 76 E -7],
+--R    [0.61,1.8378059,1.8378059135 575501759,0.1355755017 59 E -7],
+--R    [0.62,1.8144604,1.8144604306 416324911,0.3064163249 11 E -7],
+--R    [0.63,1.791947,1.7919470116 200198016,0.1162001980 16 E -7],
+--R    [0.64,1.7702262,1.7702262197 970203553,0.1979702035 53 E -7],
+--R    [0.65,1.749261,1.7492610410 306350676,0.4103063506 76 E -7],
+--R    [0.66,1.7290167,1.7290167003 061253871,0.3061253871 E -9],
+--R    [0.67,1.7094605,1.7094604947 361744339,- 0.5263825566 1 E -8],
+--R    [0.68,1.6905616,1.6905616412 966267409,0.4129662674 09 E -7],
+--R    [0.69,1.6722911,1.6722911378 028518669,0.3780285186 7 E -7],
+--R    [0.7,1.6546216,1.6546216358 026294047,0.3580262940 47 E -7],
+--R    [0.71,1.6375273,1.6375273242 106478561,0.2421064785 61 E -7],
+--R    [0.72,1.6209838,1.6209838226 402554549,0.2264025545 49 E -7],
+--R    [0.73,1.6049681,1.6049680835 025519782,- 0.1649744802 18 E -7],
+--R    [0.74,1.5894583,1.5894583020 434418638,0.2043441863 8 E -8],
+--R    [0.75,1.5744338,1.5744338335 777364887,0.3357773648 87 E -7],
+--R    [0.76,1.5598751,1.5598751172 573846082,0.1725738460 82 E -7],
+--R    [0.77,1.5457636,1.5457636057 797852649,0.5779785264 9 E -8],
+--R    [0.78,1.5320817,1.5320817005 030656562,0.5030656562 E -9],
+--R    [0.79,1.5188127,1.5188126914 891934625,- 0.8510806537 49 E -8],
+--R    [0.8,1.5059407,1.5059407020 437066212,0.2043706621 2 E -8],
+--R    [0.81,1.4934506,1.4934506373 634333213,0.3736343332 13 E -7],
+--R    [0.82,1.4813281,1.4813281369 414902969,0.3694149029 69 E -7],
+--R    [0.83,1.4695595,1.4695595304 126515262,0.3041265152 62 E -7],
+--R    [0.84,1.4581318,1.4581317965 523616189,- 0.3447638381 1 E -8],
+--R    [0.85,1.4470325,1.4470325251 696546055,0.2516965460 55 E -7],
+--R    [0.86,1.4362499,1.4362498816 584011227,- 0.1834159887 73 E -7],
+--R    [0.87,1.4257726,1.4257725739 929697346,- 0.2600703026 54 E -7],
+--R    [0.88,1.4155898,1.4155898219 738353763,0.2197383537 63 E -7],
+--R    [0.89,1.4056913,1.4056913285 461485888,0.2854614858 88 E -7],
+--R    [0.9,1.3960673,1.3960672530 300118351,- 0.4696998816 49 E -7],
+--R    [0.91,1.3867082,1.3867081861 153858453,- 0.1388461415 47 E -7],
+--R    [0.92,1.3776051,1.3776051264 873387552,0.2648733875 52 E -7],
+--R    [0.93,1.3687495,1.3687494589 589029032,- 0.4104109709 68 E -7],
+--R    [0.94,1.3601329,1.3601329339 992502041,0.3399925020 41 E -7],
+--R    [0.95,1.3517476,1.3517476485 543534541,0.4855435345 41 E -7],
+--R    [0.96,1.343586,1.3435860280 658708165,0.2806587081 65 E -7],
+--R    [0.97,1.3356408,1.3356408096 017654982,0.9601765498 2 E -8],
+--R    [0.98,1.327905,1.3279050260 192333791,0.2601923337 91 E -7],
+--R    [0.99,1.320372,1.3203719910 869302267,- 0.8913069773 26 E -8],
+--R    [1.0,1.3130353,1.3130352854 993313036,- 0.1450066869 64 E -7],
+--R    [1.01,1.3058887,1.3058887437 213768521,0.4372137685 21 E -7],
+--R    [1.02,1.2989264,1.2989264416 064081471,0.4160640814 71 E -7],
+--R    [1.03,1.2921427,1.2921426847 348261269,- 0.1526517387 31 E -7],
+--R    [1.04,1.285532,1.2855319974 249487926,- 0.2575051207 4 E -8],
+--R    [1.05,1.2790891,1.2790891123 712411139,0.1237124111 39 E -7],
+--R    [1.06,1.272809,1.2728089608 684747719,- 0.3913152522 81 E -7],
+--R    [1.07,1.2666867,1.2666866635 834740719,- 0.3641652592 81 E -7],
+--R    [1.08,1.2607175,1.2607175218 389451031,0.2183894510 31 E -7],
+--R    [1.09,1.254897,1.2548970093 764914129,0.9376491412 86 E -8],
+--R    [1.1,1.2492208,1.2492207645 683124166,- 0.3543168758 34 E -7],
+--R    [1.11,1.2436846,1.2436845830 492796933,- 0.1695072030 67 E -7],
+--R    [1.12,1.2382844,1.2382844107 43108546,0.1074310854 6 E -7],
+--R    [1.13,1.2330163,1.2330163372 582033675,0.3725820336 75 E -7],
+--R    [1.14,1.2278766,1.2278765896 304695801,- 0.1036953041 99 E -7],
+--R    [1.15,1.2228615,1.2228615263 919649847,0.2639196498 47 E -7],
+--R    [1.16,1.2179676,1.2179676319 457208259,0.3194572082 59 E -7],
+--R    [1.17,1.2131915,1.2131915112 284082387,0.1122840823 87 E -7],
+--R    [1.18,1.2085299,1.2085298846 437684793,- 0.1535623152 07 E -7],
+--R    [1.19,1.2039796,1.2039795832 508740941,- 0.1674912590 59 E -7],
+--R    [1.2,1.1995375,1.1995375441 923507667,0.4419235076 67 E -7],
+--R    [1.21,1.1952008,1.1952008063 486731313,0.6348673131 3 E -8],
+--R    [1.22,1.1909665,1.1909665062 055588292,0.6205558829 2 E -8],
+--R    [1.23,1.1868319,1.1868318739 22329409,- 0.2607767059 1 E -7],
+--R    [1.24,1.1827942,1.1827942295 898897106,0.2958988971 06 E -7],
+--R    [1.25,1.178851,1.1788509796 677040268,- 0.2033229597 32 E -7],
+--R    [1.26,1.1749996,1.1749996135 898220841,0.1358982208 41 E -7],
+--R    [1.27,1.1712377,1.1712377005 306348077,0.5306348077 E -9],
+--R    [1.28,1.1675629,1.1675628863 216226741,- 0.1367837732 59 E -7],
+--R    [1.29,1.1639729,1.1639728905 10901625,- 0.9489098374 98 E -8],
+--R    [1.3,1.1604655,1.1604655035 578761464,0.3557876146 4 E -8],
+--R    [1.31,1.1570386,1.1570385841 557790704,- 0.1584422092 96 E -7],
+--R    [1.32,1.1536901,1.1536900566 75315566,- 0.4332468443 4 E -7],
+--R    [1.33,1.1504179,1.1504179087 23037052,0.8723037051 99 E -8],
+--R    [1.34,1.1472202,1.1472201888 084516077,- 0.1119154839 23 E -7],
+--R    [1.35,1.144095,1.1440950041 142329045,0.4114232904 5 E -8],
+--R    [1.36,1.1410405,1.1410405183 642215979,0.1836422159 79 E -7],
+--R    [1.37,1.138055,1.1380549497 842232286,- 0.5021577677 14 E -7],
+--R    [1.38,1.1351366,1.1351365691 508965597,- 0.3084910344 03 E -7],
+--R    [1.39,1.1322837,1.1322836979 242973771,- 0.2075702622 9 E -8],
+--R    [1.4,1.1294947,1.1294947064 598964505,0.6459896450 5 E -8],
+--R    [1.41,1.126768,1.1267680122 961278243,0.1229612782 43 E -7],
+--R    [1.42,1.1241021,1.1241020785 137460187,- 0.2148625398 13 E -7],
+--R    [1.43,1.1214954,1.1214954121 634791355,0.1216347913 55 E -7],
+--R    [1.44,1.1189466,1.1189465627 586602379,- 0.3724133976 21 E -7],
+--R    [1.45,1.1164541,1.1164541208 297026222,0.2082970262 22 E -7],
+--R    [1.46,1.1140167,1.1140167165 374565427,0.1653745654 27 E -7],
+--R    [1.47,1.111633,1.1116330183 426463597,0.1834264635 97 E -7],
+--R    [1.48,1.1093017,1.1093017317 287386746,0.3172873867 46 E -7],
+--R    [1.49,1.1070216,1.1070215979 757344389,- 0.2024265561 1 E -8],
+--R    [1.5,1.1047914,1.1047913929 825119039,- 0.7017488096 1 E -8],
+--R    [1.51,1.1026099,1.1026099261 354731665,0.2613547316 65 E -7],
+--R    [1.52,1.100476,1.1004760392 213654987,0.3922136549 87 E -7],
+--R    [1.53,1.0983886,1.0983886053 822601088,0.5382260108 8 E -8],
+--R    [1.54,1.0963465,1.0963465281 107759211,0.2811077592 11 E -7],
+--R    [1.55,1.0943487,1.0943487402 837348018,0.4028373480 18 E -7],
+--R    [1.56,1.0923942,1.0923942032 325277949,0.3232527794 9 E -8],
+--R    [1.57,1.0904819,1.0904819058 485597183,0.5848559718 3 E -8],
+--R    [1.58,1.0886109,1.0886108637 222222566,- 0.3627777774 34 E -7],
+--R    [1.59,1.0867801,1.0867801183 139237769,0.1831392377 69 E -7],
+--R    [1.6,1.0849887,1.0849887361 557777925,0.3615577779 25 E -7],
+--R    [1.61,1.0832358,1.0832358080 826215661,0.8082621566 08 E -8],
+--R    [1.62,1.0815204,1.0815204484 911020471,0.4849110204 71 E -7],
+--R    [1.63,1.0798418,1.0798417946 256284042,- 0.5374371595 8 E -8],
+--R    [1.64,1.078199,1.0781990058 90049072,0.5890049072 E -8],
+--R    [1.65,1.0765913,1.0765912631 839666847,- 0.3681603331 53 E -7],
+--R    [1.66,1.0750178,1.0750177682 626567093,- 0.3173734329 07 E -7],
+--R    [1.67,1.0734777,1.0734777431 19605205,0.4311960520 5 E -7],
+--R    [1.68,1.0719704,1.0719704293 907280832,0.2939072808 32 E -7],
+--R    [1.69,1.0704951,1.0704950877 793786871,- 0.1222062131 29 E -7],
+--R    [1.7,1.069051,1.0690509975 012925984,- 0.2498707401 6 E -8],
+--R    [1.71,1.0676375,1.0676374557 486584477,- 0.4425134155 23 E -7],
+--R    [1.72,1.0662538,1.0662537771 725412807,- 0.2282745871 93 E -7],
+--R    [1.73,1.0648993,1.0648992933 829208457,- 0.6617079154 3 E -8],
+--R    [1.74,1.0635734,1.0635733524 656411166,- 0.4753435888 34 E -7],
+--R    [1.75,1.0622753,1.0622753185 155995724,0.1851559957 24 E -7],
+--R    [1.76,1.0610046,1.0610045711 855353057,- 0.2881446469 43 E -7],
+--R    [1.77,1.0597605,1.0597605052 49804028,0.5249804028 E -8],
+--R    [1.78,1.0585425,1.0585425301 825555676,0.3018255556 76 E -7],
+--R    [1.79,1.0573501,1.0573500697 497555933,- 0.3025024440 67 E -7],
+--R    [1.8,1.0561826,1.0561825616 145181279,- 0.3838548187 21 E -7],
+--R    [1.81,1.0550395,1.0550394569 552390051,- 0.4304476099 49 E -7],
+--R    [1.82,1.0539202,1.0539202200 960428446,0.2009604284 46 E -7],
+--R    [1.83,1.0528243,1.0528243281 490774395,0.2814907743 95 E -7],
+--R    [1.84,1.0517513,1.0517512706 682097163,- 0.2933179028 37 E -7],
+--R    [1.85,1.0507005,1.0507005493 136967107,0.4931369671 07 E -7],
+--R    [1.86,1.0496717,1.0496716775 274233515,- 0.2247257664 85 E -7],
+--R    [1.87,1.0486642,1.0486641802 183162995,- 0.1978168370 05 E -7],
+--R    [1.88,1.0476776,1.0476775934 575597155,- 0.6542440284 5 E -8],
+--R    [1.89,1.0467115,1.0467114641 832546561,- 0.3581674534 39 E -7],
+--R    [1.9,1.0457653,1.0457653499 141788749,0.4991417887 49 E -7],
+--R    [1.91,1.0448388,1.0448388184 72318166,0.1847231816 6 E -7],
+--R    [1.92,1.0439314,1.0439314477 1385408,0.4771385408 E -7],
+--R    [1.93,1.0430428,1.0430428252 683058841,0.2526830588 41 E -7],
+--R    [1.94,1.0421725,1.0421725482 855370847,0.4828553708 47 E -7],
+--R    [1.95,1.0413202,1.0413202231 90348688,0.2319034868 8 E -7],
+--R    [1.96,1.0404855,1.0404854654 443926981,- 0.3455560730 19 E -7],
+--R    [1.97,1.0396679,1.0396678993 151501465,- 0.6848498535 E -9],
+--R    [1.98,1.0388672,1.0388671576 517282541,- 0.4234827174 59 E -7],
+--R    [1.99,1.0380829,1.0380828816 672411665,- 0.1833275883 35 E -7],
+--R    [2.0,1.0373147,1.0373147207 275480959,0.2072754809 59 E -7]]
+--R                                                        Type: List List Float
+--E 2
+
+)spool 
+)lisp (bye)
+ 
+@
+
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} {\bf Digital Library of Mathematical Functions},
+http://dlmf.nist.gov
+\bibitem{2} Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp213-216
+\end{thebibliography}
+\end{document}



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Subject: [Axiom-developer] Axiom and the Numerical Mathematics Consortium
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I have been concentrating on Axiom's numerical capabilities. 
The last patch is the beginnings of regression tests against
Abramowitz and Stegun (1985) and Zwillinger's CRC Standard (2003).
I've also created firefox hyperdoc pages for the gamma function
standard from the new DLMF. I plan to fill these pages out with
Spad code and test cases as time permits.

I'm a member of the Numerical Mathematics Consortium
(http://www.nmconsortium.org).A recently published draft
standard, which I'm reviewing, is available at:
<http://www.nmconsortium.org/docs/NMC_Technical_Specification%20(9-24-2007).pdf>

The A&S handbook lists polynomial coefficients for approximation of E1,
the exponential integral. Does anyone know how these coefficients were
derived? Is it a chebyshev polynomial? I want to dynamically compute
these coefficients to the required precision.

Tim



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The A&S handbook lists polynomial coefficients for approximation of E1,
the exponential integral. Does anyone know how these coefficients were
derived? Is it a chebyshev polynomial? I want to dynamically compute
these coefficients to the required precision.

Tim



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daly@axiom-developer.org wrote:
> I have been concentrating on Axiom's numerical capabilities. 
> The last patch is the beginnings of regression tests against
> Abramowitz and Stegun (1985) and Zwillinger's CRC Standard (2003).
> I've also created firefox hyperdoc pages for the gamma function
> standard from the new DLMF. I plan to fill these pages out with
> Spad code and test cases as time permits.
>
> I'm a member of the Numerical Mathematics Consortium
> (http://www.nmconsortium.org).A recently published draft
> standard, which I'm reviewing, is available at:
> <http://www.nmconsortium.org/docs/NMC_Technical_Specification%20(9-24-2007).pdf>
>
> The A&S handbook lists polynomial coefficients for approximation of E1,
> the exponential integral. Does anyone know how these coefficients were
> derived? Is it a chebyshev polynomial? I want to dynamically compute
> these coefficients to the required precision.
>
> Tim
>
>
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer
>
>   


The exponential integral can be written as a special case of the 
incomplete gamma function 
<http://en.wikipedia.org/wiki/Incomplete_gamma_function>:

    {\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,

The exponential integral may also be generalized to

    E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt


this is from

http://en.wikipedia.org/wiki/Exponential_integral



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Subject: [Axiom-developer] Two Computational Mathematics Job Openings
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City College of New York (my prior place of employment) and the 
Center for Algorithms and Interactive Scientific Software (CAISS) has 
two tenure track job openings. Headed by Gilbert Baumslag, the designer
of Magnus <http://sourceforge.net/projects/magnus>, this is a group
dedicated to computational mathematics. There is a strong emphasis
on group-theoretic work, as Dr. Baumslag is the leading expert in
infinite group theory.

Dr. Baumslag has invented and reduced to practice a new interface idea
called the "zero learning curve" front end. It changes the way users
think about solving problems, allows parallel computations naturally,
and, as you'd expect from the title, is trivial to learn.

As a former employee I can highly recommend the people and the
working environment. There are many visitors to the group who are
known worldwide so there are many interesting connections.

Visit <http://grouptheory.org>

Tim Daly



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daly@axiom-developer.org wrote:
> The A&S handbook lists polynomial coefficients for approximation of E1,
> the exponential integral. Does anyone know how these coefficients were
> derived? Is it a chebyshev polynomial? I want to dynamically compute
> these coefficients to the required precision.
> 
> Tim
> 
> 
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer
> 

Aren't there references in A&S? I don't have my copy handy, but IIRC
everything in there was painstakingly documented. In the good old days
of expensive scientific computing, finding things that could be computed
accurately with as few multiplies and divides as possible was a very big
deal. Extra points were awarded for methods that got decent results
using the crappy floating point arithmetic most machines had prior to
IEEE 754. :)





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Doug Stewart wrote:
> daly@axiom-developer.org wrote:
>> I have been concentrating on Axiom's numerical capabilities. The last
>> patch is the beginnings of regression tests against
>> Abramowitz and Stegun (1985) and Zwillinger's CRC Standard (2003).
>> I've also created firefox hyperdoc pages for the gamma function
>> standard from the new DLMF. I plan to fill these pages out with
>> Spad code and test cases as time permits.
>>
>> I'm a member of the Numerical Mathematics Consortium
>> (http://www.nmconsortium.org).A recently published draft
>> standard, which I'm reviewing, is available at:
>> <http://www.nmconsortium.org/docs/NMC_Technical_Specification%20(9-24-2007).pdf>
>>
>>
>> The A&S handbook lists polynomial coefficients for approximation of E1,
>> the exponential integral. Does anyone know how these coefficients were
>> derived? Is it a chebyshev polynomial? I want to dynamically compute
>> these coefficients to the required precision.
>>
>> Tim
>>
>>
>> _______________________________________________
>> Axiom-developer mailing list
>> Axiom-developer@nongnu.org
>> http://lists.nongnu.org/mailman/listinfo/axiom-developer
>>
>>   
> 
> 
> The exponential integral can be written as a special case of the
> incomplete gamma function
> <http://en.wikipedia.org/wiki/Incomplete_gamma_function>:
> 
>    {\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,
> 
> The exponential integral may also be generalized to
> 
>    E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt
> 
> 
> this is from
> 
> http://en.wikipedia.org/wiki/Exponential_integral
> 
> 
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer
> 

P.S.: Another classic from those days was _Approximations for Digital
Computers_ by Cecil Hastings. That's on my list of collectibles to pick
up. :)



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see subject,

there remain 114 open items in categories Axiom
Compiler/Library/Interpreter/Documentation/User Interface, not counting
wishlist items.

Sorting by Category, one sees immediately a few bugs, namely #149, 145, 129,
36, 321, 320 which should probably be "fixed in FriCAS" (and maybe elsewhere
too).  There are other dubious ones: 204, 203, 232, 157, 330, 75, 57.

What should we do with the bugs in Axiom on Windows and Axiom On Linux that
only affect the original axiom?  Or, do they affect OpenAxiom, too?  I cannot
tell.

I'd really like to reduce the number of open bugs, because 

* it is then easier to find the real issues, but also

* it gives users the (correct) impression that axiom is being worked on.


Martin




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This patch adds the function to compute approximations to the
Exponential Integral E1 in the range x > -4. Documentation is
in the special.spad pamphlet. A regression test file e1.input
verifies the result against Abramowitz and Stegun for all
published values.

Tim

========================================================================
diff --git a/changelog b/changelog
index 45e1388..f018a38 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080119 tpd src/input/Makefile add e1.input regression test file
+20080119 tpd src/input/e1.input create A&S reference regression for E1
+20080119 tpd src/algebra/special.spad add E1 function
 20080112 tpd src/input/besselk.input complex gamma A&S reference regression
 20080110 tpd src/input/Makefile add new regression test files
 20080110 tpd src/input/asinhatanh.input create A&S reference regression
diff --git a/src/algebra/special.spad.pamphlet b/src/algebra/special.spad.pamphlet
index 0fe76ca..241d23e 100644
--- a/src/algebra/special.spad.pamphlet
+++ b/src/algebra/special.spad.pamphlet
@@ -2,7 +2,7 @@
 \usepackage{axiom}
 \begin{document}
 \title{\$SPAD/src/algebra special.spad}
-\author{Bruce W. Char, Stephen M. Watt}
+\author{Bruce W. Char, Timothy Daly, Stephen M. Watt}
 \maketitle
 \begin{abstract}
 \end{abstract}
@@ -12,9 +12,9 @@
 \section{package DFSFUN DoubleFloatSpecialFunctions}
 <<package DFSFUN DoubleFloatSpecialFunctions>>=
 )abbrev package DFSFUN DoubleFloatSpecialFunctions
-++ Author: Bruce W. Char, Stephen M. Watt
+++ Author: Bruce W. Char, Timothy Daly, Stephen M. Watt
 ++ Date Created:  1990
-++ Date Last Updated: June 25, 1991
+++ Date Last Updated: Jan 19, 2008
 ++ Basic Operations:
 ++ Related Domains:
 ++ Also See:
@@ -32,156 +32,472 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
     C   ==> Complex DoubleFloat
 
     Exports ==> with
-	Gamma: R -> R
-	    ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by
-	    ++   \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}.
-	Gamma: C -> C
-	    ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by
-	    ++   \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}.
-
-	Beta: (R, R) -> R
-	    ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by
-            ++   \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.
-	    ++ This is related to \spad{Gamma(x)} by
-	    ++   \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.
-	Beta: (C, C) -> C
-	    ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by
-            ++   \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.
-	    ++ This is related to \spad{Gamma(x)} by
-	    ++   \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.
-
-	logGamma: R -> R
-	    ++ logGamma(x) is the natural log of \spad{Gamma(x)}.
-	    ++ This can often be computed even if \spad{Gamma(x)} cannot.
-	logGamma: C -> C
-	    ++ logGamma(x) is the natural log of \spad{Gamma(x)}.
-	    ++ This can often be computed even if \spad{Gamma(x)} cannot.
-
-	digamma: R -> R
-	    ++ digamma(x) is the function, \spad{psi(x)}, defined by
-	    ++   \spad{psi(x) = Gamma'(x)/Gamma(x)}.
-	digamma: C -> C
-	    ++ digamma(x) is the function, \spad{psi(x)}, defined by
-	    ++   \spad{psi(x) = Gamma'(x)/Gamma(x)}.
-
-	polygamma: (NNI, R) -> R
-	    ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}.
-	polygamma: (NNI, C) -> C
-	    ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}.
-
-
-	besselJ: (R,R) -> R
-	    ++ besselJ(v,x) is the Bessel function of the first kind,
-            ++ \spad{J(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
-	besselJ: (C,C) -> C
-	    ++ besselJ(v,x) is the Bessel function of the first kind,
-            ++ \spad{J(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
-
-	besselY: (R, R) -> R
-	    ++ besselY(v,x) is the Bessel function of the second kind,
-            ++ \spad{Y(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
-            ++ Note: The default implementation uses the relation
-            ++   \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)}
-            ++ so is not valid for integer values of v.
-	besselY: (C, C) -> C
-	    ++ besselY(v,x) is the Bessel function of the second kind,
-            ++ \spad{Y(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
-            ++ Note: The default implementation uses the relation
-            ++   \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)}
-            ++ so is not valid for integer values of v.
-
-	besselI: (R,R) -> R
-	    ++ besselI(v,x) is the modified Bessel function of the first kind,
-            ++ \spad{I(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
-	besselI: (C,C) -> C
-	    ++ besselI(v,x) is the modified Bessel function of the first kind,
-            ++ \spad{I(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
-
-	besselK: (R, R) -> R
-	    ++ besselK(v,x) is the modified Bessel function of the second kind,
-            ++ \spad{K(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
-            ++ Note: The default implementation uses the relation
-            ++   \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)}.
-            ++ so is not valid for integer values of v.
-	besselK: (C, C) -> C
-	    ++ besselK(v,x) is the modified Bessel function of the second kind,
-            ++ \spad{K(v,x)}.
-	    ++ This function satisfies the differential equation:
-	    ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
-            ++ Note: The default implementation uses the relation
-            ++   \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)}
-            ++ so is not valid for integer values of v.
+        Gamma: R -> R
+         ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by
+         ++   \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}.
+        Gamma: C -> C
+         ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by
+         ++   \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}.
+
+        E1: R -> R
+	 ++ E1(x) is the Exponential Integral function
+         ++ The current implementation is a piecewise approximation
+         ++ involving one poly from -4..4 and a second poly for x > 4
+
+        Beta: (R, R) -> R
+         ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by
+         ++   \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.
+         ++ This is related to \spad{Gamma(x)} by
+         ++   \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.
+        Beta: (C, C) -> C
+         ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by
+         ++   \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.
+         ++ This is related to \spad{Gamma(x)} by
+         ++   \spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.
+
+        logGamma: R -> R
+         ++ logGamma(x) is the natural log of \spad{Gamma(x)}.
+         ++ This can often be computed even if \spad{Gamma(x)} cannot.
+        logGamma: C -> C
+         ++ logGamma(x) is the natural log of \spad{Gamma(x)}.
+         ++ This can often be computed even if \spad{Gamma(x)} cannot.
+
+        digamma: R -> R
+         ++ digamma(x) is the function, \spad{psi(x)}, defined by
+         ++   \spad{psi(x) = Gamma'(x)/Gamma(x)}.
+        digamma: C -> C
+         ++ digamma(x) is the function, \spad{psi(x)}, defined by
+         ++   \spad{psi(x) = Gamma'(x)/Gamma(x)}.
+
+        polygamma: (NNI, R) -> R
+         ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}.
+        polygamma: (NNI, C) -> C
+         ++ polygamma(n, x) is the n-th derivative of \spad{digamma(x)}.
+
+        besselJ: (R,R) -> R
+         ++ besselJ(v,x) is the Bessel function of the first kind,
+         ++ \spad{J(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
+        besselJ: (C,C) -> C
+         ++ besselJ(v,x) is the Bessel function of the first kind,
+         ++ \spad{J(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
+
+        besselY: (R, R) -> R
+         ++ besselY(v,x) is the Bessel function of the second kind,
+         ++ \spad{Y(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
+         ++ Note: The default implementation uses the relation
+         ++   \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)}
+         ++ so is not valid for integer values of v.
+        besselY: (C, C) -> C
+         ++ besselY(v,x) is the Bessel function of the second kind,
+         ++ \spad{Y(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.
+         ++ Note: The default implementation uses the relation
+         ++   \spad{Y(v,x) = (J(v,x) cos(v*%pi) - J(-v,x))/sin(v*%pi)}
+         ++ so is not valid for integer values of v.
+
+        besselI: (R,R) -> R
+         ++ besselI(v,x) is the modified Bessel function of the first kind,
+         ++ \spad{I(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
+        besselI: (C,C) -> C
+         ++ besselI(v,x) is the modified Bessel function of the first kind,
+         ++ \spad{I(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
+
+        besselK: (R, R) -> R
+         ++ besselK(v,x) is the modified Bessel function of the second kind,
+         ++ \spad{K(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
+         ++ Note: The default implementation uses the relation
+         ++   \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)}.
+         ++ so is not valid for integer values of v.
+        besselK: (C, C) -> C
+         ++ besselK(v,x) is the modified Bessel function of the second kind,
+         ++ \spad{K(v,x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.
+         ++ Note: The default implementation uses the relation
+         ++   \spad{K(v,x) = %pi/2*(I(-v,x) - I(v,x))/sin(v*%pi)}
+         ++ so is not valid for integer values of v.
 
         airyAi:   C -> C
-            ++ airyAi(x) is the Airy function \spad{Ai(x)}.
-            ++ This function satisfies the differential equation:
-            ++   \spad{Ai''(x) - x * Ai(x) = 0}.
+         ++ airyAi(x) is the Airy function \spad{Ai(x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{Ai''(x) - x * Ai(x) = 0}.
         airyAi:   R -> R
-            ++ airyAi(x) is the Airy function \spad{Ai(x)}.
-            ++ This function satisfies the differential equation:
-            ++   \spad{Ai''(x) - x * Ai(x) = 0}.
+         ++ airyAi(x) is the Airy function \spad{Ai(x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{Ai''(x) - x * Ai(x) = 0}.
 
         airyBi:   R -> R
-            ++ airyBi(x) is the Airy function \spad{Bi(x)}.
-            ++ This function satisfies the differential equation:
-            ++   \spad{Bi''(x) - x * Bi(x) = 0}.
+         ++ airyBi(x) is the Airy function \spad{Bi(x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{Bi''(x) - x * Bi(x) = 0}.
         airyBi:   C -> C
-            ++ airyBi(x) is the Airy function \spad{Bi(x)}.
-            ++ This function satisfies the differential equation:
-            ++   \spad{Bi''(x) - x * Bi(x) = 0}.
+         ++ airyBi(x) is the Airy function \spad{Bi(x)}.
+         ++ This function satisfies the differential equation:
+         ++   \spad{Bi''(x) - x * Bi(x) = 0}.
 
-	hypergeometric0F1: (R, R) -> R
-	    ++ hypergeometric0F1(c,z) is the hypergeometric function
-            ++ \spad{0F1(; c; z)}.
-	hypergeometric0F1: (C, C) -> C
-	    ++ hypergeometric0F1(c,z) is the hypergeometric function
-            ++ \spad{0F1(; c; z)}.
+        hypergeometric0F1: (R, R) -> R
+         ++ hypergeometric0F1(c,z) is the hypergeometric function
+         ++ \spad{0F1(; c; z)}.
+        hypergeometric0F1: (C, C) -> C
+         ++ hypergeometric0F1(c,z) is the hypergeometric function
+         ++ \spad{0F1(; c; z)}.
 
 
     Impl ==> add
-	a, v, w, z: C
-	n, x, y: R
+        a, v, w, z: C
+        n, x, y: R
 
         -- These are hooks to Bruce's boot code.
-	Gamma z         == CGAMMA(z)$Lisp
-	Gamma x         == RGAMMA(x)$Lisp
-
-	polygamma(k,z)  == CPSI(k, z)$Lisp
-	polygamma(k,x)  == RPSI(k, x)$Lisp
-
-	logGamma z      == CLNGAMMA(z)$Lisp
-	logGamma x      == RLNGAMMA(x)$Lisp
+        Gamma z         == CGAMMA(z)$Lisp
+        Gamma x         == RGAMMA(x)$Lisp
 
-	besselJ(v,z)    == CBESSELJ(v,z)$Lisp
-	besselJ(n,x)    == RBESSELJ(n,x)$Lisp
-
-	besselI(v,z)    == CBESSELI(v,z)$Lisp
-	besselI(n,x)    == RBESSELI(n,x)$Lisp
+@
+\section{The Exponential Integral}
+(Quoted from Segletes\cite{2}):
+
+A number of useful integrals exist for which no exact solutions have
+been found. In other cases, an exact solution, if found, may be
+impractical to utilize over the complete domain of the function
+because of precision limitations associated with what usually ends up
+as a series solution to the challenging integral. For many of these
+integrals, tabulated values may be published in various mathematical
+handbooks and articles. In some handbooks, fits (usually piecewise)
+also are offered. In some cases, an application may be forced to
+resort to numerical integration in order to acquire the integrated
+function. In this context, compact ({\sl i.e.} not piecewise)
+analytical fits to some of these problematic integrals, accurate to
+within a small fraction of the numerically integrated value, serve as
+a useful tool to applications requiring the results of the
+integration, especially when the integration is required numerous
+times throughout the course of the application. Furthermore, the
+ability and methodology to develop intelligent fits, in contract to
+the more traditional ``brute force'' fits, provide the means to
+minimize parameters and maximize accuracy when tackling some of these
+difficult functions. The exponential integral will be used as an
+opportunity to both demonstrate a methodology for intelligent fitting
+as well as for providing an accurate, compact, analytical fit to the
+exponential integral.
+
+The exponential integral is a useful class of functions that arise in
+a variety of applications [...]. The real branch of the family of
+exponential integrals may be defined as
+\begin{equation}
+E_n(x)=x^{n-1}\int_x^\infty{\frac{e^{-t}}{t^n}\ dt}
+\end{equation}
+where $n$, a positive integer, denotes the specific member of the
+exponential integral family. The argument of the exponential integral,
+rather than expressing a lower limit of integration as in (1),
+may be thought of as describing the exponential decay
+constant, as given in this equivalent (and perhaps more popular)
+definition of the integral:
+\begin{equation}
+E_n(x)=\int_1^\infty{\frac{e^{-xt}}{t^n}\ dt}
+\end{equation}
+
+Integration by parts permits any member of the exponential integral
+family to be converted to an adjacent member of the family, by way of 
+\begin{equation}
+\int_x^\infty{\frac{e^{-t}}{t^{n+1}}\ dt}=\frac{1}{n}
+\left(
+\frac{e^{-x}}{x^n}-\int_x^\infty{\frac{e^{-t}}{t^n}\ dt}
+\right)
+\end{equation}
+expressable in terms of $E_n$ as
+\begin{equation}
+E_{n+1}(x)=\frac{1}{n}\left[e^{-x}-xE_n(x)\right]\ (n=1,2,3)
+\end{equation}
+
+Through recursive employment of this equation, all members of the
+exponential integral family may be analytically related. However, this
+technique only allows for the transformation of one integral into
+another. There remains the problem of evaluating $E_1(x)$. There is an
+exact solution to the integral of $(e^{-t}/t)$, appearing in a number
+of mathematical references \cite{4,5} which is obtainable by
+expanding the exponential into a power series and integrating term by
+term. That exact solution, which is convergent, may be used to specify
+$E_1(x)$ as 
+\begin{equation}
+E_1(x)=-\gamma-ln(x)
++\frac{x}{1!}
+-\frac{x^2}{2\cdot 2!}
++\frac{x^3}{3\cdot 3!}
+-\ldots
+\end{equation}
+
+Euler's constant, $\gamma$, equal to $0.57721\ldots$, arises when the
+power series expansion for $(e^{-t}/t)$ is integrated and evaluated at
+its upper limit, as $x\rightarrow\infty$\cite{6}.
+
+Employing eqn (5), however, to evaluate $E_1(x)$ is problematic for
+finite $x$ significantly larger than unity. One may well ask of the
+need to evaluate the exponential integral for large $x$, since the
+function to be integrated drops off so rapidly that the integral is
+surely a very flat function. Such reasoning is true when comparing the
+integrand at large $x$ to that at small $x$. However, the definition
+of eqn (1) has as its upper limit not a small value of $x$, but rather
+that of $\infty$. Therefore, the actual values for $E_n(x)$ are
+extremely small numbers for large values of $x$. Thus, it is not
+sufficient merely to select enough terms of eqn (5) to evaluate the
+integral to within a value of, for example $\pm 0.0001$ because the
+actual integral value for large $x$ would be smaller than this
+arbitrary tolerance. To draw an analogy, it would be like saying that
+it is good enough to approximate $e^{-x}$ as 0.0 for $x>10$, since its
+actual value is within 0.0001 of zero. For some applications, such an
+approximation may be warranted. In general, though, such an
+approximation is mathematically unacceptable. Worse yet, as seen from
+eqns (1) and (2), the need to evaluate the exponential integral for
+large arguments can arise in real-world problems from either a large
+integraion limit or a large value of an exponential decay
+constant. Thus, the need to evaluate exponential integrals for large
+values of the argument is established. It is here that the practical
+problems with the evaluation of eqn (5) become manifest.
+
+First, the number of terms, $N$, required to achieve convergence rises
+rapidly with increasing $x$, making the summation an inefficient tool,
+even when expressed as a recursion relation (for three digits of
+accuracy, $N$ is observed to vary roughly as $9+1.6x$, for $1<x<7$).
+More important, however, is the fact that, for calculations of finite
+precision, the accuracy of the complete summation will be governed by
+the individual term of greatest magnitude. The source of the problem
+is that as $x$ is increased, the total summation decreases in
+magnitude more rapidly than a decaying exponential, while at the same
+time, the largest individual term in the series is observed to grow
+rapidly with increasing $x$ (
+$\tilde{}10^1$ for $x=7$, 
+$\tilde{}10^2$ for $x=10$, 
+$\tilde{}10^3$ for $x=13$, {\sl etc.}). The magnitude of this largest
+individual term consumes the available precision and, as a result,
+leaves little or none left for the ever-diminishing net sum that
+constitutes the desired integral.
+
+Literally, the use of eqn (5), even with (32-bit) double precision,
+does not permit the exponential integral to be evaluated to three
+places for $x>14$ in any case, and with the situation worsening for
+lesser precision. For these reasons, the use of eqn (5) to evaluate
+the exponential integral numerically for large $x$ is wholly
+unsuitable.
+
+\begin{equation}
+E_1(x)=e^{-x}\cdot
+\frac{1}{\displaystyle x+
+\frac{1}{\displaystyle 1+
+\frac{1}{\displaystyle x+
+\frac{2}{\displaystyle 1+
+\frac{2}{\displaystyle x+\ldots}}}}}
+\end{equation}
+
+But as $x$ becomes smaller, the number of terms required for
+convergence rises quickly. Similar arguments apply for the use of an
+asymptotic expansion for $E_1$, which also converges for large $x$. As
+such, the more typical approach employed by handbooks is that of a
+fit. While some steps are taken to make the fits intelligent ({\sl
+e.g.}, transformation of variables), the fits are all piecewise over
+the domain of the integral.
+
+Cody and Thatcher \cite{7} performed what is perhaps the definitive
+work, with the use of Chebyshev approximations to the exponential 
+integral $E_1$. Like others, they fit the integral over a piecewise
+series of subdomains (three in their case) and provide the fitting
+parameters necessary to evaluate the function to various required
+precisions, down to relative errors of $10^-20$. One of the problems
+with piecewise fitting over two or more subdomains is that functional
+value and derivatives of the spliced fits will not, in general, match
+at the domain transition point, unless special accomodations are
+made. This sort of discontinuity in functional value and/or slope,
+curvature, {\sl etc.}, may cause difficulties for some numerical
+algorithms operating upon the fitted function. Numerical
+splicing/smoothing algorithms aimed at eliminating discontinuities in
+the value and/or derivatives of a piecewise fit are not, in general,
+computationally insignificant. Problems associated with piecewise
+splicing of fits may also be obviated by obtaining an accurate enough
+fit, such that the error is on the order of magnitude of the limiting
+machine precision. This alternative, however, requires the use of
+additional fitting parameters to acquire the improved precision. Thus,
+regardless of approach, the desire to eliminate discontinuities in the
+function and its derivatives, between piecewise splices, requires
+extra computational effort. One final benefit to be had by avoiding
+the use of piecewise fits is the concomitant avoidance of conditional
+({\sl i.e.}, IF...THEN) programming statements in the coding of the
+routine. The use of conditional statements can preclude maximum
+computing efficiency on certain parallel computing architectures.
+
+Segletes constructs an analytic, non-piecewise fit to the Exponential
+Integral but the precision is on the order of 4 decimal places and is
+not sufficient to compare against the Abramowitz and Stegun Handbook.
+
+Instead we have chosen to use a two piece fitting function based on
+the Chebyshev polynomial for computing $E_1$. This agrees with the
+handbook values to almost the last published digit. See the {\tt e1.input}
+pamphlet for regression testing against the handbook tables.
+
+\subsection{E1:R$\rightarrow$R}
+The special function E1 below was originally derived from a function
+written by T.Haavie as the {\tt expint.c} function in the Numlibc library
+by Lars Erik Lund. Haavie approximates the E1 function by two 
+Chebyshev polynomials. For the range $-4 < x < 4$ the Chebyshev
+coefficients are:
+\begin{verbatim}
+ 7.8737715392882774, -8.0314874286705335,  3.8797325768522250,
+-1.6042971072992259,  0.5630905453891458, -0.1704423017433357,
+ 0.0452099390015415, -0.0106538986439085,  0.0022562638123478,
+-0.0004335700473221,  0.0000762166811878, -0.0000123417443064,
+ 0.0000018519745698, -0.0000002588698662,  0.0000000338604319,
+-0.0000000041611418,  0.0000000004821606, -0.0000000000528465,
+ 0.0000000000054945, -0.0000000000005433,  0.0000000000000512,
+-0.0000000000000046,  0.0000000000000004 
+\end{verbatim}
+and for the range $x > 4$ the Chebyshev coefficients are:
+\begin{verbatim}
+ 0.2155283776715125,  0.1028106215227030, -0.0045526707131788,
+ 0.0003571613122851, -0.0000379341616932,  0.0000049143944914,
+-0.0000007355024922,  0.0000001230603606, -0.0000000225236907,
+ 0.0000000044412375, -0.0000000009328509,  0.0000000002069297,
+-0.0000000000481502,  0.0000000000116891, -0.0000000000029474,
+ 0.0000000000007691, -0.0000000000002070,  0.0000000000000573,
+-0.0000000000000163,  0.0000000000000047, -0.0000000000000014,
+ 0.0000000000000004, -0.0000000000000001
+\end{verbatim}
+
+I've rewritten the polynomial to use precomputed coefficients
+that take into account the scaling used by Haavie. I've also
+rewritten the polynomial using Horner's method so the large
+powers of $x$ are only computed once.
 
-	hypergeometric0F1(a,z) == CHYPER0F1(a, z)$Lisp
-	hypergeometric0F1(n,x) == retract hypergeometric0F1(n::C, x::C)
+<<package DFSFUN DoubleFloatSpecialFunctions>>=
+        E1(x:R):R ==
+         x = 0.0::R => error "E1 undefined at zero"
+         x > 4.0::R =>
+          t1:R:=0.14999948967737774608E-15::R
+          t2:R:=0.9999999999993112::R
+          ta:R:=(t1*x+t2)
+          t3:R:=0.99999999953685760001::R
+          tb:R:=(ta*x-t3)
+          t4:R:=1.9999998808293376::R
+          tc:R:=(tb*x+t4)
+          t5:R:=5.999983407661056::R
+          td:R:=(tc*x-t5)
+          t6:R:=23.9985380938481664::R
+          te:R:=(td*x+t6)
+          t7:R:=119.9108830382784512::R
+          tf:R:=(te*x-t7)
+          t8:R:=716.01351020920176641::R
+          tg:R:=(tf*x+t8)
+          t9:R:=4903.3466623370985473::R
+          th:R:=(tg*x-t9)
+          t10:R:=36601.25841454446674::R
+          ti:R:=(th*x+t10)
+          t11:R:=279913.28608482691646::R
+          tj:R:=(ti*x-t11)
+          t12:R:=2060518.7020296525186::R
+          tk:R:=(tj*x+t12)
+          t13:R:=13859772.093039815059::R
+          tl:R:=(tk*x-t13)
+          t14:R:=81945572.630072918857::R
+          tm:R:=(tl*x+t14)
+          t15:R:=413965714.82128317479::R
+          tn:R:=(tm*x-t15)
+          t16:R:=1747209536.2595547568::R
+          to:R:=(tn*x+t16)
+          t17:R:=6036182333.96179427::R
+          tp:R:=(to*x-t17)
+          t18:R:=16693683576.106267572::R
+          tq:R:=(tp*x+t18)
+          t19:R:=35938625644.58286097::R
+          tr:R:=(tq*x-t19)
+          t20:R:=57888657293.609258888::R
+          ts:R:=(tr*x+t20)
+          t21:R:=65523779423.11290127::R
+          tt:R:=(ts*x-t21)
+          t22:R:=46422751473.201760309::R
+          tu:R:=(tt*x+t22)
+          t23:R:=15474250491.067253436::R
+          tv:R:=(tu*x-t23)
+          tw:R:=(-1.0::R*x)
+          (tv * exp(tw)::R)/x**22
+         x > -4.0::R => 
+          a1:R:=0.476837158203125E-22::R
+          a2:R:=0.10967254638671875E-20::R
+          aa:R:=(-a1*x+a2)
+          a3:R:=0.20217895507812500001E-19::R
+          ab:R:=(aa*x-a3)
+          a4:R:=0.42600631713867187501E-18::R
+          ac:R:=(ab*x+a4)
+          a5:R:=0.868625640869140625E-17::R
+          ad:R:=(ac*x-a5)
+          a6:R:=0.16553192138671875E-15::R
+          ae:R:=(ad*x+a6)
+          a7:R:=0.29870208740234375E-14::R
+          af:R:=(ae*x-a7)
+          a8:R:=0.5097890777587890625E-13::R
+          ag:R:=(af*x+a8)
+          a9:R:=0.81934069213867187499E-12::R
+          ah:R:=(ag*x-a9)
+          a10:R:=0.1235313123779296875E-10::R
+          ai:R:=(ah*x+a10)
+          a11:R:=0.1739729620849609375E-9::R
+          aj:R:=(ai*x-a11)
+          a12:R:=0.22774642697021484375E-8::R
+          ak:R:=(aj*x+a12)
+          a13:R:=0.275573192853515625E-7::R
+          al:R:=(ak*x-a13)
+          a14:R:=0.30619243635087890625E-6::R
+          am:R:=(al*x+a14)
+          a15:R:=0.000003100198412519140625::R
+          an:R:=(am*x-a15)
+          a16:R:=0.00002834467120045546875::R
+          ao:R:=(an*x+a16)
+          a17:R:=0.00023148148148176953125::R
+          ap:R:=(ao*x-a17)
+          a18:R:=0.0016666666666686609375::R
+          aq:R:=(ap*x+a18)
+          a19:R:=0.01041666666666646875::R
+          ar:R:=(aq*x-a19)
+          a20:R:=0.055555555555554168751::R
+          as:R:=(ar*x+a20)
+          a21:R:=0.2500000000000000375::R
+          at:R:=(as*x-a21)
+          a22:R:=1.000000000000000325::R
+          au:R:=(at*x+a22)
+          a23:R:=0.5772156649015328::R
+          av:R:=au*x-a23
+          - 1.0::R*log(abs(x)) + av
+         error "E1: no approximation available"
+
+        polygamma(k,z)  == CPSI(k, z)$Lisp
+        polygamma(k,x)  == RPSI(k, x)$Lisp
+
+        logGamma z      == CLNGAMMA(z)$Lisp
+        logGamma x      == RLNGAMMA(x)$Lisp
+
+        besselJ(v,z)    == CBESSELJ(v,z)$Lisp
+        besselJ(n,x)    == RBESSELJ(n,x)$Lisp
+
+        besselI(v,z)    == CBESSELI(v,z)$Lisp
+        besselI(n,x)    == RBESSELI(n,x)$Lisp
+
+        hypergeometric0F1(a,z) == CHYPER0F1(a, z)$Lisp
+        hypergeometric0F1(n,x) == retract hypergeometric0F1(n::C, x::C)
 
 
         -- All others are defined in terms of these.
-	digamma x == polygamma(0, x)
-	digamma z == polygamma(0, z)
+        digamma x == polygamma(0, x)
+        digamma z == polygamma(0, z)
 
-	Beta(x,y) == Gamma(x)*Gamma(y)/Gamma(x+y)
-	Beta(w,z) == Gamma(w)*Gamma(z)/Gamma(w+z)
+        Beta(x,y) == Gamma(x)*Gamma(y)/Gamma(x+y)
+        Beta(w,z) == Gamma(w)*Gamma(z)/Gamma(w+z)
 
         fuzz := (10::R)**(-7)
 
@@ -192,7 +508,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
             if integer? n then n := n + fuzz
             vp := n * pi()$R
             (cos(vp) * besselJ(n,x) - besselJ(-n,x) )/sin(vp)
-	besselY(v,z) ==
+        besselY(v,z) ==
             if integer? v then v := v + fuzz::C
             vp := v * pi()$C
             (cos(vp) * besselJ(v,z) - besselJ(-v,z) )/sin(vp)
@@ -362,13 +678,13 @@ NumberTheoreticPolynomialFunctions(R: CommutativeRing): Exports == Impl where
     Exports ==> with
 
         cyclotomic: (NNI, R) -> R
-		++ cyclotomic(n,r) \undocumented
+         ++ cyclotomic(n,r) \undocumented
 
         if R has Algebra RN then
             bernoulliB: (NNI, R) -> R
-		++ bernoulliB(n,r) \undocumented
+             ++ bernoulliB(n,r) \undocumented
             eulerE:     (NNI, R) -> R
-		++ eulerE(n,r) \undocumented
+             ++ eulerE(n,r) \undocumented
 
     Impl ==> add
 
@@ -451,6 +767,21 @@ NumberTheoreticPolynomialFunctions(R: CommutativeRing): Exports == Impl where
 @
 \eject
 \begin{thebibliography}{99}
-\bibitem{1} nothing
+\bibitem{1} Segletes, Steven, B., ``A Compact Analytical Fit to the 
+Exponential Integral $E_1(x)$'', Army Research Laboratory, ARL-TR-1758,
+September 1998
+\bibitem{2} Lund, Lars Erik, ``Numlibc'',\\
+{\bf http://www.math.ntnu.no/num/nnm/Program/Numlibc}
+\bibitem{3} Haavie, T. ``expint.c'',\\
+{\bf http://www.math.ntnu.no/num/nnm/Program/Numlibc}, June, 1988
+\bibitem{4} Abramowitz and Stegun,``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp238-243
+\bibitem{5} Beyer, W.H. (ed.) CRC Standard Mathematical Tables. 26th
+Edition, Boca Raton: CRC Press, 1981.
+\bibitem{6} Pearson, C.E. (ed.) ``Handbook of Applied Mathematics; 
+Selected Results and Methods''. New York: van Nostrand Reinhold, 1983.
+\bibitem{7} Cody, W.J., and H.C. Thatcher, Jr. ``Rational Chebyshev
+Approximations for the Exponential Integral $E_1(x)$.''
+Mathematics of Computation, 11, pp. 641-649, 1968
 \end{thebibliography}
 \end{document}
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 07fca93..fecb71d 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -302,7 +302,8 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     cycles1.regress   cycles.regress   cyfactor.regress \
     danzwill.regress  decimal.regress  defintef.regress defintrf.regress \
     derham.regress    dfloat.regress   dhtri.regress    divisor.regress \
-    dmp.regress       dpol.regress     easter.regress   efi.regress \
+    dmp.regress       dpol.regress     e1.regress       easter.regress  \
+    efi.regress \
     eigen.regress     elemfun.regress  elemnum.regress  elfuts.regress \
     elt.regress       eq.regress       eqtbl.regress    equation2.regress \
     equation.regress  evalex.regress   eval.regress     exdiff.regress \
@@ -538,7 +539,7 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/drawalg.input  ${OUT}/drawcfn.input \
        ${OUT}/drawcfun.input ${OUT}/drawcurv.input \
        ${OUT}/draw.input     ${OUT}/drawcx.input     ${OUT}/drawex.input \
-       ${OUT}/drawpoly.input ${OUT}/drawx.input    \
+       ${OUT}/drawpoly.input ${OUT}/drawx.input      ${OUT}/e1.input     \
        ${OUT}/easter.input   ${OUT}/efi.input        ${OUT}/egg.input      \
        ${OUT}/eigen.input \
        ${OUT}/elemfun.input  ${OUT}/elemnum.input    ${OUT}/elfuts.input \
@@ -772,7 +773,8 @@ DOCFILES= \
   ${DOC}/e04fdf.input.dvi      ${DOC}/e04gcf.input.dvi     \
   ${DOC}/e04jaf.input.dvi      ${DOC}/e04mbf.input.dvi     \
   ${DOC}/e04naf.input.dvi      ${DOC}/e04ucf.input.dvi     \
-  ${DOC}/e04ycf.input.dvi      ${DOC}/easter.input.dvi     \
+  ${DOC}/e04ycf.input.dvi      ${DOC}/e1.input.dvi         \
+  ${DOC}/easter.input.dvi     \
   ${DOC}/ecfact.as.dvi         ${DOC}/efi.input.dvi        \
   ${DOC}/egg.input.dvi         ${DOC}/eigen.input.dvi      \
   ${DOC}/elemfun.input.dvi     ${DOC}/elemnum.input.dvi    \
diff --git a/src/input/e1.input.pamphlet b/src/input/e1.input.pamphlet
new file mode 100644
index 0000000..0c24a64
--- /dev/null
+++ b/src/input/e1.input.pamphlet
@@ -0,0 +1,1223 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input e1.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This is a regression test for E1(x)
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool e1.output
+)set message test on
+)set message auto off
+)clear all
+
+@
+Here we enter the value of Gamma
+<<*>>=
+--S 1 of 6
+G:DFLOAT:=0.577215664901532860606512::DFLOAT
+--R
+--R   (1)  0.57721566490153287
+--R                                                            Type: DoubleFloat
+--E 1
+@
+
+Since the E1 function varies over a couple hundred orders
+of magnitude it is necessary to scale the computations to
+put the results into a reasonable range. We use the function
+listed in 
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp238
+<<*>>=
+--S 2 of 6
+f(x)==x^-1 * (E1(x) + log(x) + G)
+--R                                                                   Type: Void
+--E 2
+@
+
+This table computes the Exponential Integral $E1(x)$ for $x$
+in 0.01 to 0.50. Column 1 is the point of evaluation, column 2
+is the reference value of $E1(x)$ from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp238
+
+<<*>>=
+--S 3 of 6
+[[0.01,0.9975055452, f(0.01), f(0.01)-0.9975055452],_
+[0.02,0.9950221392, f(0.02), f(0.02)-0.9950221392],_
+[0.03,0.9925497201, f(0.03), f(0.03)-0.9925497201],_
+[0.04,0.9900882265, f(0.04), f(0.04)-0.9900882265],_
+[0.05,0.9876375971, f(0.05), f(0.05)-0.9876375971],_
+[0.06,0.9851977714, f(0.06), f(0.06)-0.9851977714],_
+[0.07,0.9827686889, f(0.07), f(0.07)-0.9827686889],_
+[0.08,0.9803502898, f(0.08), f(0.08)-0.9803502898],_
+[0.09,0.9779425142, f(0.09), f(0.09)-0.9779425142],_
+[0.10,0.9755453033, f(0.10), f(0.10)-0.9755453033],_
+[0.11,0.9731585980, f(0.11), f(0.11)-0.9731585980],_
+[0.12,0.9707823399, f(0.12), f(0.12)-0.9707823399],_
+[0.13,0.9684164710, f(0.13), f(0.13)-0.9684164710],_
+[0.14,0.9660609336, f(0.14), f(0.14)-0.9660609336],_
+[0.15,0.9637156702, f(0.15), f(0.15)-0.9637156702],_
+[0.16,0.9613806240, f(0.16), f(0.16)-0.9613806240],_
+[0.17,0.9590557383, f(0.17), f(0.17)-0.9590557383],_
+[0.18,0.9567409569, f(0.18), f(0.18)-0.9567409569],_
+[0.19,0.9544362237, f(0.19), f(0.19)-0.9544362237],_
+[0.20,0.9521414833, f(0.20), f(0.20)-0.9521414833],_
+[0.21,0.9498566804, f(0.21), f(0.21)-0.9498566804],_
+[0.22,0.9475817603, f(0.22), f(0.22)-0.9475817603],_
+[0.23,0.9453166684, f(0.23), f(0.23)-0.9453166684],_
+[0.24,0.9430613506, f(0.24), f(0.24)-0.9430613506],_
+[0.25,0.9408157528, f(0.25), f(0.25)-0.9408157528],_
+[0.26,0.9385798221, f(0.26), f(0.26)-0.9385798221],_
+[0.27,0.9363535046, f(0.27), f(0.27)-0.9363535046],_
+[0.28,0.9341367481, f(0.28), f(0.28)-0.9341367481],_
+[0.29,0.9319294997, f(0.29), f(0.29)-0.9319294997],_
+[0.30,0.9297317075, f(0.30), f(0.30)-0.9297317075],_
+[0.31,0.9275433196, f(0.31), f(0.31)-0.9275433196],_
+[0.32,0.9253642845, f(0.32), f(0.32)-0.9253642845],_
+[0.33,0.9231945510, f(0.33), f(0.33)-0.9231945510],_
+[0.34,0.9210340684, f(0.34), f(0.34)-0.9210340684],_
+[0.35,0.9188827858, f(0.35), f(0.35)-0.9188827858],_
+[0.36,0.9167406533, f(0.36), f(0.36)-0.9167406533],_
+[0.37,0.9146076209, f(0.37), f(0.37)-0.9146076209],_
+[0.38,0.9124836388, f(0.38), f(0.38)-0.9124836388],_
+[0.39,0.9103686582, f(0.39), f(0.39)-0.9103686582],_
+[0.40,0.9082626297, f(0.40), f(0.40)-0.9082626297],_
+[0.41,0.9061655048, f(0.41), f(0.41)-0.9061655048],_
+[0.42,0.9040772350, f(0.42), f(0.42)-0.9040772350],_
+[0.43,0.9019977725, f(0.43), f(0.43)-0.9019977725],_
+[0.44,0.8999270693, f(0.44), f(0.44)-0.8999270693],_
+[0.45,0.8978650778, f(0.45), f(0.45)-0.8978650778],_
+[0.46,0.8958117511, f(0.46), f(0.46)-0.8958117511],_
+[0.47,0.8937670423, f(0.47), f(0.47)-0.8937670423],_
+[0.48,0.8917309048, f(0.48), f(0.48)-0.8917309048],_
+[0.49,0.8897032920, f(0.49), f(0.49)-0.8897032920],_
+[0.50,0.8876841584, f(0.50), f(0.50)-0.8876841584]]
+--R 
+--R   Compiling function f with type Float -> DoubleFloat 
+--R
+--R   (3)
+--R   [[1.0E-2,0.99750554520000001,0.99750554515544154,- 4.455846802642327E-11],
+--R    [2.0E-2,0.99502213920000004,0.99502213915481641,- 4.5183634611589696E-11],
+--R
+--R     [2.9999999999999999E-2, 0.99254972009999998, 0.99254972009439713,
+--R      - 5.602851516073315E-12]
+--R     ,
+--R
+--R     [4.0000000000000001E-2, 0.99008822649999995, 0.99008822646530492,
+--R      - 3.4695024631048454E-11]
+--R     ,
+--R
+--R     [5.0000000000000003E-2, 0.98763759709999999, 0.98763759715033261,
+--R      5.0332626955196247E-11]
+--R     ,
+--R
+--R     [5.9999999999999998E-2, 0.98519777139999998, 0.98519777142131459,
+--R      2.1314616738266068E-11]
+--R     ,
+--R
+--R     [7.0000000000000007E-2, 0.98276868890000002, 0.98276868893648617,
+--R      3.648614743667622E-11]
+--R     ,
+--R
+--R     [8.0000000000000002E-2, 0.98035028980000005, 0.98035028973773719,
+--R      - 6.2262861533213254E-11]
+--R     ,
+--R
+--R     [8.9999999999999997E-2, 0.9779425142, 0.97794251424804735,
+--R      4.8047343881307825E-11]
+--R     ,
+--R
+--R     [0.10000000000000001, 0.9755453033, 0.97554530326877553,
+--R      - 3.1224467456070215E-11]
+--R     ,
+--R    [0.11,0.97315859800000004,0.97315859797713988,- 2.2860158210846748E-11],
+--R    [0.12,0.97078233989999996,0.97078233992354002,2.3540058791127194E-11],
+--R    [0.13,0.96841647099999995,0.96841647102903816,2.9038216275978357E-11],
+--R
+--R     [0.14000000000000001, 0.9660609336, 0.96606093358279166,
+--R      - 1.7208345859387464E-11]
+--R     ,
+--R
+--R     [0.14999999999999999, 0.96371567020000004, 0.96371567023951998,
+--R      3.9519942873766922E-11]
+--R     ,
+--R    [0.16,0.96138062400000002,0.96138062401698243,1.6982415473876245E-11],
+--R
+--R     [0.17000000000000001, 0.95905573830000002, 0.95905573829349,
+--R      - 6.5100147494945304E-12]
+--R     ,
+--R
+--R     [0.17999999999999999, 0.95674095690000005, 0.9567409568054186,
+--R      - 9.4581453780051561E-11]
+--R     ,
+--R    [0.19,0.95443622370000003,0.95443622364474956,- 5.525047086507584E-11],
+--R
+--R     [0.20000000000000001, 0.95214148330000004, 0.9521414832566294,
+--R      - 4.3370640412376815E-11]
+--R     ,
+--R
+--R     [0.20999999999999999, 0.94985668040000004, 0.94985668043693772,
+--R      3.6937675140791271E-11]
+--R     ,
+--R    [0.22,0.94758176029999996,0.94758176032988584,2.9885871555279664E-11],
+--R
+--R     [0.23000000000000001, 0.94531666839999995, 0.9453166684256229,
+--R      2.562294820762645E-11]
+--R     ,
+--R
+--R     [0.23999999999999999, 0.94306135059999996, 0.94306135055786067,
+--R      - 4.2139292055765054E-11]
+--R     ,
+--R    [0.25,0.94081575279999996,0.94081575290152264,1.015226791523105E-10],
+--R
+--R     [0.26000000000000001, 0.93857982210000002, 0.93857982197039913,
+--R      - 1.2960088557889549E-10]
+--R     ,
+--R
+--R     [0.27000000000000002, 0.9363535046, 0.93635350461483224,
+--R      1.4832246542084704E-11]
+--R     ,
+--R
+--R     [0.28000000000000003, 0.93413674810000003, 0.93413674801940527,
+--R      - 8.0594753093521376E-11]
+--R     ,
+--R
+--R     [0.28999999999999998, 0.93192949970000005, 0.93192949970065808,
+--R      6.5802918669533028E-13]
+--R     ,
+--R
+--R     [0.29999999999999999, 0.9297317075, 0.92973170750481327,
+--R      4.8132609009599037E-12]
+--R     ,
+--R    [0.31,0.92754331960000003,0.92754331960551961,5.5195847892264283E-12],
+--R
+--R     [0.32000000000000001, 0.92536428449999997, 0.92536428450162023,
+--R      1.6202594821379535E-12]
+--R     ,
+--R
+--R     [0.33000000000000002, 0.92319455100000003, 0.92319455101492243,
+--R      1.4922396651684267E-11]
+--R     ,
+--R
+--R     [0.34000000000000002, 0.92103406840000002, 0.92103406828799361,
+--R      - 1.1200640415154339E-10]
+--R     ,
+--R
+--R     [0.34999999999999998, 0.91888278580000005, 0.91888278578197524,
+--R      - 1.8024803871696804E-11]
+--R     ,
+--R
+--R     [0.35999999999999999, 0.91674065329999999, 0.91674065327439791,
+--R      - 2.5602076014763497E-11]
+--R     ,
+--R    [0.37,0.91460762090000003,0.91460762085703573,- 4.2964298785364008E-11],
+--R    [0.38,0.91248363880000005,0.91248363893375239,1.3375234253487633E-10],
+--R
+--R     [0.39000000000000001, 0.91036865820000001, 0.91036865821837942,
+--R      1.8379409105762079E-11]
+--R     ,
+--R
+--R     [0.40000000000000002, 0.90826262970000005, 0.90826262973260075,
+--R      3.2600699917395559E-11]
+--R     ,
+--R
+--R     [0.40999999999999998, 0.90616550480000002, 0.90616550480385905,
+--R      3.8590242112945816E-12]
+--R     ,
+--R
+--R     [0.41999999999999998, 0.90407723500000003, 0.90407723506326909,
+--R      6.3269056660431033E-11]
+--R     ,
+--R
+--R     [0.42999999999999999, 0.90199777250000002, 0.90199777244355628,
+--R      - 5.6443738571942959E-11]
+--R     ,
+--R    [0.44,0.89992706929999999,0.89992706917700038,- 1.2299961049677677E-10],
+--R
+--R     [0.45000000000000001, 0.89786507780000002, 0.8978650777934013,
+--R      - 6.5987215691620804E-12]
+--R     ,
+--R
+--R     [0.46000000000000002, 0.8958117511, 0.89581175111805511,
+--R      1.8055112960269071E-11]
+--R     ,
+--R
+--R     [0.46999999999999997, 0.89376704230000004, 0.89376704226974857,
+--R      - 3.0251467997288728E-11]
+--R     ,
+--R
+--R     [0.47999999999999998, 0.89173090479999995, 0.89173090465876237,
+--R      - 1.412375771892016E-10]
+--R     ,
+--R
+--R     [0.48999999999999999, 0.88970329199999998, 0.88970329198489473,
+--R      - 1.510525038384003E-11]
+--R     ,
+--R    [0.5,0.88768415840000003,0.88768415823549685,- 1.6450318884864146E-10]]
+--R                                                  Type: List List DoubleFloat
+--E 3
+@
+
+This table computes the Exponential Integral $E1(x)$ for $x$
+in 0.50 to 2.00. Column 1 is the point of evaluation, column 2
+is the reference value of $E1(x)$ from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp239-241
+<<*>>=
+--S 4 of 6
+[[0.50, 0.559773595, E1(0.50), E1(0.50)-0.559773595],_
+[0.51, 0.547822352, E1(0.51), E1(0.51)-0.547822352],_
+[0.52, 0.536219798, E1(0.52), E1(0.52)-0.536219798],_
+[0.53, 0.524951510, E1(0.53), E1(0.53)-0.524951510],_
+[0.54, 0.514003886, E1(0.54), E1(0.54)-0.514003886],_
+[0.55, 0.503364081, E1(0.55), E1(0.55)-0.503364081],_
+[0.56, 0.493019959, E1(0.56), E1(0.56)-0.493019959],_
+[0.57, 0.482960034, E1(0.57), E1(0.57)-0.482960034],_
+[0.58, 0.473173433, E1(0.58), E1(0.58)-0.473173433],_
+[0.59, 0.463649849, E1(0.59), E1(0.59)-0.463649849],_
+[0.60, 0.454379503, E1(0.60), E1(0.60)-0.454379503],_
+[0.61, 0.445353112, E1(0.61), E1(0.61)-0.445353112],_
+[0.62, 0.436561854, E1(0.62), E1(0.62)-0.436561854],_
+[0.63, 0.427997338, E1(0.63), E1(0.63)-0.427997338],_
+[0.64, 0.419651581, E1(0.64), E1(0.64)-0.419651581],_
+[0.65, 0.411516976, E1(0.65), E1(0.65)-0.411516976],_
+[0.66, 0.403586275, E1(0.66), E1(0.66)-0.403586275],_
+[0.67, 0.395852563, E1(0.67), E1(0.67)-0.395852563],_
+[0.68, 0.388309243, E1(0.68), E1(0.68)-0.388309243],_
+[0.69, 0.380950010, E1(0.69), E1(0.69)-0.380950010],_
+[0.70, 0.373768843, E1(0.70), E1(0.70)-0.373768843],_
+[0.71, 0.366759981, E1(0.71), E1(0.71)-0.366759981],_
+[0.72, 0.359917914, E1(0.72), E1(0.72)-0.359917914],_
+[0.73, 0.353237364, E1(0.73), E1(0.73)-0.353237364],_
+[0.74, 0.346713279, E1(0.74), E1(0.74)-0.346713279],_
+[0.75, 0.340340813, E1(0.75), E1(0.75)-0.340340813],_
+[0.76, 0.334115321, E1(0.76), E1(0.76)-0.334115321],_
+[0.77, 0.328032346, E1(0.77), E1(0.77)-0.328032346],_
+[0.78, 0.322087610, E1(0.78), E1(0.78)-0.322087610],_
+[0.79, 0.316277004, E1(0.79), E1(0.79)-0.316277004],_
+[0.80, 0.310596579, E1(0.80), E1(0.80)-0.310596579],_
+[0.81, 0.305042539, E1(0.81), E1(0.81)-0.305042539],_
+[0.82, 0.299611236, E1(0.82), E1(0.82)-0.299611236],_
+[0.83, 0.294299155, E1(0.83), E1(0.83)-0.294299155],_
+[0.84, 0.289102918, E1(0.84), E1(0.84)-0.289102918],_
+[0.85, 0.284019269, E1(0.85), E1(0.85)-0.284019269],_
+[0.86, 0.279045070, E1(0.86), E1(0.86)-0.279045070],_
+[0.87, 0.274177301, E1(0.87), E1(0.87)-0.274177301],_
+[0.88, 0.269413046, E1(0.88), E1(0.88)-0.269413046],_
+[0.89, 0.264749496, E1(0.89), E1(0.89)-0.264749496],_
+[0.90, 0.260183939, E1(0.90), E1(0.90)-0.260183939],_
+[0.91, 0.255713758, E1(0.91), E1(0.91)-0.255713758],_
+[0.92, 0.251336425, E1(0.92), E1(0.92)-0.251336425],_
+[0.93, 0.247049501, E1(0.93), E1(0.93)-0.247049501],_
+[0.94, 0.242850627, E1(0.94), E1(0.94)-0.242850627],_
+[0.95, 0.238737524, E1(0.95), E1(0.95)-0.238737524],_
+[0.96, 0.234707988, E1(0.96), E1(0.96)-0.234707988],_
+[0.97, 0.230759890, E1(0.97), E1(0.97)-0.230759890],_
+[0.98, 0.226891167, E1(0.98), E1(0.98)-0.226891167],_
+[0.99, 0.223099826, E1(0.99), E1(0.99)-0.223099826],_
+[1.00, 0.219383934, E1(1.00), E1(1.00)-0.219383934],_
+[1.01, 0.215741624, E1(1.01), E1(1.01)-0.215741624],_
+[1.02, 0.212171083, E1(1.02), E1(1.02)-0.212171083],_
+[1.03, 0.208670559, E1(1.03), E1(1.03)-0.208670559],_
+[1.04, 0.205238352, E1(1.04), E1(1.04)-0.205238352],_
+[1.05, 0.201872813, E1(1.05), E1(1.05)-0.201872813],_
+[1.06, 0.198572347, E1(1.06), E1(1.06)-0.198572347],_
+[1.07, 0.195335403, E1(1.07), E1(1.07)-0.195335403],_
+[1.08, 0.192160479, E1(1.08), E1(1.08)-0.192160479],_
+[1.09, 0.189046118, E1(1.09), E1(1.09)-0.189046118],_
+[1.10, 0.185990905, E1(1.10), E1(1.10)-0.185990905],_
+[1.11, 0.182993465, E1(1.11), E1(1.11)-0.182993465],_
+[1.12, 0.180052467, E1(1.12), E1(1.12)-0.180052467],_
+[1.13, 0.177166615, E1(1.13), E1(1.13)-0.177166615],_
+[1.14, 0.174334651, E1(1.14), E1(1.14)-0.174334651],_
+[1.15, 0.171555354, E1(1.15), E1(1.15)-0.171555354],_
+[1.16, 0.168827535, E1(1.16), E1(1.16)-0.168827535],_
+[1.17, 0.166150040, E1(1.17), E1(1.17)-0.166150040],_
+[1.18, 0.163521748, E1(1.18), E1(1.18)-0.163521748],_
+[1.19, 0.160941567, E1(1.19), E1(1.19)-0.160941567],_
+[1.20, 0.158408437, E1(1.20), E1(1.20)-0.158408437],_
+[1.21, 0.155921324, E1(1.21), E1(1.21)-0.155921324],_
+[1.22, 0.153479226, E1(1.22), E1(1.22)-0.153479226],_
+[1.23, 0.151081164, E1(1.23), E1(1.23)-0.151081164],_
+[1.24, 0.148726188, E1(1.24), E1(1.24)-0.148726188],_
+[1.25, 0.146413373, E1(1.25), E1(1.25)-0.146413373],_
+[1.26, 0.144141815, E1(1.26), E1(1.26)-0.144141815],_
+[1.27, 0.141910639, E1(1.27), E1(1.27)-0.141910639],_
+[1.28, 0.139718989, E1(1.28), E1(1.28)-0.139718989],_
+[1.29, 0.137566032, E1(1.29), E1(1.29)-0.137566032],_
+[1.30, 0.135450958, E1(1.30), E1(1.30)-0.135450958],_
+[1.31, 0.133372975, E1(1.31), E1(1.31)-0.133372975],_
+[1.32, 0.131331314, E1(1.32), E1(1.32)-0.131331314],_
+[1.33, 0.129325224, E1(1.33), E1(1.33)-0.129325224],_
+[1.34, 0.127353972, E1(1.34), E1(1.34)-0.127353972],_
+[1.35, 0.125416844, E1(1.35), E1(1.35)-0.125416844],_
+[1.36, 0.123513146, E1(1.36), E1(1.36)-0.123513146],_
+[1.37, 0.121642198, E1(1.37), E1(1.37)-0.121642198],_
+[1.38, 0.119803337, E1(1.38), E1(1.38)-0.119803337],_
+[1.39, 0.117995919, E1(1.39), E1(1.39)-0.117995919],_
+[1.40, 0.116219313, E1(1.40), E1(1.40)-0.116219313],_
+[1.41, 0.114472903, E1(1.41), E1(1.41)-0.114472903],_
+[1.42, 0.112756090, E1(1.42), E1(1.42)-0.112756090],_
+[1.43, 0.111068287, E1(1.43), E1(1.43)-0.111068287],_
+[1.44, 0.109408923, E1(1.44), E1(1.44)-0.109408923],_
+[1.45, 0.107777440, E1(1.45), E1(1.45)-0.107777440],_
+[1.46, 0.106173291, E1(1.46), E1(1.46)-0.106173291],_
+[1.47, 0.104595946, E1(1.47), E1(1.47)-0.104595946],_
+[1.48, 0.103044882, E1(1.48), E1(1.48)-0.103044882],_
+[1.49, 0.101519593, E1(1.49), E1(1.49)-0.101519593],_
+[1.50, 0.100019582, E1(1.50), E1(1.50)-0.100019582],_
+[1.51, 0.098544365, E1(1.51), E1(1.51)-0.098544365],_
+[1.52, 0.097093466, E1(1.52), E1(1.52)-0.097093466],_
+[1.53, 0.095666424, E1(1.53), E1(1.53)-0.095666424],_
+[1.54, 0.094262786, E1(1.54), E1(1.54)-0.094262786],_
+[1.55, 0.092882108, E1(1.55), E1(1.55)-0.092882108],_
+[1.56, 0.091523960, E1(1.56), E1(1.56)-0.091523960],_
+[1.57, 0.090187917, E1(1.57), E1(1.57)-0.090187917],_
+[1.58, 0.088873566, E1(1.58), E1(1.58)-0.088873566],_
+[1.59, 0.087580504, E1(1.59), E1(1.59)-0.087580504],_
+[1.60, 0.086308334, E1(1.60), E1(1.60)-0.086308334],_
+[1.61, 0.085056670, E1(1.61), E1(1.61)-0.085056670],_
+[1.62, 0.083825133, E1(1.62), E1(1.62)-0.083825133],_
+[1.63, 0.082613354, E1(1.63), E1(1.63)-0.082613354],_
+[1.64, 0.081420970, E1(1.64), E1(1.64)-0.081420970],_
+[1.65, 0.080247627, E1(1.65), E1(1.65)-0.080247627],_
+[1.66, 0.079092978, E1(1.66), E1(1.66)-0.079092978],_
+[1.67, 0.077956684, E1(1.67), E1(1.67)-0.077956684],_
+[1.68, 0.076838412, E1(1.68), E1(1.68)-0.076838412],_
+[1.69, 0.075737839, E1(1.69), E1(1.69)-0.075737839],_
+[1.70, 0.074654644, E1(1.70), E1(1.70)-0.074654644],_
+[1.71, 0.073588518, E1(1.71), E1(1.71)-0.073588518],_
+[1.72, 0.072539154, E1(1.72), E1(1.72)-0.072539154],_
+[1.73, 0.071506255, E1(1.73), E1(1.73)-0.071506255],_
+[1.74, 0.070489527, E1(1.74), E1(1.74)-0.070489527],_
+[1.75, 0.069488685, E1(1.75), E1(1.75)-0.069488685],_
+[1.76, 0.068503447, E1(1.76), E1(1.76)-0.068503447],_
+[1.77, 0.067533539, E1(1.77), E1(1.77)-0.067533539],_
+[1.78, 0.066578691, E1(1.78), E1(1.78)-0.066578691],_
+[1.79, 0.065638641, E1(1.79), E1(1.79)-0.065638641],_
+[1.80, 0.064713129, E1(1.80), E1(1.80)-0.064713129],_
+[1.81, 0.063801903, E1(1.81), E1(1.81)-0.063801903],_
+[1.82, 0.062904715, E1(1.82), E1(1.82)-0.062904715],_
+[1.83, 0.062021320, E1(1.83), E1(1.83)-0.062021320],_
+[1.84, 0.061151482, E1(1.84), E1(1.84)-0.061151482],_
+[1.85, 0.060294967, E1(1.85), E1(1.85)-0.060294967],_
+[1.86, 0.059451545, E1(1.86), E1(1.86)-0.059451545],_
+[1.87, 0.058620994, E1(1.87), E1(1.87)-0.058620994],_
+[1.88, 0.057803091, E1(1.88), E1(1.88)-0.057803091],_
+[1.89, 0.056997623, E1(1.89), E1(1.89)-0.056997623],_
+[1.90, 0.056204378, E1(1.90), E1(1.90)-0.056204378],_
+[1.91, 0.055423149, E1(1.91), E1(1.91)-0.055423149],_
+[1.92, 0.054653731, E1(1.92), E1(1.92)-0.054653731],_
+[1.93, 0.053895927, E1(1.93), E1(1.93)-0.053895927],_
+[1.94, 0.053149540, E1(1.94), E1(1.94)-0.053149540],_
+[1.95, 0.052414380, E1(1.95), E1(1.95)-0.052414380],_
+[1.96, 0.051690257, E1(1.96), E1(1.96)-0.051690257],_
+[1.97, 0.050976988, E1(1.97), E1(1.97)-0.050976988],_
+[1.98, 0.050274392, E1(1.98), E1(1.98)-0.050274392],_
+[1.99, 0.049582291, E1(1.99), E1(1.99)-0.049582291],_
+[2.00, 0.048900511, E1(2.00), E1(2.00)-0.048900511]]
+--R 
+--R
+--R   (4)
+--R   [[0.5,0.55977359500000001,0.55977359477616084,- 2.2383916942203541E-10],
+--R
+--R     [0.51000000000000001, 0.54782235199999996, 0.54782235178082872,
+--R      - 2.1917123671499894E-10]
+--R     ,
+--R
+--R     [0.52000000000000002, 0.53621979799999997, 0.53621979784563623,
+--R      - 1.5436374400934483E-10]
+--R     ,
+--R
+--R     [0.53000000000000003, 0.52495150999999995, 0.52495151011486541,
+--R      1.148654504845581E-10]
+--R     ,
+--R
+--R     [0.54000000000000004, 0.51400388600000002, 0.51400388570224909,
+--R      - 2.9775093501882566E-10]
+--R     ,
+--R
+--R     [0.55000000000000004, 0.50336408099999996, 0.50336408139239386,
+--R      3.9239389515444145E-10]
+--R     ,
+--R
+--R     [0.56000000000000005, 0.49301995900000001, 0.49301995877649291,
+--R      - 2.2350710171537003E-10]
+--R     ,
+--R
+--R     [0.56999999999999995, 0.48296003399999998, 0.48296003424511297,
+--R      2.451129854641465E-10]
+--R     ,
+--R
+--R     [0.57999999999999996, 0.47317343299999998, 0.47317343333112627,
+--R      3.3112629305165342E-10]
+--R     ,
+--R
+--R     [0.58999999999999997, 0.463649849, 0.46364984895652972,
+--R      - 4.3470282928836923E-11]
+--R     ,
+--R
+--R     [0.59999999999999998, 0.45437950300000002, 0.45437950318940223,
+--R      1.8940221613306107E-10]
+--R     ,
+--R    [0.60999999999999999,0.445353112,0.44535311216282059,1.628205903436708E-10],
+--R    [0.62,0.43656185400000003,0.43656185384719148,- 1.5280854359644991E-10],
+--R    [0.63,0.427997338,0.42799733840201848,4.0201847406606817E-10],
+--R
+--R     [0.64000000000000001, 0.419651581, 0.41965158086333326,
+--R      - 1.366667334856686E-10]
+--R     ,
+--R
+--R     [0.65000000000000002, 0.41151697599999998, 0.41151697594947956,
+--R      - 5.0520421179811592E-11]
+--R     ,
+--R
+--R     [0.66000000000000003, 0.40358627499999999, 0.40358627479116588,
+--R      - 2.088341166661678E-10]
+--R     ,
+--R
+--R     [0.67000000000000004, 0.39585256299999999, 0.39585256341213687,
+--R      4.1213688017904815E-10]
+--R     ,
+--R
+--R     [0.68000000000000005, 0.38830924300000003, 0.38830924280482559,
+--R      - 1.9517443217154096E-10]
+--R     ,
+--R
+--R     [0.68999999999999995, 0.38095001000000001, 0.38095001046125104,
+--R      4.6125103736471829E-10]
+--R     ,
+--R
+--R     [0.69999999999999996, 0.37376884300000002, 0.37376884323350923,
+--R      2.3350921196652052E-10]
+--R     ,
+--R
+--R     [0.70999999999999996, 0.36675998100000001, 0.36675998141067723,
+--R      4.1067721445742222E-10]
+--R     ,
+--R
+--R     [0.71999999999999997, 0.35991791400000001, 0.35991791391003464,
+--R      - 8.9965368488265085E-11]
+--R     ,
+--R    [0.72999999999999998,0.353237364,0.35323736449036641,4.9036641414090809E-10]
+--R     ,
+--R
+--R     [0.73999999999999999, 0.34671327899999999, 0.34671327890389447,
+--R      - 9.6105512437105745E-11]
+--R     ,
+--R    [0.75,0.34034081300000002,0.34034081291123008,- 8.8769935846499948E-11],
+--R
+--R     [0.76000000000000001, 0.33411532100000002, 0.33411532109074837,
+--R      9.0748353276381977E-11]
+--R     ,
+--R
+--R     [0.77000000000000002, 0.32803234599999997, 0.3280323463800649,
+--R      3.8006492397713032E-10]
+--R     ,
+--R
+--R     [0.78000000000000003, 0.32208761000000002, 0.32208761029292271,
+--R      2.9292268610703331E-10]
+--R     ,
+--R
+--R     [0.79000000000000004, 0.31627700399999997, 0.31627700375985612,
+--R      - 2.4014384925052923E-10]
+--R     ,
+--R
+--R     [0.80000000000000004, 0.31059657899999998, 0.31059657854554301,
+--R      - 4.5445697205437341E-10]
+--R     ,
+--R    [0.81000000000000005,0.305042539,0.30504253919985258,1.9985257893040398E-10]
+--R     ,
+--R
+--R     [0.81999999999999995, 0.299611236, 0.29961123550328894,
+--R      - 4.967110611708847E-10]
+--R     ,
+--R
+--R     [0.82999999999999996, 0.29429915499999998, 0.29429915537086676,
+--R      3.7086678172926213E-10]
+--R     ,
+--R
+--R     [0.83999999999999997, 0.28910291799999999, 0.28910291818146794,
+--R      1.8146795177642616E-10]
+--R     ,
+--R
+--R     [0.84999999999999998, 0.28401926900000002, 0.2840192685024614,
+--R      - 4.975386214134403E-10]
+--R     ,
+--R
+--R     [0.85999999999999999, 0.27904507000000001, 0.27904507018183955,
+--R      1.818395434227682E-10]
+--R     ,
+--R    [0.87,0.27417730099999998,0.27417730078237224,- 2.1762774915501382E-10],
+--R    [0.88,0.26941304599999999,0.26941304633432023,3.343202381600463E-10],
+--R
+--R     [0.89000000000000001, 0.26474949599999997, 0.26474949638510148,
+--R      3.8510150623949357E-10]
+--R     ,
+--R
+--R     [0.90000000000000002, 0.26018393899999998, 0.26018393932599954,
+--R      3.259995606796906E-10]
+--R     ,
+--R
+--R     [0.91000000000000003, 0.25571375800000001, 0.25571375797753926,
+--R      - 2.2460755477737848E-11]
+--R     ,
+--R
+--R     [0.92000000000000004, 0.25133642499999997, 0.25133642541656154,
+--R      4.1656156302138925E-10]
+--R     ,
+--R    [0.93000000000000005,0.247049501,0.24704950102931605,2.9316049587890802E-11]
+--R     ,
+--R
+--R     [0.93999999999999995, 0.24285062700000001, 0.24285062677606084,
+--R      - 2.2393917276097852E-10]
+--R     ,
+--R
+--R     [0.94999999999999996, 0.23873752400000001, 0.23873752365373468,
+--R      - 3.4626532197101767E-10]
+--R     ,
+--R
+--R     [0.95999999999999996, 0.23470798800000001, 0.23470798834425491,
+--R      3.4425490236245082E-10]
+--R     ,
+--R    [0.96999999999999997,0.23075989,0.23075989003689171,3.6891711907571789E-11],
+--R    [0.97999999999999998,0.226891167,0.22689116741400336,4.1400335937247235E-10]
+--R     ,
+--R
+--R     [0.98999999999999999, 0.223099826, 0.22309982579017718,
+--R      - 2.0982282578074773E-10]
+--R     ,
+--R    [1.,0.219383934,0.21938393439552029,3.9552028319178589E-10],
+--R    [1.01,0.21574162399999999,0.21574162379448991,- 2.0551008117486447E-10],
+--R    [1.02,0.21217108300000001,0.2121710834322488,4.3224879231473778E-10],
+--R    [1.03,0.20867055900000001,0.20867055930107367,3.0107366599807506E-10],
+--R    [1.04,0.20523835200000001,0.20523835171985597,- 2.8014404684917338E-10],
+--R    [1.05,0.20187281300000001,0.20187281322019657,2.2019655543381589E-10],
+--R
+--R     [1.0600000000000001, 0.19857234700000001, 0.19857234653302808,
+--R      - 4.6697193334388487E-10]
+--R     ,
+--R
+--R     [1.0700000000000001, 0.19533540299999999, 0.19533540267009863,
+--R      - 3.2990135623300887E-10]
+--R     ,
+--R    [1.0800000000000001,0.192160479,0.19216047909501838,9.5018382051392791E-11],
+--R
+--R     [1.0900000000000001, 0.18904611800000001, 0.18904611797891213,
+--R      - 2.1087881441061995E-11]
+--R     ,
+--R
+--R     [1.1000000000000001, 0.18599090500000001, 0.18599090453604011,
+--R      - 4.6395989827807682E-10]
+--R     ,
+--R
+--R     [1.1100000000000001, 0.18299346499999999, 0.1829934654350395,
+--R      4.3503950442058681E-10]
+--R     ,
+--R
+--R     [1.1200000000000001, 0.18005246699999999, 0.18005246728171573,
+--R      2.8171573407398398E-10]
+--R     ,
+--R    [1.1299999999999999,0.177166615,0.17716661516956422,1.6956422377312208E-10],
+--R
+--R     [1.1399999999999999, 0.17433465100000001, 0.17433465129443812,
+--R      2.9443811278007104E-10]
+--R     ,
+--R
+--R     [1.1499999999999999, 0.17155535399999999, 0.1715553536299986,
+--R      - 3.7000139063714244E-10]
+--R     ,
+--R
+--R     [1.1599999999999999, 0.168827535, 0.16882753466078662,
+--R      - 3.3921337960762799E-10]
+--R     ,
+--R    [1.1699999999999999,0.16615004,0.16615004016994619,1.6994619600474437E-10],
+--R
+--R     [1.1799999999999999, 0.16352174799999999, 0.16352174807880468,
+--R      7.8804684999767005E-11]
+--R     ,
+--R
+--R     [1.1899999999999999, 0.16094156700000001, 0.1609415673356836,
+--R      3.3568359203428599E-10]
+--R     ,
+--R    [1.2,0.15840843700000001,0.15840843685146253,- 1.4853748786514132E-10],
+--R    [1.21,0.155921324,0.15592132447956802,4.7956802418092082E-10],
+--R    [1.22,0.153479226,0.15347922603818942,3.8189423845480519E-11],
+--R    [1.23,0.15108116399999999,0.15108116437265298,3.7265299179800593E-10],
+--R    [1.24,0.14872618800000001,0.14872618845599739,4.559973787454652E-10],
+--R    [1.25,0.14641337300000001,0.14641337252591019,- 4.7408982295493729E-10],
+--R    [1.26,0.14414181500000001,0.14414181525628297,2.5628296707047582E-10],
+--R    [1.27,0.14191063900000001,0.14191063896174164,- 3.8258368695309741E-11],
+--R    [1.28,0.13971898899999999,0.1397189888335964,- 1.6640358535546795E-10],
+--R    [1.29,0.137566032,0.13756603220574354,2.0574353332136752E-10],
+--R    [1.3,0.13545095800000001,0.13545095784912908,- 1.5087092686272285E-10],
+--R    [1.3100000000000001,0.133372975,0.13337297529345732,2.9345731400454156E-10],
+--R    [1.3200000000000001,0.131331314,0.13133131417489974,1.7489973358486566E-10],
+--R
+--R     [1.3300000000000001, 0.12932522399999999, 0.12932522360862764,
+--R      - 3.9137235119390823E-10]
+--R     ,
+--R
+--R     [1.3400000000000001, 0.12735397200000001, 0.12735397158504419,
+--R      - 4.1495581970529827E-10]
+--R     ,
+--R    [1.3500000000000001,0.125416844,0.12541684438866441,3.8866440621454501E-10],
+--R    [1.3600000000000001,0.123513146,0.12351314603863212,3.8632111398761992E-11],
+--R
+--R     [1.3700000000000001, 0.12164219800000001, 0.1216421977499248,
+--R      - 2.5007521053943549E-10]
+--R     ,
+--R    [1.3799999999999999,0.119803337,0.11980333741433752,4.1433752262509671E-10],
+--R    [1.3899999999999999,0.117995919,0.11799591910039325,1.0039324926935933E-10],
+--R
+--R     [1.3999999999999999, 0.116219313, 0.11621931257135804,
+--R      - 4.2864196914127461E-10]
+--R     ,
+--R
+--R     [1.4099999999999999, 0.114472903, 0.11447290282058709,
+--R      - 1.7941291508005719E-10]
+--R     ,
+--R    [1.4199999999999999,0.11275609,0.11275608962347,- 3.7653000162229944E-10],
+--R    [1.4299999999999999,0.111068287,0.11106828710526567,1.0526567117974395E-10],
+--R
+--R     [1.4399999999999999, 0.10940892300000001, 0.10940892332417007,
+--R      3.2417006579077423E-10]
+--R     ,
+--R    [1.45,0.10777744,0.10777743986897642,- 1.3102358087380139E-10],
+--R    [1.46,0.106173291,0.10617329147072579,4.7072579167917183E-10],
+--R    [1.47,0.104595946,0.10459594562777519,- 3.7222480653298362E-10],
+--R    [1.48,0.103044882,0.10304488224373387,2.4373386642295714E-10],
+--R    [1.49,0.10151959300000001,0.10151959327774779,2.7774778310618586E-10],
+--R    [1.5,0.100019582,0.10001958240663256,4.0663256095641032E-10],
+--R    [1.51,9.8544364999999995E-2,9.85443646983854E-2,- 3.0161459441124805E-10],
+--R    [1.52,9.7093466000000003E-2,9.7093466296618358E-2,2.9661835487804211E-10],
+--R    [1.53,9.5666424E-2,9.5666424115486592E-2,1.1548659251126026E-10],
+--R    [1.54,9.4262786000000001E-2,9.4262785544698136E-2,- 4.5530186565390096E-10],
+--R    [1.55,9.2882108000000005E-2,9.2882108164209165E-2,1.6420916015835729E-10],
+--R
+--R     [1.5600000000000001, 9.1523960000000001E-2, 9.152395946823666E-2,
+--R      - 5.3176334169346973E-10]
+--R     ,
+--R
+--R     [1.5700000000000001, 9.0187917000000006E-2, 9.0187916598222728E-2,
+--R      - 4.0177727811396835E-10]
+--R     ,
+--R
+--R     [1.5800000000000001, 8.8873566000000001E-2, 8.8873566084412048E-2,
+--R      8.441204679687786E-11]
+--R     ,
+--R
+--R     [1.5900000000000001, 8.7580504000000003E-2, 8.7580503595714843E-2,
+--R      - 4.0428516090429412E-10]
+--R     ,
+--R
+--R     [1.6000000000000001, 8.6308334E-2, 8.6308333697539708E-2,
+--R      - 3.0246029292246845E-10]
+--R     ,
+--R
+--R     [1.6100000000000001, 8.5056670000000001E-2, 8.5056669617302794E-2,
+--R      - 3.8269720725736533E-10]
+--R     ,
+--R
+--R     [1.6200000000000001, 8.3825132999999996E-2, 8.3825133017319142E-2,
+--R      1.7319146117245054E-11]
+--R     ,
+--R
+--R     [1.6299999999999999, 8.2613354E-2, 8.2613353774808662E-2,
+--R      - 2.25191337799302E-10]
+--R     ,
+--R
+--R     [1.6399999999999999, 8.1420969999999995E-2, 8.1420969768751517E-2,
+--R      - 2.3124847869926413E-10]
+--R     ,
+--R
+--R     [1.6499999999999999, 8.0247627000000002E-2, 8.0247626673343175E-2,
+--R      - 3.266568265880565E-10]
+--R     ,
+--R
+--R     [1.6599999999999999, 7.9092977999999994E-2, 7.9092977757806437E-2,
+--R      - 2.4219355687638E-10]
+--R     ,
+--R
+--R     [1.6699999999999999, 7.7956683999999998E-2, 7.7956683692333661E-2,
+--R      - 3.0766633685175293E-10]
+--R     ,
+--R
+--R     [1.6799999999999999, 7.6838411999999995E-2, 7.6838412359934938E-2,
+--R      3.5993494296171491E-10]
+--R     ,
+--R
+--R     [1.6899999999999999, 7.5737839000000001E-2, 7.5737838673983093E-2,
+--R      - 3.2601690791445037E-10]
+--R     ,
+--R    [1.7,7.4654644000000006E-2,7.4654644401252912E-2,4.0125290590165008E-10],
+--R    [1.71,7.3588518000000006E-2,7.358851799025623E-2,- 9.7437752311080317E-12],
+--R    [1.72,7.2539153999999995E-2,7.2539154404693273E-2,4.0469327888814632E-10],
+--R    [1.73,7.1506255000000005E-2,7.150625496183538E-2,- 3.8164624238667955E-11],
+--R    [1.74,7.0489526999999996E-2,7.0489527175668809E-2,1.7566881282959912E-10],
+--R    [1.75,6.9488684999999994E-2,6.9488684604638751E-2,- 3.9536124374350834E-10],
+--R    [1.76,6.8503446999999995E-2,6.8503446703828352E-2,- 2.9617164276629637E-10],
+--R    [1.77,6.7533539000000004E-2,6.7533538681429195E-2,- 3.1857080862174314E-10],
+--R    [1.78,6.6578690999999995E-2,6.657869135934702E-2,3.5934702435902466E-10],
+--R    [1.79,6.5638640999999998E-2,6.5638641037815026E-2,3.7815028886001301E-11],
+--R    [1.8,6.4713128999999994E-2,6.4713129363868749E-2,3.638687545715058E-10],
+--R
+--R     [1.8100000000000001, 6.3801902999999993E-2, 6.3801903203559385E-2,
+--R      2.0355939156502245E-10]
+--R     ,
+--R
+--R     [1.8200000000000001, 6.2904715E-2, 6.2904714517779237E-2,
+--R      - 4.8222076332038455E-10]
+--R     ,
+--R
+--R     [1.8300000000000001, 6.2021319999999998E-2, 6.2021320241580469E-2,
+--R      2.4158047090550028E-10]
+--R     ,
+--R
+--R     [1.8400000000000001, 6.1151482E-2, 6.1151482166870164E-2,
+--R      1.6687016352046058E-10]
+--R     ,
+--R
+--R     [1.8500000000000001, 6.0294966999999998E-2, 6.0294966828373431E-2,
+--R      - 1.7162656712477187E-10]
+--R     ,
+--R
+--R     [1.8600000000000001, 5.9451545000000001E-2, 5.9451545392755656E-2,
+--R      3.9275565438812166E-10]
+--R     ,
+--R
+--R     [1.8700000000000001, 5.8620994000000003E-2, 5.8620993550804079E-2,
+--R      - 4.4919592351311266E-10]
+--R     ,
+--R
+--R     [1.8799999999999999, 5.7803091000000001E-2, 5.7803091412567897E-2,
+--R      4.1256789651278325E-10]
+--R     ,
+--R
+--R     [1.8899999999999999, 5.6997622999999997E-2, 5.6997623405359299E-2,
+--R      4.0535930168061896E-10]
+--R     ,
+--R
+--R     [1.8999999999999999, 5.6204377999999999E-2, 5.6204378174534608E-2,
+--R      1.7453460898764206E-10]
+--R     ,
+--R
+--R     [1.9099999999999999, 5.5423148999999998E-2, 5.5423148486950402E-2,
+--R      - 5.1304959586273569E-10]
+--R     ,
+--R
+--R     [1.9199999999999999, 5.4653730999999997E-2, 5.4653731137026984E-2,
+--R      1.3702698697937166E-10]
+--R     ,
+--R
+--R     [1.9299999999999999, 5.3895927000000003E-2, 5.3895926855324849E-2,
+--R      - 1.4467515380145457E-10]
+--R     ,
+--R
+--R     [1.9399999999999999, 5.3149540000000002E-2, 5.3149540219563307E-2,
+--R      2.1956330503725141E-10]
+--R     ,
+--R    [1.95,5.2414380000000003E-2,5.2414379567998548E-2,- 4.3200145544153301E-10],
+--R    [1.96,5.1690257000000003E-2,5.1690256915094213E-2,- 8.4905790731504283E-11],
+--R    [1.97,5.0976988000000001E-2,5.0976987869409074E-2,- 1.3059092696110497E-10],
+--R    [1.98,5.0274392000000001E-2,5.0274391553638775E-2,- 4.4636122575880677E-10],
+--R    [1.99,4.9582291000000001E-2,4.9582290526736128E-2,- 4.732638725357674E-10],
+--R    [2.,4.8900511000000001E-2,4.8900510708061007E-2,- 2.9193899381274591E-10]]
+--R                                                  Type: List List DoubleFloat
+--E 4
+@
+
+Now that we move into larger numbers we need a new scaling function.
+<<*>>=
+--S 5 of 6
+g(x)==x * %e^x * E1(x)
+--R                                                                   Type: Void
+--E 5
+@
+
+And we compute the scaled value of E1(x) in the range 2.0 to 10.0
+from Abramowitz and Stegun,``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp242-243
+<<*>>=
+--S 6 of 6
+[[2.0,0.722657234,g(2.0),g(2.0)-0.722657234],_
+[2.1,0.730791502,g(2.1),g(2.1)-0.730791502],_
+[2.2,0.738431132,g(2.2),g(2.2)-0.738431132],_
+[2.3,0.745622149,g(2.3),g(2.3)-0.745622149],_
+[2.4,0.752404829,g(2.4),g(2.4)-0.752404829],_
+[2.5,0.758814592,g(2.5),g(2.5)-0.758814592],_
+[2.6,0.764882722,g(2.6),g(2.6)-0.764882722],_
+[2.7,0.770636987,g(2.7),g(2.7)-0.770636987],_
+[2.8,0.776102123,g(2.8),g(2.8)-0.776102123],_
+[2.9,0.781300252,g(2.9),g(2.9)-0.781300252],_
+[3.0,0.786251221,g(3.0),g(3.0)-0.786251221],_
+[3.1,0.790972800,g(3.1),g(3.1)-0.790972800],_
+[3.2,0.795481422,g(3.2),g(3.2)-0.795481422],_
+[3.3,0.799791408,g(3.3),g(3.3)-0.799791408],_
+[3.4,0.803916127,g(3.4),g(3.4)-0.803916127],_
+[3.5,0.807867661,g(3.5),g(3.5)-0.807867661],_
+[3.6,0.811657037,g(3.6),g(3.6)-0.811657037],_
+[3.7,0.815294342,g(3.7),g(3.7)-0.815294342],_
+[3.8,0.818788821,g(3.8),g(3.8)-0.818788821],_
+[3.9,0.822148967,g(3.9),g(3.9)-0.822148967],_
+[4.0,0.825382500,g(4.0),g(4.0)-0.825382500],_
+[4.1,0.828496926,g(4.1),g(4.1)-0.828496926],_
+[4.2,0.831498602,g(4.2),g(4.2)-0.831498602],_
+[4.3,0.834393794,g(4.3),g(4.3)-0.834393794],_
+[4.4,0.837188207,g(4.4),g(4.4)-0.837188207],_
+[4.5,0.839887144,g(4.5),g(4.5)-0.839887144],_
+[4.6,0.842495539,g(4.6),g(4.6)-0.842495539],_
+[4.7,0.845017971,g(4.7),g(4.7)-0.845017971],_
+[4.8,0.847458721,g(4.8),g(4.8)-0.847458721],_
+[4.9,0.849821778,g(4.9),g(4.9)-0.849821778],_
+[5.0,0.852110880,g(5.0),g(5.0)-0.852110880],_
+[5.1,0.854329519,g(5.1),g(5.1)-0.854329519],_
+[5.2,0.856480958,g(5.2),g(5.2)-0.856480958],_
+[5.3,0.858568275,g(5.3),g(5.3)-0.858568275],_
+[5.4,0.860594348,g(5.4),g(5.4)-0.860594348],_
+[5.5,0.862561885,g(5.5),g(5.5)-0.862561885],_
+[5.6,0.864473436,g(5.6),g(5.6)-0.864473436],_
+[5.7,0.866331399,g(5.7),g(5.7)-0.866331399],_
+[5.8,0.868138040,g(5.8),g(5.8)-0.868138040],_
+[5.9,0.869895494,g(5.9),g(5.9)-0.869895494],_
+[6.0,0.871605775,g(6.0),g(6.0)-0.871605775],_
+[6.1,0.873270793,g(6.1),g(6.1)-0.873270793],_
+[6.2,0.874892347,g(6.2),g(6.2)-0.874892347],_
+[6.3,0.876472150,g(6.3),g(6.3)-0.876472150],_
+[6.4,0.878011816,g(6.4),g(6.4)-0.878011816],_
+[6.5,0.879512881,g(6.5),g(6.5)-0.879512881],_
+[6.6,0.880976797,g(6.6),g(6.6)-0.880976797],_
+[6.7,0.882404955,g(6.7),g(6.7)-0.882404955],_
+[6.8,0.883798662,g(6.8),g(6.8)-0.883798662],_
+[6.9,0.885159176,g(6.9),g(6.9)-0.885159176],_
+[7.0,0.886487675,g(7.0),g(7.0)-0.886487675],_
+[7.1,0.887785294,g(7.1),g(7.1)-0.887785294],_
+[7.2,0.889053119,g(7.2),g(7.2)-0.889053119],_
+[7.3,0.890292173,g(7.3),g(7.3)-0.890292173],_
+[7.4,0.891503440,g(7.4),g(7.4)-0.891503440],_
+[7.5,0.892687854,g(7.5),g(7.5)-0.892687854],_
+[7.6,0.893846312,g(7.6),g(7.6)-0.893846312],_
+[7.7,0.894979666,g(7.7),g(7.7)-0.894979666],_
+[7.8,0.896088737,g(7.8),g(7.8)-0.896088737],_
+[7.9,0.897174302,g(7.9),g(7.9)-0.897174302],_
+[8.0,0.898237113,g(8.0),g(8.0)-0.898237113],_
+[8.1,0.899277888,g(8.1),g(8.1)-0.899277888],_
+[8.2,0.900297306,g(8.2),g(8.2)-0.900297306],_
+[8.3,0.901296033,g(8.3),g(8.3)-0.901296033],_
+[8.4,0.902274695,g(8.4),g(8.4)-0.902274695],_
+[8.5,0.903233900,g(8.5),g(8.5)-0.903233900],_
+[8.6,0.904174228,g(8.6),g(8.6)-0.904174228],_
+[8.7,0.905096235,g(8.7),g(8.7)-0.905096235],_
+[8.8,0.906000459,g(8.8),g(8.8)-0.906000459],_
+[8.9,0.906887414,g(8.9),g(8.9)-0.906887414],_
+[9.0,0.907757602,g(9.0),g(9.0)-0.907757602],_
+[9.1,0.908611483,g(9.1),g(9.1)-0.908611483],_
+[9.2,0.909449530,g(9.2),g(9.2)-0.909449530],_
+[9.3,0.910272177,g(9.3),g(9.3)-0.910272177],_
+[9.4,0.911079850,g(9.4),g(9.4)-0.911079850],_
+[9.5,0.911872958,g(9.5),g(9.5)-0.911872958],_
+[9.6,0.912651897,g(9.6),g(9.6)-0.912651897],_
+[9.7,0.913417043,g(9.7),g(9.7)-0.913417043],_
+[9.8,0.914168766,g(9.8),g(9.8)-0.914168766],_
+[9.9,0.914907418,g(9.9),g(9.9)-0.914907418],_
+[10.0,0.915633339,g(10.0),g(10.0)-0.915633339]]
+--R 
+--R   Compiling function g with type Float -> Expression DoubleFloat 
+--R
+--R   (6)
+--R   [[2.,0.72265723400000004,0.72265723377644353,- 2.2355650663996585E-10],
+--R
+--R     [2.1000000000000001, 0.73079150199999998, 0.73079150228850298,
+--R      2.8850299926830303E-10]
+--R     ,
+--R
+--R     [2.2000000000000002, 0.73843113199999999, 0.73843113069659072,
+--R      - 1.3034092694041988E-9]
+--R     ,
+--R
+--R     [2.2999999999999998, 0.74562214900000001, 0.74562214881923961,
+--R      - 1.8076040664283255E-10]
+--R     ,
+--R
+--R     [2.3999999999999999, 0.75240482900000005, 0.75240483025618621,
+--R      1.2561861550963727E-9]
+--R     ,
+--R    [2.5,0.75881459200000001,0.75881459121494477,- 7.850552430355151E-10],
+--R
+--R     [2.6000000000000001, 0.76488272199999996, 0.76488272217978248,
+--R      1.7978252220274271E-10]
+--R     ,
+--R
+--R     [2.7000000000000002, 0.77063698700000005, 0.77063698825333671,
+--R      1.2533366566813697E-9]
+--R     ,
+--R
+--R     [2.7999999999999998, 0.77610212300000003, 0.77610212535832457,
+--R      2.3583245356562088E-9]
+--R     ,
+--R
+--R     [2.8999999999999999, 0.78130025199999997, 0.78130025314743023,
+--R      1.1474302619163268E-9]
+--R     ,
+--R    [3.,0.786251221,0.78625122076592868,- 2.3407131788388824E-10],
+--R
+--R     [3.1000000000000001, 0.79097280000000003, 0.79097289808240101,
+--R      9.8082400978860562E-8]
+--R     ,
+--R
+--R     [3.2000000000000002, 0.79548142200000005, 0.7954814223275547,
+--R      3.2755465007028306E-10]
+--R     ,
+--R
+--R     [3.2999999999999998, 0.79979140800000004, 0.79979140803702831,
+--R      3.7028269339600683E-11]
+--R     ,
+--R
+--R     [3.3999999999999999, 0.80391612700000004, 0.80391612661323797,
+--R      - 3.8676206681742542E-10]
+--R     ,
+--R    [3.5,0.80786766099999996,0.80786766059300552,- 4.0699443815128689E-10],
+--R
+--R     [3.6000000000000001, 0.81165703700000003, 0.81165703674711587,
+--R      - 2.5288415805846398E-10]
+--R     ,
+--R
+--R     [3.7000000000000002, 0.81529434199999995, 0.81529434137285406,
+--R      - 6.2714589077472738E-10]
+--R     ,
+--R
+--R     [3.7999999999999998, 0.81878882100000006, 0.81878882054050406,
+--R      - 4.5949599680739084E-10]
+--R     ,
+--R
+--R     [3.8999999999999999, 0.82214896699999995, 0.8221489675670105,
+--R      5.6701054962360331E-10]
+--R     ,
+--R    [4.,0.82538250000000002,0.82538259960411076,9.9604110737061546E-8],
+--R
+--R     [4.0999999999999996, 0.82849692600000002, 0.82849692490970006,
+--R      - 1.0902999614259556E-9]
+--R     ,
+--R
+--R     [4.2000000000000002, 0.83149860200000003, 0.83149860211639337,
+--R      1.1639333941104724E-10]
+--R     ,
+--R
+--R     [4.2999999999999998, 0.83439379400000002, 0.83439379260257418,
+--R      - 1.397425841709321E-9]
+--R     ,
+--R
+--R     [4.4000000000000004, 0.83718820699999996, 0.83718820689462137,
+--R      - 1.0537859473913613E-10]
+--R     ,
+--R    [4.5,0.83988714399999997,0.83988714589085944,1.8908594690003611E-9],
+--R
+--R     [4.5999999999999996, 0.84249553899999996, 0.84249553757701434,
+--R      - 1.4229856182268463E-9]
+--R     ,
+--R
+--R     [4.7000000000000002, 0.84501797099999998, 0.84501796980531307,
+--R      - 1.1946869049594966E-9]
+--R     ,
+--R
+--R     [4.7999999999999998, 0.84745872099999997, 0.84745871962692243,
+--R      - 1.3730775405562667E-9]
+--R     ,
+--R
+--R     [4.9000000000000004, 0.84982177800000003, 0.84982177959827732,
+--R      1.5982772882949803E-9]
+--R     ,
+--R    [5.,0.85211088000000001,0.85211088142366165,1.4236616330265406E-9],
+--R
+--R     [5.0999999999999996, 0.85432951899999998, 0.85432951724709605,
+--R      - 1.7529039331165563E-9]
+--R     ,
+--R
+--R     [5.2000000000000002, 0.85648095800000001, 0.85648095886487274,
+--R      8.6487272898949641E-10]
+--R     ,
+--R
+--R     [5.2999999999999998, 0.85856827499999999, 0.85856827509448064,
+--R      9.4480645529415597E-11]
+--R     ,
+--R
+--R     [5.4000000000000004, 0.86059434800000001, 0.86059434750532948,
+--R      - 4.946705267627749E-10]
+--R     ,
+--R    [5.5,0.86256188499999997,0.86256188469070161,- 3.0929836469795191E-10],
+--R
+--R     [5.5999999999999996, 0.86447343600000004, 0.86447343523800879,
+--R      - 7.6199124787734718E-10]
+--R     ,
+--R    [5.7000000000000002,0.866331399,0.8663313995352464,5.3524640275526281E-10],
+--R
+--R     [5.7999999999999998, 0.86813804000000006, 0.86813804053493893,
+--R      5.3493887097744164E-10]
+--R     ,
+--R
+--R     [5.9000000000000004, 0.86989549399999999, 0.86989549358247464,
+--R      - 4.1752534762906635E-10]
+--R     ,
+--R    [6.,0.87160577500000003,0.87160577540332174,4.0332170936352441E-10],
+--R
+--R     [6.0999999999999996, 0.87327079299999999, 0.8732707923327413,
+--R      - 6.6725869274364413E-10]
+--R     ,
+--R
+--R     [6.2000000000000002, 0.87489234699999996, 0.8748923478621643,
+--R      8.6216433992092334E-10]
+--R     ,
+--R
+--R     [6.2999999999999998, 0.87647215000000001, 0.87647214956817143,
+--R      - 4.3182857289991716E-10]
+--R     ,
+--R
+--R     [6.4000000000000004, 0.878011816, 0.87801181548273155,
+--R      - 5.1726845029520518E-10]
+--R     ,
+--R    [6.5,0.87951288100000002,0.87951287995710281,- 1.0428972130327452E-9],
+--R
+--R     [6.5999999999999996, 0.88097679699999998, 0.88097679906610149,
+--R      2.0661015120992943E-9]
+--R     ,
+--R
+--R     [6.7000000000000002, 0.88240495500000005, 0.8824049555946607,
+--R      5.9466065405189283E-10]
+--R     ,
+--R
+--R     [6.7999999999999998, 0.88379866200000001, 0.88379866364416337,
+--R      1.6441633610142503E-9]
+--R     ,
+--R
+--R     [6.9000000000000004, 0.88515917600000005, 0.88515917289225332,
+--R      - 3.1077467266271697E-9]
+--R     ,
+--R    [7.,0.88648767500000003,0.88648767253642946,- 2.4635705697662047E-9],
+--R
+--R     [7.0999999999999996, 0.88778529399999995, 0.8877852949486843,
+--R      9.4868435329686918E-10]
+--R     ,
+--R
+--R     [7.2000000000000002, 0.88905311899999995, 0.88905311906582485,
+--R      6.5824901085420606E-11]
+--R     ,
+--R
+--R     [7.2999999999999998, 0.89029217299999996, 0.89029217353766843,
+--R      5.3766846530578505E-10]
+--R     ,
+--R
+--R     [7.4000000000000004, 0.89150344000000004, 0.89150343965322676,
+--R      - 3.4677327676035929E-10]
+--R     ,
+--R    [7.5,0.89268785399999995,0.89268785406308437,6.308442657143587E-11],
+--R
+--R     [7.5999999999999996, 0.89384631199999998, 0.89384631131444836,
+--R      - 6.8555161547578791E-10]
+--R     ,
+--R
+--R     [7.7000000000000002, 0.89497966600000001, 0.89497966621388148,
+--R      2.1388146809186992E-10]
+--R     ,
+--R
+--R     [7.7999999999999998, 0.89608873700000002, 0.89608873603132189,
+--R      - 9.686781377027387E-10]
+--R     ,
+--R
+--R     [7.9000000000000004, 0.89717430200000003, 0.8971743025577732,
+--R      5.5777316099181462E-10]
+--R     ,
+--R    [8.,0.89823711299999998,0.89823711402799578,1.0279957995962263E-9],
+--R
+--R     [8.0999999999999996, 0.89927788799999997, 0.89927788691844668,
+--R      - 1.0815532913710513E-9]
+--R     ,
+--R
+--R     [8.1999999999999993, 0.90029730600000002, 0.90029730762992677,
+--R      1.629926749124877E-9]
+--R     ,
+--R
+--R     [8.3000000000000007, 0.90129603300000005, 0.90129603406349046,
+--R      1.0634904068496098E-9]
+--R     ,
+--R
+--R     [8.4000000000000004, 0.90227469500000002, 0.90227469709753316,
+--R      2.0975331471717595E-9]
+--R     ,
+--R    [8.5,0.90323390000000003,0.90323390197320852,1.9732084854950926E-9],
+--R
+--R     [8.5999999999999996, 0.90417422800000002, 0.9041742295948515,
+--R      1.5948514731078944E-9]
+--R     ,
+--R
+--R     [8.6999999999999993, 0.90509623500000003, 0.90509623775141723,
+--R      2.7514172051823493E-9]
+--R     ,
+--R
+--R     [8.8000000000000007, 0.90600045900000004, 0.90600046226454201,
+--R      3.264541970082746E-9]
+--R     ,
+--R
+--R     [8.9000000000000004, 0.90688741399999995, 0.90688741806836271,
+--R      4.0683627577919879E-9]
+--R     ,
+--R    [9.,0.907757602,0.90775760022576812,- 1.7742318725311179E-9],
+--R
+--R     [9.0999999999999996, 0.90861148300000005, 0.90861148488548271,
+--R      1.8854826588921014E-9]
+--R     ,
+--R
+--R     [9.1999999999999993, 0.90944952999999995, 0.90944953018395813,
+--R      1.8395818202066039E-10]
+--R     ,
+--R
+--R     [9.3000000000000007, 0.91027217699999996, 0.91027217709579178,
+--R      9.5791818921497907E-11]
+--R     ,
+--R
+--R     [9.4000000000000004, 0.91107985000000002, 0.91107985023607185,
+--R      2.3607182875196031E-10]
+--R     ,
+--R    [9.5,0.91187295800000001,0.91187295861782769,6.1782767790674598E-10],
+--R
+--R     [9.5999999999999996, 0.91265189700000005, 0.91265189636747834,
+--R      - 6.3252170168226485E-10]
+--R     ,
+--R
+--R     [9.6999999999999993, 0.91341704300000004, 0.91341704340103613,
+--R      4.010360932227286E-10]
+--R     ,
+--R
+--R     [9.8000000000000007, 0.91416876599999997, 0.91416876606351216,
+--R      6.3512195502823943E-11]
+--R     ,
+--R
+--R     [9.9000000000000004, 0.91490741799999997, 0.9149074177339066,
+--R      - 2.6609336956084917E-10]
+--R     ,
+--R    [10.,0.91563333899999999,0.91563333939788116,3.9788117245365129E-10]]
+--R                                       Type: List List Expression DoubleFloat
+--E 6
+
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun,``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp238-243
+\bibitem{2} Segletes, Steven, B., ``A Compact Analytical Fit to the 
+Exponential Integral $E_1(x)$'', Army Research Laboratory, ARL-TR-1758,
+September 1998
+\bibitem{3} Lund, Lars Erik, ``Numlibc'',\\
+{\bf http://www.math.ntnu.no/num/nnm/Program/Numlibc}
+\bibitem{4} Haavie, T. ``expint.c'',\\
+{\bf http://www.math.ntnu.no/num/nnm/Program/Numlibc}, June, 1988
+\end{thebibliography}
+\end{document}



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If you'd like to see bug fixes applied to Axiom please submit a
diff-Naur patch against silver. 

Bugs are being fixed. Look for patches that have trailing numbers
in the title. One of the numbers is the Axiom bug number and, if
there is a corresponding published bug, there is a second number
corresponding to the published number.

I hesitate to "fix" things in the algebra without a great deal of
testing. I've applied a few fixes and backed them out, for example, 
trivially, the Gamma overbar display or bug 370 which incorrectly
changed the algebra rather than the compiler, and rejected others as
not being the correct fix.

One of the primary reasons why these bugs are not being fixed faster
is that there is so little documentation of what the bug is, what the
root cause is, how the change fixes the bug, and regression tests that
show the bug was fixed and did not break other things. All of this
takes a great deal of time and limits the pace of bug fixing. I'm 
reluctant to change the algebra until I understand the issues.
Without documentation or even a reference to external documentation
it just takes time.

I realize that FriCAS does not require documentation and can apply
changes without comment. Axiom requires documentation.




That said, I also would like to reduce the number of open bugs.

Axiom is clearly being worked on. Simply dividing the number of
one-line changes in the changelog by the number of days since the
first entry gives approximately 2.8 changes per day over the last 4
years or so. Hardly signs of a dead project.

Tim




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daly@axiom-developer.org writes:

> If you'd like to see bug fixes applied to Axiom please submit a diff-Naur
> patch against silver.

Not as long as you insist on your axiom being the only axiom.

> I realize that FriCAS does not require documentation and can apply
> changes without comment.

I think you are quite mistaken here.

Martin




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The FreeAbelianGroup bug fix was actually submitted by Franz Lehner.
This corrects the credits in the changelog.

Tim

========================================================================
diff --git a/changelog b/changelog
index f018a38..e2767b4 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080119 tpd changelog credit FreeAbelianGroup fix to Franz Lehner
 20080119 tpd src/input/Makefile add e1.input regression test file
 20080119 tpd src/input/e1.input create A&S reference regression for E1
 20080119 tpd src/algebra/special.spad add E1 function
@@ -19,7 +20,8 @@
 20080107 tpd src/input/pfaffian.input missing file added
 20080107 tpd src/input/Makefile add free to test bug 7089/343
 20080107 tpd src/input/free.input added to test bug 7089/343
-20080107 mxr src/input/free.spad fix FreeAbelianGroup bug 7089/343
+20080107 fxl src/input/free.spad fix FreeAbelianGroup bug 7089/343
+20080107 fxl Franz Lehner <lehner@finanz.math.tu-graz.ac.at>
 20080106 tpd src/algebra/combfunc.spad \mathbb -> \bf
 20080106 tpd src/algebra/op.spad revert dgamma change, reject patch
 20080106 tpd src/scripts/tex/axiom.sty add binom for combfunc



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E1(0.0) is infinity. This fixes the special case to give correct results.
e1.input has a new regression test for infinity at 0.0

Because of the infinity the range of E1 is OnePointCompletion(DoubleFloat)
and we need to add some explicit coercions.

Tim

===========================================================================
diff --git a/changelog b/changelog
index e2767b4..2a361bf 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080119 tpd src/input/e1.input regression test E1(0.0) 
+20080119 tpd src/algebra/special.spad handle E1(0.0) properly
 20080119 tpd changelog credit FreeAbelianGroup fix to Franz Lehner
 20080119 tpd src/input/Makefile add e1.input regression test file
 20080119 tpd src/input/e1.input create A&S reference regression for E1
diff --git a/src/algebra/special.spad.pamphlet b/src/algebra/special.spad.pamphlet
index 241d23e..0744b97 100644
--- a/src/algebra/special.spad.pamphlet
+++ b/src/algebra/special.spad.pamphlet
@@ -30,6 +30,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
     NNI ==> NonNegativeInteger
     R   ==> DoubleFloat
     C   ==> Complex DoubleFloat
+    OPR ==> OnePointCompletion R
 
     Exports ==> with
         Gamma: R -> R
@@ -39,7 +40,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
          ++ Gamma(x) is the Euler gamma function, \spad{Gamma(x)}, defined by
          ++   \spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..%infinity)}.
 
-        E1: R -> R
+        E1: R -> OPR
 	 ++ E1(x) is the Exponential Integral function
          ++ The current implementation is a piecewise approximation
          ++ involving one poly from -4..4 and a second poly for x > 4
@@ -377,8 +378,8 @@ rewritten the polynomial using Horner's method so the large
 powers of $x$ are only computed once.
 
 <<package DFSFUN DoubleFloatSpecialFunctions>>=
-        E1(x:R):R ==
-         x = 0.0::R => error "E1 undefined at zero"
+        E1(x:R):OPR ==
+         x = 0.0::R => infinity()
          x > 4.0::R =>
           t1:R:=0.14999948967737774608E-15::R
           t2:R:=0.9999999999993112::R
@@ -426,7 +427,11 @@ powers of $x$ are only computed once.
           t23:R:=15474250491.067253436::R
           tv:R:=(tu*x-t23)
           tw:R:=(-1.0::R*x)
-          (tv * exp(tw)::R)/x**22
+          tx:R:=exp(tw)
+          ty:R:=tv*tx
+          tz:R:=x**22
+          taz:R:=ty/tz
+          taz::OPR
          x > -4.0::R => 
           a1:R:=0.476837158203125E-22::R
           a2:R:=0.10967254638671875E-20::R
@@ -473,7 +478,8 @@ powers of $x$ are only computed once.
           au:R:=(at*x+a22)
           a23:R:=0.5772156649015328::R
           av:R:=au*x-a23
-          - 1.0::R*log(abs(x)) + av
+          aw:R:=- 1.0::R*log(abs(x)) + av
+          aw::OPR
          error "E1: no approximation available"
 
         polygamma(k,z)  == CPSI(k, z)$Lisp
diff --git a/src/input/e1.input.pamphlet b/src/input/e1.input.pamphlet
index 0c24a64..329cb01 100644
--- a/src/input/e1.input.pamphlet
+++ b/src/input/e1.input.pamphlet
@@ -19,7 +19,7 @@ This is a regression test for E1(x)
 @
 Here we enter the value of Gamma
 <<*>>=
---S 1 of 6
+--S 1 of 7
 G:DFLOAT:=0.577215664901532860606512::DFLOAT
 --R
 --R   (1)  0.57721566490153287
@@ -34,8 +34,8 @@ listed in
 Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp238
 <<*>>=
---S 2 of 6
-f(x)==x^-1 * (E1(x) + log(x) + G)
+--S 2 of 7
+f(x)==x^-1 * (E1(x)::DFLOAT + log(x) + G)
 --R                                                                   Type: Void
 --E 2
 @
@@ -47,7 +47,7 @@ Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp238
 
 <<*>>=
---S 3 of 6
+--S 3 of 7
 [[0.01,0.9975055452, f(0.01), f(0.01)-0.9975055452],_
 [0.02,0.9950221392, f(0.02), f(0.02)-0.9950221392],_
 [0.03,0.9925497201, f(0.03), f(0.03)-0.9925497201],_
@@ -97,7 +97,7 @@ Dover Publications, Inc. New York 1965. pp238
 [0.47,0.8937670423, f(0.47), f(0.47)-0.8937670423],_
 [0.48,0.8917309048, f(0.48), f(0.48)-0.8917309048],_
 [0.49,0.8897032920, f(0.49), f(0.49)-0.8897032920],_
-[0.50,0.8876841584, f(0.50), f(0.50)-0.8876841584]]
+[0.50,0.8876841584, f(0.50), f(0.50)-0.8876841584]]::LIST(LIST(DFLOAT))
 --R 
 --R   Compiling function f with type Float -> DoubleFloat 
 --R
@@ -270,7 +270,7 @@ is the reference value of $E1(x)$ from the book
 Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp239-241
 <<*>>=
---S 4 of 6
+--S 4 of 7
 [[0.50, 0.559773595, E1(0.50), E1(0.50)-0.559773595],_
 [0.51, 0.547822352, E1(0.51), E1(0.51)-0.547822352],_
 [0.52, 0.536219798, E1(0.52), E1(0.52)-0.536219798],_
@@ -421,7 +421,7 @@ Dover Publications, Inc. New York 1965. pp239-241
 [1.97, 0.050976988, E1(1.97), E1(1.97)-0.050976988],_
 [1.98, 0.050274392, E1(1.98), E1(1.98)-0.050274392],_
 [1.99, 0.049582291, E1(1.99), E1(1.99)-0.049582291],_
-[2.00, 0.048900511, E1(2.00), E1(2.00)-0.048900511]]
+[2.00, 0.048900511, E1(2.00), E1(2.00)-0.048900511]]::LIST(LIST(DFLOAT))
 --R 
 --R
 --R   (4)
@@ -835,8 +835,8 @@ Dover Publications, Inc. New York 1965. pp239-241
 
 Now that we move into larger numbers we need a new scaling function.
 <<*>>=
---S 5 of 6
-g(x)==x * %e^x * E1(x)
+--S 5 of 7
+g(x)==x * %e^x * E1(x)::DFLOAT
 --R                                                                   Type: Void
 --E 5
 @
@@ -845,7 +845,7 @@ And we compute the scaled value of E1(x) in the range 2.0 to 10.0
 from Abramowitz and Stegun,``Handbook of Mathematical Functions'',
 Dover Publications, Inc. New York 1965. pp242-243
 <<*>>=
---S 6 of 6
+--S 6 of 7
 [[2.0,0.722657234,g(2.0),g(2.0)-0.722657234],_
 [2.1,0.730791502,g(2.1),g(2.1)-0.730791502],_
 [2.2,0.738431132,g(2.2),g(2.2)-0.738431132],_
@@ -1204,6 +1204,12 @@ Dover Publications, Inc. New York 1965. pp242-243
 --R                                       Type: List List Expression DoubleFloat
 --E 6
 
+--S 7 of 7
+E1(0.0)
+--R
+--R   (7)  infinity
+--R                                         Type: OnePointCompletion DoubleFloat
+--E 7
 )spool 
 )lisp (bye)
  



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Hyperdoc does not allow changes in input fields when the num-lock key is on.
This patch by Greg Vanuxem fixes that problem.

Tim

========================================================================
diff --git a/changelog b/changelog
index 2a361bf..7fdcca6 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080119 gxv src/hyper/keyin handle numlock in hyperdoc correctly	
+20080119 gxv src/hyper/dialog handle numlock in hyperdoc correctly	
 20080119 tpd src/input/e1.input regression test E1(0.0) 
 20080119 tpd src/algebra/special.spad handle E1(0.0) properly
 20080119 tpd changelog credit FreeAbelianGroup fix to Franz Lehner
diff --git a/src/hyper/dialog.pamphlet b/src/hyper/dialog.pamphlet
index fc973f5..8478694 100644
--- a/src/hyper/dialog.pamphlet
+++ b/src/hyper/dialog.pamphlet
@@ -1224,7 +1224,7 @@ dialog(XEvent *event, KeySym keysym, char *buffer)
     {
         /* only handle normal keys */
 
-        if (event->xkey.state & ShiftModMask)
+        if (event->xkey.state & UnsupportedModMask)
             BeepAtTheUser();
         else
             add_buffer_to_sym(buffer, item);
diff --git a/src/hyper/keyin.pamphlet b/src/hyper/keyin.pamphlet
index 3f2a59b..b1252da 100644
--- a/src/hyper/keyin.pamphlet
+++ b/src/hyper/keyin.pamphlet
@@ -28,7 +28,7 @@ extern int simple_box_width;
 extern int gInInsertMode;
 
 extern unsigned int ModifiersMask;
-extern unsigned int ShiftModMask;
+extern unsigned int UnsupportedModMask;
 
 
 #endif
@@ -75,7 +75,7 @@ unsigned int ModifiersMask = ShiftMask | LockMask | ControlMask
     | Mod1Mask | Mod2Mask | Mod3Mask
     | Mod4Mask | Mod5Mask;
 
-unsigned int ShiftModMask = LockMask | ControlMask
+unsigned int UnsupportedModMask = LockMask | ControlMask
     | Mod1Mask | Mod2Mask | Mod3Mask
     | Mod4Mask | Mod5Mask;
 
@@ -233,6 +233,40 @@ handle_key(XEvent *event)
   }
 }
 
+/*
+ * This routine returns the modifier mask associated
+ * to a key symbol
+ */
+
+static unsigned int
+get_modifier_mask(KeySym sym)
+{
+    unsigned int       i, mask;
+    XModifierKeymap    *mod;
+    KeyCode            kcode;
+    const int          masks[8] = {
+        ShiftMask, LockMask, ControlMask,
+            Mod1Mask, Mod2Mask, Mod3Mask, Mod4Mask, Mod5Mask
+    };
+
+    mod = XGetModifierMapping(gXDisplay);
+    kcode = XKeysymToKeycode(gXDisplay,sym);
+
+    if (mod) {
+        for (i = 0; i < (8 * mod->max_keypermod); i++){
+             if (!mod->modifiermap[i]) continue;
+             else if (kcode == mod->modifiermap[i]){
+                 mask = masks[i / mod->max_keypermod];
+                 XFreeModifiermap(mod);
+                 return mask;
+             }
+        }
+        XFreeModifiermap(mod);
+    }
+    return 0;
+}
+
+
 
 /*
  * This routine initializes some of the variables needed by the input
@@ -243,6 +277,11 @@ void
 init_keyin(void)
 {
     char *prop;
+    unsigned int nlm;
+
+    nlm = get_modifier_mask(XK_Num_Lock);
+    UnsupportedModMask &= ~nlm;
+    ModifiersMask &= ~nlm;
 
     /*
      * First set all the values for when the active cursor is in the window



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This fixes an unnecessary load of the MoreSystemCommands domain during testing.

Tim
=========================================================================
diff --git a/changelog b/changelog
index 7fdcca6..2d037de 100644
--- a/changelog
+++ b/changelog
@@ -1,5 +1,6 @@
-20080119 gxv src/hyper/keyin handle numlock in hyperdoc correctly	
-20080119 gxv src/hyper/dialog handle numlock in hyperdoc correctly	
+20080120 tpd src/input/Makefile fix MoreSystemCommand loading (7101/204)
+20080119 gxv src/hyper/keyin handle numlock in hyperdoc correctly (7100/149)
+20080119 gxv src/hyper/dialog handle numlock in hyperdoc correctly (7100/149)
 20080119 tpd src/input/e1.input regression test E1(0.0) 
 20080119 tpd src/algebra/special.spad handle E1(0.0) properly
 20080119 tpd changelog credit FreeAbelianGroup fix to Franz Lehner
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index fecb71d..533002e 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -396,10 +396,10 @@ identical output.
 	@ echo ')read $*' >> tmp.input 
 	@ echo ')lisp (bye)' >> tmp.input 
 	@ if [ -z "${NOISE}" ] ; then \
-	   echo 'systemCommand "read tmp.input"' | ${TESTSYS} \
+	   echo ")read tmp.input" | ${TESTSYS} \
                 | egrep -v '(Timestamp|Version)' | tee $*.output ; \
 	  else \
-	   echo 'systemCommand "read tmp.input"' | ${TESTSYS} \
+	   echo ")read tmp.input" | ${TESTSYS} \
                 | egrep -v '(Timestamp|Version)' > $*.output ; \
 	  fi 
 	@ rm tmp.input 



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This patch by Martin fixes equality testing in TableAggregate.
The problem is that the syntax " => false" causes the loop to 
iterate, skipping any following expressions in the loop. However,
the intention of this loop is clearly to cause an early exit from
the function when it can be determined that the arguments are not
equal. As originally written, the loop always returned true.

There is a regression test file, tbagg.input, added.

Tim

===========================================================================
diff --git a/changelog b/changelog
index 2d037de..665492c 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080120 tpd src/input/Makefile add tbagg.input regression test (7102/412)
+20080120 tpd src/input/tbagg.input regression test equality in TBAGG (7102/412)
+20080120 mxr src/algebra/aggcat.spad fix equality in TBAGG (7102/412)
 20080120 tpd src/input/Makefile fix MoreSystemCommand loading (7101/204)
 20080119 gxv src/hyper/keyin handle numlock in hyperdoc correctly (7100/149)
 20080119 gxv src/hyper/dialog handle numlock in hyperdoc correctly (7100/149)
diff --git a/src/algebra/aggcat.spad.pamphlet b/src/algebra/aggcat.spad.pamphlet
index 1363c02..ae221f4 100644
--- a/src/algebra/aggcat.spad.pamphlet
+++ b/src/algebra/aggcat.spad.pamphlet
@@ -1270,7 +1270,7 @@ TableAggregate(Key:SetCategory, Entry:SetCategory): Category ==
        eq?(s,t) => true
        #s ^= #t => false
        for k in keys s repeat
-	 (e := search(k, t)) case "failed" or (e::Entry) ^= s.k => false
+	 (e := search(k, t)) case "failed" or (e::Entry) ^= s.k => return false
        true
 
      map(f: Record(key:Key,entry:Entry)->Record(key:Key,entry:Entry), t: %): % ==
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 533002e..65741f8 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -363,7 +363,8 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     stbl.regress      stream2.regress  stream.regress   streams.regress \
     string.regress    strtbl.regress   summation.regress \
     symbol.regress    t111293.regress  table.regress    tancot.regress \
-    tanhcoth.regress  tanatan.regress  textfile.regress torus.regress \
+    tanhcoth.regress  tanatan.regress  tbagg.regress \
+    textfile.regress  torus.regress \
     triglim.regress   tsetcatvermeer.regress            tutchap1.regress \
     void.regress      uniseg.regress
 
@@ -638,7 +639,7 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/symbol.input \
        ${OUT}/synonym.input  ${OUT}/t111293.input \
        ${OUT}/table.input    ${OUT}/tancot.input     ${OUT}/tanhcoth.input \
-       ${OUT}/tanatan.input  ${OUT}/test.input \
+       ${OUT}/tanatan.input  ${OUT}/tbagg.input      ${OUT}/test.input \
        ${OUT}/tetra.input    ${OUT}/textfile.input   ${OUT}/tknot.input \
        ${OUT}/tree.input \
        ${OUT}/triglim.input  ${OUT}/tschirn.input ${OUT}/tsetcatbutcher.input \
@@ -960,6 +961,7 @@ DOCFILES= \
   ${DOC}/synonym.input.dvi     ${DOC}/t111293.input.dvi    \
   ${DOC}/table.input.dvi       ${DOC}/tanatan.input.dvi    \
   ${DOC}/tancot.input.dvi      ${DOC}/tanhcoth.input.dvi   \
+  ${DOC}/tbagg.input.dvi       \
   ${DOC}/test.input.dvi        ${DOC}/tetra.input.dvi      \
   ${DOC}/textfile.input.dvi    ${DOC}/tknot.input.dvi      \
   ${DOC}/torus.input.dvi       ${DOC}/tree.input.dvi       \
diff --git a/src/input/tbagg.input.pamphlet b/src/input/tbagg.input.pamphlet
new file mode 100644
index 0000000..c33dd1f
--- /dev/null
+++ b/src/input/tbagg.input.pamphlet
@@ -0,0 +1,92 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input tbagg.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+Equality testing was broken in TableAggregate (TBAGG)
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+The equality function int TableAggregate used to read:
+\begin{verbatim}
+     s:% = t:% ==
+       eq?(s,t) => true
+       #s ^= #t => false
+       for k in keys s repeat
+	 (e := search(k, t)) case "failed" or (e::Entry) ^= s.k => false
+       true
+\end{verbatim}
+but this always returns true. Using the ``$=>$'' operator in a loop
+just causes the loop to move to the next iteration. Thus the loop
+will not take an early exit from the function and the code will always
+fall thru to the ``true'' case. This is fixed by calling the return function.
+\begin{verbatim}
+     s:% = t:% ==
+       eq?(s,t) => true
+       #s ^= #t => false
+       for k in keys s repeat
+	 (e := search(k, t)) case "failed" or (e::Entry) ^= s.k => return false
+       true
+\end{verbatim}
+<<*>>=
+)spool tbagg.output
+)set message test on
+)set message auto off
+)clear all
+ 
+--S 1 of 7
+R ==> Record(key: Symbol, entry: String)
+--R                                                                   Type: Void
+--E 1
+
+--S 2 of 7
+T ==> AssociationList(Symbol, String)
+--R                                                                   Type: Void
+--E 2
+
+--S 3 of 7
+t1:=construct([[x,"ix"]$R])$T
+--R
+--R   (3)  table(x= "ix")
+--R                                         Type: AssociationList(Symbol,String)
+--E 3
+
+--S 4 of 7
+t2:=construct([[y,"iy"]$R])$T
+--R
+--R   (4)  table(y= "iy")
+--R                                         Type: AssociationList(Symbol,String)
+--E 4
+
+--S 5 of 7
+(t1=t2)::Boolean
+--R
+--R   (5)  false
+--R                                                                Type: Boolean
+--E 5
+
+--S 6 of 7
+t3:=construct([[y,"iy"]$R])$T
+--R
+--R   (6)  table(y= "iy")
+--R                                         Type: AssociationList(Symbol,String)
+--E 6
+
+--S 7 of 7
+(t3=t2)::Boolean
+--R
+--R   (7)  true
+--R                                                                Type: Boolean
+--E 7
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}



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Subject: Re: [Axiom-developer] Nice Fixes! Another go on IssueTracker
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Hi Tim,

I know that documentation is one of the most important things for the 
Axiom project, but your approach to fixing bugs is just strange.

Suppose you have software A which has very few documentation and it has 
a bug B.

You can produce from that two systems A1 and A2.

A1: Fix the bug today. Fix the documentation in 1 year.
A2: Fix the bug together with the documentation in 1 year.
A3: Fix the bug today. Fix or don't fix the documention.

Clearly, in the long term A2 is the best, but actually A1 is better 
since it contains a partial fix of the code tomorrow.

Now what about A3?

Software A does not have enough documentation. So in some way A3 is as 
worse as A. But A3 is better since it has fewer bugs.

I know that every bugfix introduces new bugs, but since A3 fixes a known 
bug and passes all tests, it only might introduce a new yet unknown bug.
And one should not forget... there are many eyes at the mailing list 
watching what happens to the code. Some people might even be able to say 
something about a possible (yet unknown) bug.

Just some thoughts that don't count ...

Ralf




daly@axiom-developer.org wrote:
> If you'd like to see bug fixes applied to Axiom please submit a
> diff-Naur patch against silver. 
> 
> Bugs are being fixed. Look for patches that have trailing numbers
> in the title. One of the numbers is the Axiom bug number and, if
> there is a corresponding published bug, there is a second number
> corresponding to the published number.
> 
> I hesitate to "fix" things in the algebra without a great deal of
> testing. I've applied a few fixes and backed them out, for example, 
> trivially, the Gamma overbar display or bug 370 which incorrectly
> changed the algebra rather than the compiler, and rejected others as
> not being the correct fix.
> 
> One of the primary reasons why these bugs are not being fixed faster
> is that there is so little documentation of what the bug is, what the
> root cause is, how the change fixes the bug, and regression tests that
> show the bug was fixed and did not break other things. All of this
> takes a great deal of time and limits the pace of bug fixing. I'm 
> reluctant to change the algebra until I understand the issues.
> Without documentation or even a reference to external documentation
> it just takes time.
> 
> I realize that FriCAS does not require documentation and can apply
> changes without comment. Axiom requires documentation.
> 
> 
> 
> 
> That said, I also would like to reduce the number of open bugs.
> 
> Axiom is clearly being worked on. Simply dividing the number of
> one-line changes in the changelog by the number of days since the
> first entry gives approximately 2.8 changes per day over the last 4
> years or so. Hardly signs of a dead project.



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Date: Thu, 17 Jan 2008 09:32:54 +0000
From: "Peter Broadbery" <p.broadbery@googlemail.com>
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Subject: [Axiom-developer] aldor/axiom interoperability
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Finally got round to looking at this.
The code attached can be used to build the axiom/aldor interopability
library (libaxiom.al).

New features:
Less abuse of make features (though I've only used a recent version of
make, and have used some newer features)
Dependency on java removed.
Cleaner build process - all the dependency analysis is done in the
aldor code.  It should be more data driven, but it isn't.

There's a small aldor patch that I'll send along this evening.

Peter

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Subject: Re: [Axiom-developer] Nice Fixes! Another go on IssueTracker
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>I know that documentation is one of the most important things for the 
>Axiom project, but your approach to fixing bugs is just strange.

My approach to code development in Axiom is strange. But I'm fighting
against the "get it done" crowd and trying to go for "get it right"
and "get it documented".




>Suppose you have software A which has very few documentation and it has 
>a bug B.
>
>You can produce from that two systems A1 and A2.
>
>A1: Fix the bug today. Fix the documentation in 1 year.
>A2: Fix the bug together with the documentation in 1 year.
>A3: Fix the bug today. Fix or don't fix the documention.


A4: Fix the program. Which, in Axiom, means fix the literate
    document, which implies fixing the documentation as well
    as the code.





>Clearly, in the long term A2 is the best, but actually A1 is better 
>since it contains a partial fix of the code tomorrow.

That is the kind of short-term thinking that I'm trying to avoid.  You
are optimizing ("the best") the wrong function.  Every developer I
know worries about "tomorrow" and doesn't care about next month. 

If you take a 30-year horizon view, the answer to the question about
"what is best" changes. It is no longer vital that the change happen
quickly. It is vital that the change is correct and that the function
is documented well enough so that anyone can see that the change is
correct. Achieving "the best" is a good thing, but you have to know
what you are trying to optimize for it to make sense. As a
mathematician I know you understand this.

Axiom is trying to optimize for the next generation. Thus a bug
fix is "good" but a documented function which is clearly corrected
by this bug fix is "best". After all, it might turn out that the
bug exists in a function that shouldn't exist at all, or in a
function that is poorly defined, or in a function that could be
decomposed into several more general functions, etc.. Unless you
lift your eyes away from your desk you won't see the larger, longer
term picture.





>Now what about A3?
>
>Software A does not have enough documentation. So in some way A3 is as 
>worse as A. But A3 is better since it has fewer bugs.

Axiom will always have bugs. And the known bugs won't disappear off
the bug list. However, a "quick fix" to a non-critical bug means that
no-one will ever look at the problem again. If you know enough to fix
it you know enough to document the block of code containing the bug.

In non-algebra code I'm going a bit further. When I change things now
I "tree shake" the code, pick it up, add it to the proper book volume
along with all supporting code, and document the functions.

In algebra code I'm generally trying to understand the function well
enough to find good documentation sources since no-one who is actually
a domain expert seems to care enough to document. When I do write new
algebra code (e.g. the Exponential Integral E1 function I just added)
I find sources and write documentation.




>
>I know that every bugfix introduces new bugs, but since A3 fixes a known 
>bug and passes all tests, it only might introduce a new yet unknown bug.
>And one should not forget... there are many eyes at the mailing list 
>watching what happens to the code. Some people might even be able to say 
>something about a possible (yet unknown) bug.

Well, my eyes are the "eyes of last resort" and I'm pretty careful 
about changing things, especially algebra code I don't understand.
Since Axiom code runs in many different domains a "simple" fix that
works for integers might be wrong for finite fields. We don't have
peer-reviewed literature for bug fixes. The algebra code was written
by experts in their respective domains and I really look critically
at bugs that "work for me" without proper understanding.





>
>Just some thoughts that don't count ...

When I'm doing bug fixing I need to understand that code that is 
being fixed. Since I need to understand the code I can clearly
explain at least the parts I understand. Given the huge amount
of code in Axiom it is likely that the buggy code is going to
be visited just once. If there is no explanation added at the
time I'm debugging, there will be no explanation later. All that
will remain will be a small code change with no explanation.

Take, for instance, bug 370. There the original bug fix was to
change the algebra code from (something like)

  f(a+b)

to 

  c:=a+b
  f(c)

When I went to apply this patch it didn't look right. But there
was no documentation about what the patch was intended to do.
It eventually turned out that the correct patch was to fix the
spad compiler.



The long term goal of Axiom is to be understandable and maintainable
by developers we've never met. If I can't understand why a change is
being made now due to lack of documentation why would I expect that
someone 30 years from now will understand it? If it is minimally
documented now so that I can understand it, then there is some hope
that future developers can also understand it.

When changing an algebra function there is an implicit assumption
that the developer has a fair amount of domain knowledge, not generally
shared by all of the developers. Given that you understand a function
it is not unreasonable to ask that the fix at least explain a little
bit about the function, either why it was wrong or how it works now.





After 36 years of programming I know that nobody is going to sit
down and "document" all of Axiom. It has to be done incrementally
both because the task is too large and because no one person has
all of the expertise.

The people who worked on Axiom are dying off. One of the original
group members died Friday and I'm going to the funeral tomorrow.
We need to make the code live longer than the developers.

Tim







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root <daly@axiom-developer.org> writes:

> Take, for instance, bug 370. There the original bug fix was to change the
> algebra code from (something like)
> 
>   f(a+b)
> 
> to 
> 
>   c:=a+b
>   f(c)
> 
> When I went to apply this patch it didn't look right. But there was no
> documentation about what the patch was intended to do.  It eventually turned
> out that the correct patch was to fix the spad compiler.

Sorry, but do you really mean 

http://axiom-wiki.newsynthesis.org/370ImproperCodeEmmitedByMkUnionFunListImpactsDefintrfSpad

? I cannot see any algebra code there - of course, possibly only the proper fix
(by Stephen Wilson) was sent to IssueTracker, but I'd like to make sure.

Many thanks for clarification,

Martin




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>> Take, for instance, bug 370. There the original bug fix was to change the
>> algebra code from (something like)
>> 
>>   f(a+b)
>> 
>> to 
>> 
>>   c:=a+b
>>   f(c)
>> 
>> When I went to apply this patch it didn't look right. But there was no
>> documentation about what the patch was intended to do.  It eventually turned
>> out that the correct patch was to fix the spad compiler.
>
>Sorry, but do you really mean 
>
>http://axiom-wiki.newsynthesis.org/370ImproperCodeEmmitedByMkUnionFunListImpactsDefintrfSpad
>
>? I cannot see any algebra code there - of course, possibly only the proper fix
>(by Stephen Wilson) was sent to IssueTracker, but I'd like to make sure.
>
>Many thanks for clarification,

Yes, that's the bug. 
The original fix was in the algebra.
I didn't think the original patch made sense and I started 
debugging the generated code, leading to the compiler.
Stephen Wilson found and fixed the compiler bug.

Tim



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>>Suppose you have software A which has very few documentation and it has 
>>a bug B.
>>
>>You can produce from that two systems A1 and A2.
>>
>>A1: Fix the bug today. Fix the documentation in 1 year.
>>A2: Fix the bug together with the documentation in 1 year.
>>A3: Fix the bug today. Fix or don't fix the documention.
>
>
>A4: Fix the program. Which, in Axiom, means fix the literate
>    document, which implies fixing the documentation as well
>    as the code.

Just to give you an analogy from last week.

Last week I went to my mom's funeral. Because she had been sick
for a while her house was shut down. I arrived and turned on the
kitchen appliances. Everything worked but the microwave. I finally
figured out that the outlet was dead even though all other outlets
worked. The fuse box in the kitchen was fine.

My older brother arrived and pointed out that the outlet goes thru
the fusebox in the back of the house (about 20 meters away). When
I enabled that box the outlet worked.

So what is the proper action in this case?
  a) having found the bug and not understood it I could call an
     expert (an electrician)
  b) having understood the bug from my brother I could have fixed
     the problem by enabling the distant fusebox
  c) having understood the problem and fixed it I could have 
     "documented" the problem by writing "fusebox A5, shop" on
     the coverplate so the next owner knows what my brother knew.

What is the proper fix? It depends on what you want to optimize.
Axiom is optimizing for the long term. Why? Because I want this
code to live.

Tim



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On Jan 21, 2008 9:30 PM, root <daly@axiom-developer.org> wrote:
> >>Suppose you have software A which has very few documentation and it has
> >>a bug B.
> >>
> >>You can produce from that two systems A1 and A2.
> >>
> >>A1: Fix the bug today. Fix the documentation in 1 year.
> >>A2: Fix the bug together with the documentation in 1 year.
> >>A3: Fix the bug today. Fix or don't fix the documention.
> >
> >
> >A4: Fix the program. Which, in Axiom, means fix the literate
> >    document, which implies fixing the documentation as well
> >    as the code.
>
> Just to give you an analogy from last week.
>
> Last week I went to my mom's funeral. Because she had been sick
> for a while her house was shut down. I arrived and turned on the
> kitchen appliances. Everything worked but the microwave. I finally
> figured out that the outlet was dead even though all other outlets
> worked. The fuse box in the kitchen was fine.
>
> My older brother arrived and pointed out that the outlet goes thru
> the fusebox in the back of the house (about 20 meters away). When
> I enabled that box the outlet worked.
>
> So what is the proper action in this case?
>   a) having found the bug and not understood it I could call an
>      expert (an electrician)
>   b) having understood the bug from my brother I could have fixed
>      the problem by enabling the distant fusebox
>   c) having understood the problem and fixed it I could have
>      "documented" the problem by writing "fusebox A5, shop" on
>      the coverplate so the next owner knows what my brother knew.
>
> What is the proper fix? It depends on what you want to optimize.
> Axiom is optimizing for the long term. Why? Because I want this
> code to live.

I think this should be written on the front page of the axiom project,
because my personal opinion is, that many people want to know the
strategy behind the project. Also, I think, but maybe I am wrong, that
most people prefer the "fix it now, instead of tomorrow" strategy. So
one should then ask - do I want to create a project that most of the
people would agree with the course of development (strategy of fixing
bugs)? Or do I want to create a project that most people disagree with
the strategy, but maybe in 30 years there is some chance that axiom
will become very popular. I, personally, prefer the former, because I
(but that's just my opinion) prefer my project to be joined by a lot
of people now, not in 30 years.

And with this perspective, I think (again just my opinion) that even
if you want to optimize the project over 30 years, the strategy is
wrong too, because it's imho essential that people join the
development. I think the best strategy is to optimize for the needs
and opinions of potential developers in 2008, not 2038.

Ondrej



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From: Martin Rubey <martin.rubey@univie.ac.at>
Date: 25 Jan 2008 08:42:52 +0100
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Dear bug masters :-),

so far I made all (well, mostly algebra) bugs that are fixed in FriCAS "fix
proposed" on IssueTracker.  The reason being that I would like to see at a
glance which bugs are open, and with the philosophy that everybody could just
diff the algebra sources and get the fix from the FriCAS sources.

However, there are also important bug fixes in open-axiom.  Gaby has added a
comment "fixed in OpenAxiom", which is fine.  But I really think that it is
unfair treatment not to change status to "fix proposed" there, even if it may
be more difficult to port them to the other axioms.

I see two possibilities:

1) mark them also "fix proposed"

2) have more possible statuses.  My earlier proposal (with a total of 2^3 = 8
   statuses) was rejected.  Maybe somebody has a better idea, but I really
   would like to be able to filter the bugs which have been fixed in OpenAxiom
   but not in FriCAS.

In any case, I dislike the situation as it is now - I do not want to see
OpenAxiom treated differently then OpenAxiom.

Martin

mathaction@newsynthesis.org (gdr) writes:

> fixed in OpenAxiom
> --
> forwarded from http://axiom-wiki.newsynthesis.org/311RepeatedOutputOfTheShowSystemCommandForCategoriesThatDoNotExportOperations#msg20080124221858-0800@axiom-wiki.newsynthesis.org




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On Fri, 25 Jan 2008, Martin Rubey wrote:

| 
| Dear bug masters :-),
| 
| so far I made all (well, mostly algebra) bugs that are fixed in FriCAS "fix
| proposed" on IssueTracker.  The reason being that I would like to see at a
| glance which bugs are open, and with the philosophy that everybody could just
| diff the algebra sources and get the fix from the FriCAS sources.

I believe that produces more work than actually help.  There are
growing differences that you cannot just require people to diff
trees.  If you want to make the Axiom Wiki The Issue Tracker for
FriCAS, that is fine.  But, don't tell us to go and diff it against
other trees. 

| However, there are also important bug fixes in open-axiom.  Gaby has added a
| comment "fixed in OpenAxiom", which is fine. 

It is far more accurate than saying fix proposed -- because I don't
think it makes sense for me to tell people to go and diff their trees
against OpenAxiom to extrapolate a `proposed fix' -- I'm not saying
they should not be doing it, but I don't think that qualifies as `fix
proposed'. 

[...]

| In any case, I dislike the situation as it is now - I do not want to see
| OpenAxiom treated differently then OpenAxiom.

But, you're already actively doing it.

-- Gaby



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Gabriel Dos Reis <gdr@cs.tamu.edu> writes:

> Hi,
> 
>   I would suggest to use `fix proposed' on the Axiom Wiki only when
> a patch is effectly proposed as an attachment in the tracker, as
> opposed to the mere fact that the bug is fixed in an Axiom flavor.
> That would help reduce confusion.

Oh, sorry, I overlooked this mail.  I think that's a good suggestion, although
it does not make me happy, because it may be a lot of work to do so.  In fact,
it only makes sense where the code bases are still close enough (i.e., the
algebra, I guess)

How about splitting status fix proposed into

1) fix proposed                              (i.e, patch attached)
2) fixed in one of OpenAxiom, FriCAS, Axiom  (a shorter name would be needed here)

?

In any case, the current situation is very unsatisfactory, as I wrote also in
the other email.

Martin




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From: Martin Rubey <martin.rubey@univie.ac.at>
Date: 25 Jan 2008 08:57:35 +0100
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Subject: [Axiom-developer] Re: [open-axiom-devel] [fricas-devel] "fixed
	proposed"
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Gabriel Dos Reis <gdr@cs.tamu.edu> writes:

> | In any case, I dislike the situation as it is now - I do not want to see
> | OpenAxiom treated differently then OpenAxiom.

(change one "OpenAxiom" to FriCAS, of course)

> But, you're already actively doing it.

Yes, I know.  And I feel bad about it.  For my defense, I did propose a
solution and there was very little reaction.  I like the proposal you made -
let's continue discussion on the other thread, however.

Thanks for looking into this,

Martin




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On Fri, 25 Jan 2008, Martin Rubey wrote:

| 
| Gabriel Dos Reis <gdr@cs.tamu.edu> writes:
| 
| > Hi,
| > 
| >   I would suggest to use `fix proposed' on the Axiom Wiki only when
| > a patch is effectly proposed as an attachment in the tracker, as
| > opposed to the mere fact that the bug is fixed in an Axiom flavor.
| > That would help reduce confusion.
| 
| Oh, sorry, I overlooked this mail.  I think that's a good suggestion, although
| it does not make me happy, because it may be a lot of work to do so.

It may be a lot of work, when you do effectively propose a fix -- not
just a comment that it is fixed somewhere.  So, you could still
maintain the level of work you're doing now by just saying it is fixed
somewhere, without chaning to `fix proposed'.

| In fact,
| it only makes sense where the code bases are still close enough (i.e., the
| algebra, I guess)
| 
| How about splitting status fix proposed into
| 
| 1) fix proposed                              (i.e, patch attached)
| 2) fixed in one of OpenAxiom, FriCAS, Axiom  (a shorter name would be needed here)
| 
| ?

That sounds good.

-- Gaby



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Gabriel Dos Reis <gdr@cs.tamu.edu> writes:

> | How about splitting status fix proposed into
> | 
> | 1) fix proposed                              (i.e, patch attached)
> | 2) fixed in one of OpenAxiom, FriCAS, Axiom  (a shorter name would be needed here)
> | 
> | ?
> 
> That sounds good.

OK, I'll implement this if there is no protest.  (I must say I'm very happy
that you made me think of this possibility!)

When I implement it, please bear with me: I will be unable to upload patches to
every Issue that is marked "fix proposed".

About usage, what do you think:


-------------------------------------------------------------------------------
A:

1) fix proposed

   would be used if a patch is attached, no matter whether the patch has been
   included in one of the systems

2) fixed in one of OpenAxiom, FriCAS, Axiom  

   would be used if no patch is attached, but the issue is fixed somewhere
-------------------------------------------------------------------------------

or

-------------------------------------------------------------------------------
B:

1) fix proposed

   would be used if a patch is attached, but it hasn't been fixed in one of the
   systems

2) fixed in one of OpenAxiom, FriCAS, Axiom  

   would be used if the issue is fixed somewhere, regardless whether a patch is
   included or not.
-------------------------------------------------------------------------------


I vote (from the top of my head) for B, but I'm not sure.  In A, I would have
to move an element from (2) to (1) when I include a patch. B seems to be a
little less work for those who fix an issue in "their" system.



Martin




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This patch adds the Exponential Integral for higher orders $E_n$.
It also adds a regression test that checks the results against the
results published in Abramowitz and Stegun.

Tim

===========================================================================
diff --git a/changelog b/changelog
index 665492c..9cfc6cf 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080125 tpd src/input/Makefile add En regression test 
+20080125 tpd src/input/en.input regression test En
+20080125 tpd src/algebra/special.spad add special function En 
 20080120 tpd src/input/Makefile add tbagg.input regression test (7102/412)
 20080120 tpd src/input/tbagg.input regression test equality in TBAGG (7102/412)
 20080120 mxr src/algebra/aggcat.spad fix equality in TBAGG (7102/412)
diff --git a/src/algebra/special.spad.pamphlet b/src/algebra/special.spad.pamphlet
index 0744b97..6b173ae 100644
--- a/src/algebra/special.spad.pamphlet
+++ b/src/algebra/special.spad.pamphlet
@@ -28,6 +28,7 @@
 
 DoubleFloatSpecialFunctions(): Exports == Impl where
     NNI ==> NonNegativeInteger
+    PI  ==> Integer
     R   ==> DoubleFloat
     C   ==> Complex DoubleFloat
     OPR ==> OnePointCompletion R
@@ -45,6 +46,9 @@ DoubleFloatSpecialFunctions(): Exports == Impl where
          ++ The current implementation is a piecewise approximation
          ++ involving one poly from -4..4 and a second poly for x > 4
 
+        En: (PI,R) -> OPR
+         ++ En(n,x) is the nth Exponential Integral Function
+
         Beta: (R, R) -> R
          ++ Beta(x, y) is the Euler beta function, \spad{B(x,y)}, defined by
          ++   \spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.
@@ -344,7 +348,7 @@ the Chebyshev polynomial for computing $E_1$. This agrees with the
 handbook values to almost the last published digit. See the {\tt e1.input}
 pamphlet for regression testing against the handbook tables.
 
-\subsection{E1:R$\rightarrow$R}
+\subsection{E1:R$\rightarrow$OPR}
 The special function E1 below was originally derived from a function
 written by T.Haavie as the {\tt expint.c} function in the Numlibc library
 by Lars Erik Lund. Haavie approximates the E1 function by two 
@@ -377,6 +381,10 @@ that take into account the scaling used by Haavie. I've also
 rewritten the polynomial using Horner's method so the large
 powers of $x$ are only computed once.
 
+The result can be either a double float or, or if the argument
+is zero, infinity. Thus we need to extend the result to be a
+one-point completion to include infinity.
+
 <<package DFSFUN DoubleFloatSpecialFunctions>>=
         E1(x:R):OPR ==
          x = 0.0::R => infinity()
@@ -481,6 +489,20 @@ powers of $x$ are only computed once.
           aw:R:=- 1.0::R*log(abs(x)) + av
           aw::OPR
          error "E1: no approximation available"
+@
+\subsection{En:(PI,R)$\rightarrow$OPR}
+The $E_n$ function is computed using the recurrence relation:
+$$E_{n+1}(z)=\frac{1}{n}\left(e^{-z}-zE_n(z)\right)\ \ \ (n=1,2,3,\ldots)$$
+
+The base case of the recursion depends on E1 above.
+
+The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}.
+<<package DFSFUN DoubleFloatSpecialFunctions>>=
+        En(n:PI,x:R):OPR == 
+          n=1 => E1(x) 
+          v:R:=retract(En((n-1)::PI,x))
+          w:R:=1/(n-1)*(exp(-x)-x*v)
+          w::OPR
 
         polygamma(k,z)  == CPSI(k, z)$Lisp
         polygamma(k,x)  == RPSI(k, x)$Lisp
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 65741f8..6cc1513 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -305,7 +305,8 @@ REGRES= algaggr.regress algbrbf.regress  algfacob.regress alist.regress  \
     dmp.regress       dpol.regress     e1.regress       easter.regress  \
     efi.regress \
     eigen.regress     elemfun.regress  elemnum.regress  elfuts.regress \
-    elt.regress       eq.regress       eqtbl.regress    equation2.regress \
+    elt.regress       en.regress \
+    eq.regress        eqtbl.regress    equation2.regress \
     equation.regress  evalex.regress   eval.regress     exdiff.regress \
     exint.regress     exit.regress     exlap.regress    exlimit.regress \
     exp.regress \
@@ -544,7 +545,7 @@ FILES= ${OUT}/algaggr.input  ${OUT}/algbrbf.input    ${OUT}/algfacob.input \
        ${OUT}/easter.input   ${OUT}/efi.input        ${OUT}/egg.input      \
        ${OUT}/eigen.input \
        ${OUT}/elemfun.input  ${OUT}/elemnum.input    ${OUT}/elfuts.input \
-       ${OUT}/elt.input \
+       ${OUT}/elt.input      ${OUT}/en.input \
        ${OUT}/eq.input       ${OUT}/eqtbl.input      ${OUT}/equation.input \
        ${OUT}/errortrap.input \
        ${OUT}/eval.input     ${OUT}/exit.input       ${OUT}/exp.input \
@@ -780,6 +781,7 @@ DOCFILES= \
   ${DOC}/egg.input.dvi         ${DOC}/eigen.input.dvi      \
   ${DOC}/elemfun.input.dvi     ${DOC}/elemnum.input.dvi    \
   ${DOC}/elfuts.input.dvi      ${DOC}/elt.input.dvi        \
+  ${DOC}/en.input.dvi \
   ${DOC}/eq.input.dvi          ${DOC}/eqtbl.input.dvi      \
   ${DOC}/equation2.input.dvi   ${DOC}/equation.input.dvi   \
   ${DOC}/errortrap.input.dvi   ${DOC}/evalex.input.dvi     \
diff --git a/src/input/en.input.pamphlet b/src/input/en.input.pamphlet
new file mode 100644
index 0000000..1ae9749
--- /dev/null
+++ b/src/input/en.input.pamphlet
@@ -0,0 +1,3805 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input en.input}
+\author{Timothy Daly}
+\maketitle
+\begin{abstract}
+This compares Axiom's $E_n$ function with the tables published in
+Abramowitz and Stegun.
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+In the following table there are 4 columns. The first column
+is the argument of $E_n$, ranging from 0.01 to 2.00. The
+second column is the reference value of function from the book
+Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp245-248
+
+<<*>>=
+)spool en.output
+)set message test on
+)set message auto off
+)clear all
+
+@
+This table expresses the value of 
+$E_2$ with an argument range of 0.01 to 0.50.
+The handbook tabulates these values using this function.
+<<*>>=
+--S 1 of 7
+f(x)==En(2,x)-x*log(x)
+--E 1
+
+--S 2 of 7
+[[0.01,0.9957222,f(0.01),f(0.01)-0.9957222],_
+[0.02,0.9913450,f(0.02),f(0.02)-0.9913450],_
+[0.03,0.9868687,f(0.03),f(0.03)-0.9868687],_
+[0.04,0.9822939,f(0.04),f(0.04)-0.9822939],_
+[0.05,0.9776211,f(0.05),f(0.05)-0.9776211],_
+[0.06,0.9728508,f(0.06),f(0.06)-0.9728508],_
+[0.07,0.9679834,f(0.07),f(0.07)-0.9679834],_
+[0.08,0.9630194,f(0.08),f(0.08)-0.9630194],_
+[0.09,0.9579593,f(0.09),f(0.09)-0.9579593],_
+[0.10,0.9528035,f(0.10),f(0.10)-0.9528035],_
+[0.11,0.9475526,f(0.11),f(0.11)-0.9475526],_
+[0.12,0.9422071,f(0.12),f(0.12)-0.9422071],_
+[0.13,0.9367672,f(0.13),f(0.13)-0.9367672],_
+[0.14,0.9312336,f(0.14),f(0.14)-0.9312336],_
+[0.15,0.9256067,f(0.15),f(0.15)-0.9256067],_
+[0.16,0.9198870,f(0.16),f(0.16)-0.9198870],_
+[0.17,0.9140748,f(0.17),f(0.17)-0.9140748],_
+[0.18,0.9081706,f(0.18),f(0.18)-0.9081706],_
+[0.19,0.9021750,f(0.19),f(0.19)-0.9021750],_
+[0.20,0.8960882,f(0.20),f(0.20)-0.8960882],_
+[0.21,0.8899109,f(0.21),f(0.21)-0.8899109],_
+[0.22,0.8836433,f(0.22),f(0.22)-0.8836433],_
+[0.23,0.8772860,f(0.23),f(0.23)-0.8772860],_
+[0.24,0.8708393,f(0.24),f(0.24)-0.8708393],_
+[0.25,0.8643037,f(0.25),f(0.25)-0.8643037],_
+[0.26,0.8576797,f(0.26),f(0.26)-0.8576797],_
+[0.27,0.8509676,f(0.27),f(0.27)-0.8509676],_
+[0.28,0.8441678,f(0.28),f(0.28)-0.8441678],_
+[0.29,0.8372808,f(0.29),f(0.29)-0.8372808],_
+[0.30,0.8303071,f(0.30),f(0.30)-0.8303071],_
+[0.31,0.8232469,f(0.31),f(0.31)-0.8232469],_
+[0.32,0.8161007,f(0.32),f(0.32)-0.8161007],_
+[0.33,0.8088690,f(0.33),f(0.33)-0.8088690],_
+[0.34,0.8015521,f(0.34),f(0.34)-0.8015521],_
+[0.35,0.7941504,f(0.35),f(0.35)-0.7941504],_
+[0.36,0.7866644,f(0.36),f(0.36)-0.7866644],_
+[0.37,0.7790943,f(0.37),f(0.37)-0.7790943],_
+[0.38,0.7714407,f(0.38),f(0.38)-0.7714407],_
+[0.39,0.7637039,f(0.39),f(0.39)-0.7637039],_
+[0.40,0.7558843,f(0.40),f(0.40)-0.7558843],_
+[0.41,0.7479823,f(0.41),f(0.41)-0.7479823],_
+[0.42,0.7399982,f(0.42),f(0.42)-0.7399982],_
+[0.43,0.7319324,f(0.43),f(0.43)-0.7319324],_
+[0.44,0.7237854,f(0.44),f(0.44)-0.7237854],_
+[0.45,0.7155575,f(0.45),f(0.45)-0.7155575],_
+[0.46,0.7072491,f(0.46),f(0.46)-0.7072491],_
+[0.47,0.6988605,f(0.47),f(0.47)-0.6988605],_
+[0.48,0.6903921,f(0.48),f(0.48)-0.6903921],_
+[0.49,0.6818443,f(0.49),f(0.49)-0.6818443],_
+[0.50,0.6732175,f(0.50),f(0.50)-0.6732175]]
+--R 
+--R   Compiling function f with type Float -> OnePointCompletion 
+--R      DoubleFloat 
+--R
+--R   (2)
+--R   [[1.0E-2,0.9957222,0.99572223984366792,3.9843667920536063E-8],
+--R    [2.0E-2,0.99134500000000003,0.991344977749124,- 2.2250876030582845E-8],
+--R
+--R     [2.9999999999999999E-2, 0.98686870000000004, 0.98686870874746913,
+--R      8.7474690824151935E-9]
+--R     ,
+--R
+--R     [4.0000000000000001E-2, 0.98229390000000005, 0.98229392458604003,
+--R      2.4586039981677743E-8]
+--R     ,
+--R
+--R     [5.0000000000000003E-2, 0.97762110000000002, 0.97762111375291483,
+--R      1.3752914806985927E-8]
+--R     ,
+--R
+--R     [5.9999999999999998E-2, 0.97285080000000002, 0.97285076150122396,
+--R      - 3.8498776055995165E-8]
+--R     ,
+--R
+--R     [7.0000000000000007E-2, 0.96798340000000005, 0.96798334987326684,
+--R      - 5.0126733208699648E-8]
+--R     ,
+--R
+--R     [8.0000000000000002E-2, 0.96301939999999997, 0.96301935772443681,
+--R      - 4.2275563161275898E-8]
+--R     ,
+--R
+--R     [8.9999999999999997E-2, 0.95795929999999996, 0.95795926074695703,
+--R      - 3.9253042927178683E-8]
+--R     ,
+--R
+--R     [0.10000000000000001, 0.95280350000000003, 0.95280353149342489,
+--R      3.149342486796769E-8]
+--R     ,
+--R    [0.11,0.94755259999999997,0.94755263940017342,3.9400173457160292E-8],
+--R    [0.12,0.94220709999999996,0.94220705081044254,- 4.9189557427276043E-8],
+--R    [0.13,0.93676720000000002,0.93676722899736986,2.8997369838634768E-8],
+--R
+--R     [0.14000000000000001, 0.93123359999999999, 0.93123363418679772,
+--R      3.4186797726043494E-8]
+--R     ,
+--R    [0.14999999999999999,0.9256067,0.92560672357989859,2.3579898589254356E-8],
+--R    [0.16,0.91988700000000001,0.91988695137562182,- 4.8624378190531559E-8],
+--R
+--R     [0.17000000000000001, 0.91407479999999997, 0.91407476879296246,
+--R      - 3.1207037509695112E-8]
+--R     ,
+--R
+--R     [0.17999999999999999, 0.90817060000000005, 0.90817062409305238,
+--R      2.4093052330975695E-8]
+--R     ,
+--R    [0.19,0.90217499999999995,0.90217496260107799,- 3.7398921959308495E-8],
+--R    [0.20000000000000001,0.8960882,0.89608822672802324,2.6728023239108722E-8],
+--R
+--R     [0.20999999999999999, 0.88991089999999995, 0.88991085599224007,
+--R      - 4.400775988067096E-8]
+--R     ,
+--R    [0.22,0.88364330000000002,0.88364328704084927,- 1.2959150752500648E-8],
+--R
+--R     [0.23000000000000001, 0.87728600000000001, 0.87728595367097117,
+--R      - 4.6329028835501163E-8]
+--R     ,
+--R
+--R     [0.23999999999999999, 0.87083929999999998, 0.8708392868507886,
+--R      - 1.3149211386398463E-8]
+--R     ,
+--R    [0.25,0.86430370000000001,0.86430371474044287,1.4740442866312264E-8],
+--R
+--R     [0.26000000000000001, 0.85767970000000004, 0.85767966271276586,
+--R      - 3.7287234189165019E-8]
+--R     ,
+--R
+--R     [0.27000000000000002, 0.85096760000000005, 0.85096755337384578,
+--R      - 4.6626154270867914E-8]
+--R     ,
+--R
+--R     [0.28000000000000003, 0.84416780000000002, 0.84416780658343327,
+--R      6.5834332430014797E-9]
+--R     ,
+--R
+--R     [0.28999999999999998, 0.83728080000000005, 0.8372808394751845,
+--R      3.9475184454573764E-8]
+--R     ,
+--R
+--R     [0.29999999999999999, 0.83030709999999996, 0.83030706647674468,
+--R      - 3.3523255282830178E-8]
+--R     ,
+--R    [0.31,0.8232469,0.82324689932967399,- 6.7032601691607852E-10],
+--R
+--R     [0.32000000000000001, 0.81610070000000001, 0.81610074710921554,
+--R      4.7109215528529091E-8]
+--R     ,
+--R
+--R     [0.33000000000000002, 0.80886899999999995, 0.80886901624390695,
+--R      1.6243906997281954E-8]
+--R     ,
+--R
+--R     [0.34000000000000002, 0.80155209999999999, 0.80155211053503872,
+--R      1.0535038730274948E-8]
+--R     ,
+--R
+--R     [0.34999999999999998, 0.79415040000000003, 0.79415043117595796,
+--R      3.1175957926343756E-8]
+--R     ,
+--R
+--R     [0.35999999999999999, 0.78666440000000004, 0.78666437677122092,
+--R      - 2.3228779122419496E-8]
+--R     ,
+--R    [0.37,0.77909430000000002,0.77909434335559369,4.3355593670746373E-8],
+--R    [0.38,0.77144069999999998,0.77144072441290445,2.4412904475745734E-8],
+--R
+--R     [0.39000000000000001, 0.76370389999999999, 0.76370391089474698,
+--R      1.089474699345061E-8]
+--R     ,
+--R
+--R     [0.40000000000000002, 0.75588429999999995, 0.75588429123903633,
+--R      - 8.7609636212349074E-9]
+--R     ,
+--R
+--R     [0.40999999999999998, 0.74798229999999999, 0.74798225138841923,
+--R      - 4.8611580760749007E-8]
+--R     ,
+--R
+--R     [0.41999999999999998, 0.73999820000000005, 0.73999817480853991,
+--R      - 2.5191460140128186E-8]
+--R     ,
+--R
+--R     [0.42999999999999999, 0.73193240000000004, 0.73193244250616207,
+--R      4.2506162034605666E-8]
+--R     ,
+--R    [0.44,0.72378540000000002,0.7237854330471486,3.3047148573217555E-8],
+--R
+--R     [0.45000000000000001, 0.71555749999999996, 0.71555752257429939,
+--R      2.2574299429756195E-8]
+--R     ,
+--R
+--R     [0.46000000000000002, 0.70724909999999996, 0.70724908482505056,
+--R      - 1.517494940816988E-8]
+--R     ,
+--R    [0.46999999999999997,0.6988605,0.6988604911490337,- 8.8509662932167998E-9],
+--R
+--R     [0.47999999999999998, 0.69039209999999995, 0.69039211052549776,
+--R      1.0525497806668227E-8]
+--R     ,
+--R
+--R     [0.48999999999999999, 0.68184429999999996, 0.68184430958059394,
+--R      9.5805939848148114E-9]
+--R     ,
+--R    [0.5,0.67321750000000002,0.67321745260452559,- 4.739547443399772E-8]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 2
+
+@
+This table expresses the value of 
+$E_2$ with an argument range of 0.50 to 2.00.
+<<*>>=
+--S 3 of 7
+[[0.50,0.3266439,En(2,0.50),En(2,0.50)-0.3266439],_
+[0.51,0.3211062,En(2,0.51),En(2,0.51)-0.3211062],_
+[0.52,0.3156863,En(2,0.52),En(2,0.52)-0.3156863],_
+[0.53,0.3103807,En(2,0.53),En(2,0.53)-0.3103807],_
+[0.54,0.3051862,En(2,0.54),En(2,0.54)-0.3051862],_
+[0.55,0.3000996,En(2,0.55),En(2,0.55)-0.3000996],_
+[0.56,0.2951179,En(2,0.56),En(2,0.56)-0.2951179],_
+[0.57,0.2902382,En(2,0.57),En(2,0.57)-0.2902382],_
+[0.58,0.2854578,En(2,0.58),En(2,0.58)-0.2854578],_
+[0.59,0.2807739,En(2,0.59),En(2,0.59)-0.2807739],_
+[0.60,0.2761839,En(2,0.60),En(2,0.60)-0.2761839],_
+[0.61,0.2716855,En(2,0.61),En(2,0.61)-0.2716855],_
+[0.62,0.2672761,En(2,0.62),En(2,0.62)-0.2672761],_
+[0.63,0.2629535,En(2,0.63),En(2,0.63)-0.2629535],_
+[0.64,0.2587154,En(2,0.64),En(2,0.64)-0.2587154],_
+[0.65,0.2545597,En(2,0.65),En(2,0.65)-0.2545597],_
+[0.66,0.2504844,En(2,0.66),En(2,0.66)-0.2504844],_
+[0.67,0.2464874,En(2,0.67),En(2,0.67)-0.2464874],_
+[0.68,0.2425667,En(2,0.68),En(2,0.68)-0.2425667],_
+[0.69,0.2387206,En(2,0.69),En(2,0.69)-0.2387206],_
+[0.70,0.2349471,En(2,0.70),En(2,0.70)-0.2349471],_
+[0.71,0.2312446,En(2,0.71),En(2,0.71)-0.2312446],_
+[0.72,0.2276114,En(2,0.72),En(2,0.72)-0.2276114],_
+[0.73,0.2240457,En(2,0.73),En(2,0.73)-0.2240457],_
+[0.74,0.2205461,En(2,0.74),En(2,0.74)-0.2205461],_
+[0.75,0.2171109,En(2,0.75),En(2,0.75)-0.2171109],_
+[0.76,0.2137388,En(2,0.76),En(2,0.76)-0.2137388],_
+[0.77,0.2104282,En(2,0.77),En(2,0.77)-0.2104282],_
+[0.78,0.2071777,En(2,0.78),En(2,0.78)-0.2071777],_
+[0.79,0.2039860,En(2,0.79),En(2,0.79)-0.2039860],_
+[0.80,0.2008517,En(2,0.80),En(2,0.80)-0.2008517],_
+[0.81,0.1977736,En(2,0.81),En(2,0.81)-0.1977736],_
+[0.82,0.1947504,En(2,0.82),En(2,0.82)-0.1947504],_
+[0.83,0.1917810,En(2,0.83),En(2,0.83)-0.1917810],_
+[0.84,0.1888641,En(2,0.84),En(2,0.84)-0.1888641],_
+[0.85,0.1859986,En(2,0.85),En(2,0.85)-0.1859986],_
+[0.86,0.1831833,En(2,0.86),En(2,0.86)-0.1831833],_
+[0.87,0.1804173,En(2,0.87),En(2,0.87)-0.1804173],_
+[0.88,0.1776994,En(2,0.88),En(2,0.88)-0.1776994],_
+[0.89,0.1750287,En(2,0.89),En(2,0.89)-0.1750287],_
+[0.90,0.1724041,En(2,0.90),En(2,0.90)-0.1724041],_
+[0.91,0.1698247,En(2,0.91),En(2,0.91)-0.1698247],_
+[0.92,0.1672895,En(2,0.92),En(2,0.92)-0.1672895],_
+[0.93,0.1647977,En(2,0.93),En(2,0.93)-0.1647977],_
+[0.94,0.1623482,En(2,0.94),En(2,0.94)-0.1623482],_
+[0.95,0.1599404,En(2,0.95),En(2,0.95)-0.1599404],_
+[0.96,0.1575732,En(2,0.96),En(2,0.96)-0.1575732],_
+[0.97,0.1552459,En(2,0.97),En(2,0.97)-0.1552459],_
+[0.98,0.1529578,En(2,0.98),En(2,0.98)-0.1529578],_
+[0.99,0.1507079,En(2,0.99),En(2,0.99)-0.1507079],_
+[1.00,0.1484955,En(2,1.00),En(2,1.00)-0.1484955],_
+[1.01,0.1463199,En(2,1.01),En(2,1.01)-0.1463199],_
+[1.02,0.1441804,En(2,1.02),En(2,1.02)-0.1441804],_
+[1.03,0.1420763,En(2,1.03),En(2,1.03)-0.1420763],_
+[1.04,0.1400068,En(2,1.04),En(2,1.04)-0.1400068],_
+[1.05,0.1379713,En(2,1.05),En(2,1.05)-0.1379713],_
+[1.06,0.1359691,En(2,1.06),En(2,1.06)-0.1359691],_
+[1.07,0.1339996,En(2,1.07),En(2,1.07)-0.1339996],_
+[1.08,0.1320622,En(2,1.08),En(2,1.08)-0.1320622],_
+[1.09,0.1301562,En(2,1.09),En(2,1.09)-0.1301562],_
+[1.10,0.1282811,En(2,1.10),En(2,1.10)-0.1282811],_
+[1.11,0.1264362,En(2,1.11),En(2,1.11)-0.1264362],_
+[1.12,0.1246210,En(2,1.12),En(2,1.12)-0.1246210],_
+[1.13,0.1228350,En(2,1.13),En(2,1.13)-0.1228350],_
+[1.14,0.1210775,En(2,1.14),En(2,1.14)-0.1210775],_
+[1.15,0.1193481,En(2,1.15),En(2,1.15)-0.1193481],_
+[1.16,0.1176462,En(2,1.16),En(2,1.16)-0.1176462],_
+[1.17,0.1159714,En(2,1.17),En(2,1.17)-0.1159714],_
+[1.18,0.1143231,En(2,1.18),En(2,1.18)-0.1143231],_
+[1.19,0.1127008,En(2,1.19),En(2,1.19)-0.1127008],_
+[1.20,0.1111041,En(2,1.20),En(2,1.20)-0.1111041],_
+[1.21,0.1095325,En(2,1.21),En(2,1.21)-0.1095325],_
+[1.22,0.1079855,En(2,1.22),En(2,1.22)-0.1079855],_
+[1.23,0.1064627,En(2,1.23),En(2,1.23)-0.1064627],_
+[1.24,0.1049637,En(2,1.24),En(2,1.24)-0.1049637],_
+[1.25,0.1034881,En(2,1.25),En(2,1.25)-0.1034881],_
+[1.26,0.1020353,En(2,1.26),En(2,1.26)-0.1020353],_
+[1.27,0.1006051,En(2,1.27),En(2,1.27)-0.1006051],_
+[1.28,0.0991970,En(2,1.28),En(2,1.28)-0.0991970],_
+[1.29,0.0978106,En(2,1.29),En(2,1.29)-0.0978106],_
+[1.30,0.0964455,En(2,1.30),En(2,1.30)-0.0964455],_
+[1.31,0.0951015,En(2,1.31),En(2,1.31)-0.0951015],_
+[1.32,0.0937780,En(2,1.32),En(2,1.32)-0.0937780],_
+[1.33,0.0924747,En(2,1.33),En(2,1.33)-0.0924747],_
+[1.34,0.0911913,En(2,1.34),En(2,1.34)-0.0911913],_
+[1.35,0.0899275,En(2,1.35),En(2,1.35)-0.0899275],_
+[1.36,0.0886829,En(2,1.36),En(2,1.36)-0.0886829],_
+[1.37,0.0874571,En(2,1.37),En(2,1.37)-0.0874571],_
+[1.38,0.0862499,En(2,1.38),En(2,1.38)-0.0862499],_
+[1.39,0.0850610,En(2,1.39),En(2,1.39)-0.0850610],_
+[1.40,0.0838899,En(2,1.40),En(2,1.40)-0.0838899],_
+[1.41,0.0827365,En(2,1.41),En(2,1.41)-0.0827365],_
+[1.42,0.0816004,En(2,1.42),En(2,1.42)-0.0816004],_
+[1.43,0.0804813,En(2,1.43),En(2,1.43)-0.0804813],_
+[1.44,0.0793789,En(2,1.44),En(2,1.44)-0.0793789],_
+[1.45,0.0782930,En(2,1.45),En(2,1.45)-0.0782930],_
+[1.46,0.0772233,En(2,1.46),En(2,1.46)-0.0772233],_
+[1.47,0.0761694,En(2,1.47),En(2,1.47)-0.0761694],_
+[1.48,0.0751313,En(2,1.48),En(2,1.48)-0.0751313],_
+[1.49,0.0741085,En(2,1.49),En(2,1.49)-0.0741085],_
+[1.50,0.0731008,En(2,1.50),En(2,1.50)-0.0731008],_
+[1.51,0.0721080,En(2,1.51),En(2,1.51)-0.0721080],_
+[1.52,0.0711298,En(2,1.52),En(2,1.52)-0.0711298],_
+[1.53,0.0701660,En(2,1.53),En(2,1.53)-0.0701660],_
+[1.54,0.0692164,En(2,1.54),En(2,1.54)-0.0692164],_
+[1.55,0.0682807,En(2,1.55),En(2,1.55)-0.0682807],_
+[1.56,0.0673587,En(2,1.56),En(2,1.56)-0.0673587],_
+[1.57,0.0664502,En(2,1.57),En(2,1.57)-0.0664502],_
+[1.58,0.0655549,En(2,1.58),En(2,1.58)-0.0655549],_
+[1.59,0.0646726,En(2,1.59),En(2,1.59)-0.0646726],_
+[1.60,0.0638032,En(2,1.60),En(2,1.60)-0.0638032],_
+[1.61,0.0629464,En(2,1.61),En(2,1.61)-0.0629464],_
+[1.62,0.0621020,En(2,1.62),En(2,1.62)-0.0621020],_
+[1.63,0.0612698,En(2,1.63),En(2,1.63)-0.0612698],_
+[1.64,0.0604497,En(2,1.64),En(2,1.64)-0.0604497],_
+[1.65,0.0596413,En(2,1.65),En(2,1.65)-0.0596413],_
+[1.66,0.0588446,En(2,1.66),En(2,1.66)-0.0588446],_
+[1.67,0.0580594,En(2,1.67),En(2,1.67)-0.0580594],_
+[1.68,0.0572854,En(2,1.68),En(2,1.68)-0.0572854],_
+[1.69,0.0565226,En(2,1.69),En(2,1.69)-0.0565226],_
+[1.70,0.0557706,En(2,1.70),En(2,1.70)-0.0557706],_
+[1.71,0.0550294,En(2,1.71),En(2,1.71)-0.0550294],_
+[1.72,0.0542988,En(2,1.72),En(2,1.72)-0.0542988],_
+[1.73,0.0535786,En(2,1.73),En(2,1.73)-0.0535786],_
+[1.74,0.0528686,En(2,1.74),En(2,1.74)-0.0528686],_
+[1.75,0.0521687,En(2,1.75),En(2,1.75)-0.0521687],_
+[1.76,0.0514788,En(2,1.76),En(2,1.76)-0.0514788],_
+[1.77,0.0507986,En(2,1.77),En(2,1.77)-0.0507986],_
+[1.78,0.0501281,En(2,1.78),En(2,1.78)-0.0501281],_
+[1.79,0.0494670,En(2,1.79),En(2,1.79)-0.0494670],_
+[1.80,0.0488153,En(2,1.80),En(2,1.80)-0.0488153],_
+[1.81,0.0481727,En(2,1.81),En(2,1.81)-0.0481727],_
+[1.82,0.0475392,En(2,1.82),En(2,1.82)-0.0475392],_
+[1.83,0.0469146,En(2,1.83),En(2,1.83)-0.0469146],_
+[1.84,0.0462987,En(2,1.84),En(2,1.84)-0.0462987],_
+[1.85,0.0456915,En(2,1.85),En(2,1.85)-0.0456915],_
+[1.86,0.0450928,En(2,1.86),En(2,1.86)-0.0450928],_
+[1.87,0.0445024,En(2,1.87),En(2,1.87)-0.0445024],_
+[1.88,0.0439203,En(2,1.88),En(2,1.88)-0.0439203],_
+[1.89,0.0433463,En(2,1.89),En(2,1.89)-0.0433463],_
+[1.90,0.0427803,En(2,1.90),En(2,1.90)-0.0427803],_
+[1.91,0.0422222,En(2,1.91),En(2,1.91)-0.0422222],_
+[1.92,0.0416718,En(2,1.92),En(2,1.92)-0.0416718],_
+[1.93,0.0411291,En(2,1.93),En(2,1.93)-0.0411291],_
+[1.94,0.0405938,En(2,1.94),En(2,1.94)-0.0405938],_
+[1.95,0.0400660,En(2,1.95),En(2,1.95)-0.0400660],_
+[1.96,0.0395455,En(2,1.96),En(2,1.96)-0.0395455],_
+[1.97,0.0390322,En(2,1.97),En(2,1.97)-0.0390322],_
+[1.98,0.0385259,En(2,1.98),En(2,1.98)-0.0385259],_
+[1.99,0.0380267,En(2,1.99),En(2,1.99)-0.0380267],_
+[2.00,0.0375343,En(2,2.00),En(2,2.00)-0.0375343]]
+--R 
+--R
+--R   (3)
+--R   [[0.5,0.32664389999999999,0.326643862324553,- 3.7675446984408723E-8],
+--R
+--R     [0.51000000000000001, 0.32110620000000001, 0.32110617940404323,
+--R      - 2.0595956773394875E-8]
+--R     ,
+--R
+--R     [0.52000000000000002, 0.31568629999999998, 0.31568625309046355,
+--R      - 4.6909536421946285E-8]
+--R     ,
+--R
+--R     [0.53000000000000003, 0.31038070000000001, 0.31038066931747649,
+--R      - 3.0682523521896599E-8]
+--R     ,
+--R
+--R     [0.54000000000000004, 0.30518620000000002, 0.30518615409477512,
+--R      - 4.5905224899112085E-8]
+--R     ,
+--R
+--R     [0.55000000000000004, 0.30009960000000002, 0.30009956561466999,
+--R      - 3.4385330027753014E-8]
+--R     ,
+--R
+--R     [0.56000000000000005, 0.29511789999999999, 0.29511788693397883,
+--R      - 1.3066021153917973E-8]
+--R     ,
+--R    [0.56999999999999995,0.2902382,0.29023821917982273,1.9179822730031049E-8],
+--R
+--R     [0.57999999999999996, 0.28545779999999998, 0.28545777523334881,
+--R      - 2.4766651174346066E-8]
+--R     ,
+--R
+--R     [0.58999999999999997, 0.28077390000000002, 0.28077387385015457,
+--R      - 2.6149845455680776E-8]
+--R     ,
+--R
+--R     [0.59999999999999998, 0.27618389999999998, 0.27618393418038506,
+--R      3.4180385077853259E-8]
+--R     ,
+--R
+--R     [0.60999999999999999, 0.27168550000000002, 0.27168547065517928,
+--R      - 2.9344820740018207E-8]
+--R     ,
+--R    [0.62,0.26727610000000002,0.26727608820941573,- 1.1790584286686112E-8],
+--R    [0.63,0.26295350000000001,0.26295347781362555,- 2.2186374459831626E-8],
+--R
+--R     [0.64000000000000001, 0.25871539999999998, 0.25871541229051526,
+--R      1.2290515272184166E-8]
+--R     ,
+--R    [0.65000000000000002,0.2545597,0.25455974239385432,4.2393854315925239E-8],
+--R    [0.66000000000000003,0.2504844,0.25048439312952969,- 6.8704703037880677E-9],
+--R    [0.67000000000000004,0.2464874,0.24648736030041074,- 3.969958925487127E-8],
+--R    [0.68000000000000005,0.2425667,0.24256670725830815,7.2583081489607792E-9],
+--R
+--R     [0.68999999999999995, 0.23872060000000001, 0.23872056184779239,
+--R      - 3.8152207615382849E-8]
+--R     ,
+--R
+--R     [0.69999999999999996, 0.23494709999999999, 0.23494711352795306,
+--R      1.3527953063308118E-8]
+--R     ,
+--R
+--R     [0.70999999999999996, 0.23124459999999999, 0.23124461065938429,
+--R      1.0659384291900054E-8]
+--R     ,
+--R
+--R     [0.71999999999999997, 0.22761139999999999, 0.22761135794474674,
+--R      - 4.2055253252071267E-8]
+--R     ,
+--R
+--R     [0.72999999999999998, 0.22404569999999999, 0.22404571401223494,
+--R      1.4012234955673719E-8]
+--R     ,
+--R
+--R     [0.73999999999999999, 0.22054609999999999, 0.22054608913215246,
+--R      - 1.0867847538564845E-8]
+--R     ,
+--R    [0.75,0.2171109,0.21711094305759215,4.3057592158390889E-8],
+--R
+--R     [0.76000000000000001, 0.21373880000000001, 0.21373878298094046,
+--R      - 1.7019059550538174E-8]
+--R     ,
+--R
+--R     [0.77000000000000002, 0.21042820000000001, 0.21042816159857808,
+--R      - 3.8401421931233415E-8]
+--R     ,
+--R
+--R     [0.78000000000000003, 0.20717769999999999, 0.2071776752767438,
+--R      - 2.4723256192293874E-8]
+--R     ,
+--R    [0.79000000000000004,0.203986,0.20398596231206947,- 3.7687930526386637E-8],
+--R
+--R     [0.80000000000000004, 0.20085169999999999, 0.20085170128078714,
+--R      1.2807871430098317E-9]
+--R     ,
+--R
+--R     [0.81000000000000005, 0.19777359999999999, 0.19777360947106051,
+--R      9.4710605191838937E-9]
+--R     ,
+--R
+--R     [0.81999999999999995, 0.19475039999999999, 0.19475044139330239,
+--R      4.1393302396830478E-8]
+--R     ,
+--R
+--R     [0.82999999999999996, 0.19178100000000001, 0.19178098736371621,
+--R      - 1.2636283797107239E-8]
+--R     ,
+--R
+--R     [0.83999999999999997, 0.18886410000000001, 0.18886407215664666,
+--R      - 2.7843353345158306E-8]
+--R     ,
+--R
+--R     [0.84999999999999998, 0.18599859999999999, 0.18599855372163451,
+--R      - 4.627836547310693E-8]
+--R     ,
+--R
+--R     [0.85999999999999999, 0.18318329999999999, 0.1831833219613668,
+--R      2.1961366808431748E-8]
+--R     ,
+--R    [0.87,0.1804173,0.18041729756697517,- 2.4330248371562391E-9],
+--R    [0.88,0.17769940000000001,0.17769943090737958,3.0907379572342464E-8],
+--R
+--R     [0.89000000000000001, 0.17502870000000001, 0.17502870096960518,
+--R      9.696051739283007E-10]
+--R     ,
+--R    [0.90000000000000002,0.1724041,0.17240411434719952,1.4347199511766107E-8],
+--R    [0.91000000000000003,0.1698247,0.16982470427407523,4.2740752326242415E-9],
+--R
+--R     [0.92000000000000004, 0.16728950000000001, 0.1672895297012775,
+--R      2.9701277493021649E-8]
+--R     ,
+--R
+--R     [0.93000000000000005, 0.16479769999999999, 0.16479767441433715,
+--R      - 2.5585662838389922E-8]
+--R     ,
+--R    [0.93999999999999995,0.1623482,0.16234824618902396,4.6189023966691778E-8],
+--R
+--R     [0.94999999999999996, 0.15994040000000001, 0.15994037598345329,
+--R      - 2.4016546723570897E-8]
+--R     ,
+--R    [0.95999999999999996,0.1575732,0.15757321716462735,1.7164627358345896E-8],
+--R
+--R     [0.96999999999999997, 0.15524589999999999, 0.15524594476761389,
+--R      4.4767613893714753E-8]
+--R     ,
+--R    [0.97999999999999998,0.1529578,0.15295775478567628,- 4.5214323729503292E-8],
+--R
+--R     [0.98999999999999999, 0.15070790000000001, 0.15070786348977031,
+--R      - 3.6510229700636998E-8]
+--R     ,
+--R    [1.,0.1484955,0.14849550677592205,6.7759220456764524E-9],
+--R    [1.01,0.1463199,0.14631993953908851,3.9539088503293129E-8],
+--R    [1.02,0.14418039999999999,0.14418043507218453,3.5072184545459351E-8],
+--R    [1.03,0.14207629999999999,0.1420762844890415,- 1.551095848983941E-8],
+--R    [1.04,0.14000679999999999,0.14000679617012995,- 3.8298700322236812E-9],
+--R    [1.05,0.13797129999999999,0.13797129522994891,- 4.7700510763526438E-9],
+--R
+--R     [1.0600000000000001, 0.13596910000000001, 0.13596912300504763,
+--R      2.3005047616875274E-8]
+--R     ,
+--R    [1.0700000000000001,0.1339996,0.13399963656170108,3.6561701083348552E-8],
+--R
+--R     [1.0800000000000001, 0.13206219999999999, 0.13206220822231926,
+--R      8.2223192698904768E-9]
+--R     ,
+--R    [1.0900000000000001,0.1301562,0.13015622510971911,2.5109719109517314E-8],
+--R
+--R     [1.1000000000000001, 0.12828110000000001, 0.12828108870843541,
+--R      - 1.1291564600002246E-8]
+--R     ,
+--R    [1.1100000000000001,0.1264362,0.12643621444229519,1.4442295193095589E-8],
+--R    [1.1200000000000001,0.124621,0.12462103126751783,3.1267517838773351E-8],
+--R    [1.1299999999999999,0.122835,0.12283498128064541,- 1.871935459418772E-8],
+--R    [1.1399999999999999,0.1210775,0.12107751934064451,1.934064450259676E-8],
+--R    [1.1499999999999999,0.1193481,0.11934811270455489,1.270455489421618E-8],
+--R
+--R     [1.1599999999999999, 0.11764620000000001, 0.11764624067609286,
+--R      4.0676092855074231E-8]
+--R     ,
+--R    [1.1699999999999999,0.1159714,0.11597139426664799,- 5.7333520153690642E-9],
+--R    [1.1799999999999999,0.1143231,0.11432307586814175,- 2.4131858247788962E-8],
+--R    [1.1899999999999999,0.1127008,0.11270079893724061,- 1.0627593943768332E-9],
+--R    [1.2,0.1111041,0.11110408769044711,- 1.2309552890887865E-8],
+--R    [1.21,0.1095325,0.10953247680961009,- 2.3190389913940734E-8],
+--R    [1.22,0.1079855,0.10798551115742314,1.1157423138175027E-8],
+--R    [1.23,0.10646269999999999,0.10646274550249626,4.5502496270888315E-8],
+--R    [1.24,0.10496369999999999,0.10496374425361385,4.4253613856737317E-8],
+--R    [1.25,0.1034881,0.10348808120280234,- 1.8797197659514708E-8],
+--R    [1.26,0.1020353,0.10203533927685385,3.9276853852632243E-8],
+--R    [1.27,0.1006051,0.10060511029696789,1.0296967889455999E-8],
+--R    [1.28,9.9196999999999994E-2,9.9196994746190731E-2,- 5.2538092626397415E-9],
+--R    [1.29,9.7810599999999998E-2,9.7810601544343179E-2,1.5443431811146269E-9],
+--R    [1.3,9.6445500000000003E-2,9.6445547830144779E-2,4.7830144775384831E-8],
+--R
+--R     [1.3100000000000001, 9.5101500000000005E-2, 9.5101458750257717E-2,
+--R      - 4.1249742288584912E-8]
+--R     ,
+--R    [1.3200000000000001,9.3778E-2,9.3777967254982664E-2,- 3.2745017336521798E-8]
+--R     ,
+--R
+--R     [1.3300000000000001, 9.2474700000000007E-2, 9.2474713900349198E-2,
+--R      1.3900349191131589E-8]
+--R     ,
+--R
+--R     [1.3400000000000001, 9.1191300000000003E-2, 9.1191346656366773E-2,
+--R      4.6656366770037039E-8]
+--R     ,
+--R
+--R     [1.3500000000000001, 8.9927499999999994E-2, 8.9927520721194559E-2,
+--R      2.072119456575372E-8]
+--R     ,
+--R
+--R     [1.3600000000000001, 8.8682899999999995E-2, 8.8682898341016198E-2,
+--R      - 1.6589837975589106E-9]
+--R     ,
+--R
+--R     [1.3700000000000001, 8.7457099999999996E-2, 8.7457148635403298E-2,
+--R      4.8635403301910962E-8]
+--R     ,
+--R
+--R     [1.3799999999999999, 8.6249900000000004E-2, 8.6249947427970763E-2,
+--R      4.7427970759073013E-8]
+--R     ,
+--R
+--R     [1.3899999999999999, 8.5060999999999998E-2, 8.5060977082121597E-2,
+--R      - 2.2917878400585678E-8]
+--R     ,
+--R
+--R     [1.3999999999999999, 8.3889900000000003E-2, 8.3889926341705251E-2,
+--R      2.6341705247623359E-8]
+--R     ,
+--R
+--R     [1.4099999999999999, 8.2736500000000004E-2, 8.2736490176409327E-2,
+--R      - 9.8235906770272052E-9]
+--R     ,
+--R
+--R     [1.4199999999999999, 8.1600400000000003E-2, 8.1600369631709052E-2,
+--R      - 3.0368290951376942E-8]
+--R     ,
+--R
+--R     [1.4299999999999999, 8.0481300000000006E-2, 8.0481271683224637E-2,
+--R      - 2.8316775368963931E-8]
+--R     ,
+--R
+--R     [1.4399999999999999, 7.9378900000000002E-2, 7.9378909095316863E-2,
+--R      9.0953168607743606E-9]
+--R     ,
+--R    [1.45,7.8293000000000001E-2,7.8293000283781833E-2,2.8378183136723578E-10],
+--R    [1.46,7.7223299999999995E-2,7.7223269182499166E-2,- 3.0817500829005695E-8],
+--R    [1.47,7.6169399999999998E-2,7.6169445113894313E-2,4.5113894314718905E-8],
+--R    [1.48,7.5131299999999998E-2,7.5131262663086618E-2,- 3.7336913380481285E-8],
+--R    [1.49,7.4108499999999994E-2,7.4108461555594529E-2,- 3.844440546463268E-8],
+--R    [1.5,7.3100799999999994E-2,7.3100786538480983E-2,- 1.3461519010604661E-8],
+--R    [1.51,7.2108000000000005E-2,7.2107987264816237E-2,- 1.2735183768652902E-8],
+--R    [1.52,7.1129800000000007E-2,7.1129818181354831E-2,1.8181354824453599E-8],
+--R    [1.53,7.0166000000000006E-2,7.0166038419312571E-2,3.8419312564652586E-8],
+--R    [1.54,6.9216399999999997E-2,6.9216411688142798E-2,1.168814280083641E-8],
+--R    [1.55,6.82807E-2,6.8280706172218819E-2,6.172218819378017E-9],
+--R
+--R     [1.5600000000000001, 6.7358699999999994E-2, 6.735869443031553E-2,
+--R      - 5.5696844636354825E-9]
+--R     ,
+--R
+--R     [1.5700000000000001, 6.6450200000000001E-2, 6.6450153297810788E-2,
+--R      - 4.670218921309921E-8]
+--R     ,
+--R
+--R     [1.5800000000000001, 6.5554899999999999E-2, 6.5554863791512402E-2,
+--R      - 3.6208487597111372E-8]
+--R     ,
+--R
+--R     [1.5900000000000001, 6.4672599999999997E-2, 6.4672611017026815E-2,
+--R      1.1017026818604947E-8]
+--R     ,
+--R
+--R     [1.6000000000000001, 6.3803200000000004E-2, 6.380318407859184E-2,
+--R      - 1.5921408164087936E-8]
+--R     ,
+--R    [1.6100000000000001,6.29464E-2,6.2946375991286996E-2,- 2.400871300378693E-8]
+--R     ,
+--R
+--R     [1.6200000000000001, 6.2101999999999997E-2, 6.210198359555763E-2,
+--R      - 1.6404442367001781E-8]
+--R     ,
+--R
+--R     [1.6299999999999999, 6.1269799999999999E-2, 6.1269807473971261E-2,
+--R      7.4739712613292042E-9]
+--R     ,
+--R
+--R     [1.6399999999999999, 6.0449700000000002E-2, 6.0449651870139426E-2,
+--R      - 4.8129860576173744E-8]
+--R     ,
+--R
+--R     [1.6499999999999999, 5.9641300000000001E-2, 5.9641324609737895E-2,
+--R      2.460973789336629E-8]
+--R     ,
+--R
+--R     [1.6599999999999999, 5.8844599999999997E-2, 5.884463702356188E-2,
+--R      3.702356188295397E-8]
+--R     ,
+--R
+--R     [1.6699999999999999, 5.8059399999999997E-2, 5.8059403872549598E-2,
+--R      3.8725496004365922E-9]
+--R     ,
+--R    [1.6799999999999999,5.72854E-2,5.7285443274719294E-2,4.3274719294106312E-8],
+--R
+--R     [1.6899999999999999, 5.6522599999999999E-2, 5.6522576633957849E-2,
+--R      - 2.3366042149752797E-8]
+--R     ,
+--R    [1.7,5.5770599999999997E-2,5.5770628570604719E-2,2.8570604722333304E-8],
+--R    [1.71,5.5029399999999999E-2,5.5029426853783953E-2,2.685378395345106E-8],
+--R    [1.72,5.4298800000000001E-2,5.4298802335420795E-2,2.3354207942527516E-9],
+--R    [1.73,5.3578599999999997E-2,5.3578588885902612E-2,- 1.1114097385467314E-8],
+--R    [1.74,5.2868600000000002E-2,5.2868623331333131E-2,2.3331333129372744E-8],
+--R    [1.75,5.2168699999999998E-2,5.2168745392327326E-2,4.539232732747589E-8],
+--R    [1.76,5.1478799999999998E-2,5.1478797624312636E-2,- 2.375687362110579E-9],
+--R    [1.77,5.0798599999999999E-2,5.0798625359279745E-2,2.5359279745562624E-8],
+--R    [1.78,5.0128100000000002E-2,5.0128076648957798E-2,- 2.3351042204022843E-8],
+--R    [1.79,4.9466999999999997E-2,4.9467002209351796E-2,2.209351798732051E-9],
+--R    [1.8,4.8815299999999999E-2,4.8815255366622776E-2,- 4.4633377223324278E-8],
+--R
+--R     [1.8100000000000001, 4.8172699999999999E-2, 4.8172692004261553E-2,
+--R      - 7.995738446342493E-9]
+--R     ,
+--R
+--R     [1.8200000000000001, 4.7539199999999997E-2, 4.7539170511522538E-2,
+--R      - 2.9488477458483597E-8]
+--R     ,
+--R
+--R     [1.8300000000000001, 4.6914600000000001E-2, 4.691455173308047E-2,
+--R      - 4.8266919530637331E-8]
+--R     ,
+--R
+--R     [1.8400000000000001, 4.6298699999999998E-2, 4.6298698919879575E-2,
+--R      - 1.0801204236576822E-9]
+--R     ,
+--R
+--R     [1.8500000000000001, 4.5691500000000003E-2, 4.5691477681136752E-2,
+--R      - 2.2318863250603282E-8]
+--R     ,
+--R
+--R     [1.8600000000000001, 4.5092800000000002E-2, 4.5092755937471779E-2,
+--R      - 4.4062528223309805E-8]
+--R     ,
+--R
+--R     [1.8700000000000001, 4.4502399999999998E-2, 4.4502403875127772E-2,
+--R      3.8751277742221646E-9]
+--R     ,
+--R
+--R     [1.8799999999999999, 4.3920300000000002E-2, 4.3920293901256249E-2,
+--R      - 6.0987437530690691E-9]
+--R     ,
+--R
+--R     [1.8899999999999999, 4.3346299999999997E-2, 4.3346300600241791E-2,
+--R      6.0024179376894082E-10]
+--R     ,
+--R    [1.8999999999999999,4.27803E-2,4.2780300691019318E-2,6.9101931815529483E-10]
+--R     ,
+--R
+--R     [1.9099999999999999, 4.2222200000000001E-2, 4.2222172985387205E-2,
+--R      - 2.701461279674966E-8]
+--R     ,
+--R
+--R     [1.9199999999999999, 4.1671800000000002E-2, 4.1671798347258343E-2,
+--R      - 1.6527416588085764E-9]
+--R     ,
+--R
+--R     [1.9299999999999999, 4.1129100000000002E-2, 4.1129059652846772E-2,
+--R      - 4.0347153229536303E-8]
+--R     ,
+--R
+--R     [1.9399999999999999, 4.0593799999999999E-2, 4.0593841751750112E-2,
+--R      4.1751750112783981E-8]
+--R     ,
+--R    [1.95,4.0065999999999997E-2,4.0066031428916418E-2,3.1428916420772612E-8],
+--R    [1.96,3.9545499999999997E-2,3.9545517367460342E-2,1.7367460344863694E-8],
+--R    [1.97,3.9032200000000003E-2,3.9032190112315068E-2,- 9.8876849347062645E-9],
+--R    [1.98,3.8525900000000002E-2,3.8525942034688052E-2,4.2034688049752411E-8],
+--R    [1.99,3.8026699999999997E-2,3.8026667297318961E-2,- 3.2702681035390313E-8],
+--R    [2.,3.75343E-2,3.7534261820490689E-2,- 3.817950931100933E-8]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 3
+
+@
+This table expresses the value of 
+$E_3$ with an argument range of 0.01 to 2.00.
+<<*>>=
+--S 4 of 7
+[[0.01,0.4902766,En(3,0.01),En(3,0.01)-0.4902766],_
+[0.02,0.4809683,En(3,0.02),En(3,0.02)-0.4809683],_
+[0.03,0.4719977,En(3,0.03),En(3,0.03)-0.4719977],_
+[0.04,0.4633239,En(3,0.04),En(3,0.04)-0.4633239],_
+[0.05,0.4549188,En(3,0.05),En(3,0.05)-0.4549188],_
+[0.06,0.4467609,En(3,0.06),En(3,0.06)-0.4467609],_
+[0.07,0.4388327,En(3,0.07),En(3,0.07)-0.4388327],_
+[0.08,0.4311197,En(3,0.08),En(3,0.08)-0.4311197],_
+[0.09,0.4236096,En(3,0.09),En(3,0.09)-0.4236096],_
+[0.10,0.4162915,En(3,0.10),En(3,0.10)-0.4162915],_
+[0.11,0.4091557,En(3,0.11),En(3,0.11)-0.4091557],_
+[0.12,0.4021937,En(3,0.12),En(3,0.12)-0.4021937],_
+[0.13,0.3953977,En(3,0.13),En(3,0.13)-0.3953977],_
+[0.14,0.3887607,En(3,0.14),En(3,0.14)-0.3887607],_
+[0.15,0.3822761,En(3,0.15),En(3,0.15)-0.3822761],_
+[0.16,0.3759380,En(3,0.16),En(3,0.16)-0.3759380],_
+[0.17,0.3697408,En(3,0.17),En(3,0.17)-0.3697408],_
+[0.18,0.3636795,En(3,0.18),En(3,0.18)-0.3636795],_
+[0.19,0.3577491,En(3,0.19),En(3,0.19)-0.3577491],_
+[0.20,0.3519453,En(3,0.20),En(3,0.20)-0.3519453],_
+[0.21,0.3462638,En(3,0.21),En(3,0.21)-0.3462638],_
+[0.22,0.3407005,En(3,0.22),En(3,0.22)-0.3407005],_
+[0.23,0.3352518,En(3,0.23),En(3,0.23)-0.3352518],_
+[0.24,0.3299142,En(3,0.24),En(3,0.24)-0.3299142],_
+[0.25,0.3246841,En(3,0.25),En(3,0.25)-0.3246841],_
+[0.26,0.3195585,En(3,0.26),En(3,0.26)-0.3195585],_
+[0.27,0.3145343,En(3,0.27),En(3,0.27)-0.3145343],_
+[0.28,0.3096086,En(3,0.28),En(3,0.28)-0.3096086],_
+[0.29,0.3047787,En(3,0.29),En(3,0.29)-0.3047787],_
+[0.30,0.3000418,En(3,0.30),En(3,0.30)-0.3000418],_
+[0.31,0.2953956,En(3,0.31),En(3,0.31)-0.2953956],_
+[0.32,0.2908374,En(3,0.32),En(3,0.32)-0.2908374],_
+[0.33,0.2863652,En(3,0.33),En(3,0.33)-0.2863652],_
+[0.34,0.2819765,En(3,0.34),En(3,0.34)-0.2819765],_
+[0.35,0.2776693,En(3,0.35),En(3,0.35)-0.2776693],_
+[0.36,0.2734416,En(3,0.36),En(3,0.36)-0.2734416],_
+[0.37,0.2692913,En(3,0.37),En(3,0.37)-0.2692913],_
+[0.38,0.2652165,En(3,0.38),En(3,0.38)-0.2652165],_
+[0.39,0.2612155,En(3,0.39),En(3,0.39)-0.2612155],_
+[0.40,0.2572864,En(3,0.40),En(3,0.40)-0.2572864],_
+[0.41,0.2534276,En(3,0.41),En(3,0.41)-0.2534276],_
+[0.42,0.2496373,En(3,0.42),En(3,0.42)-0.2496373],_
+[0.43,0.2459141,En(3,0.43),En(3,0.43)-0.2459141],_
+[0.44,0.2422563,En(3,0.44),En(3,0.44)-0.2422563],_
+[0.45,0.2386625,En(3,0.45),En(3,0.45)-0.2386625],_
+[0.46,0.2351313,En(3,0.46),En(3,0.46)-0.2351313],_
+[0.47,0.2316612,En(3,0.47),En(3,0.47)-0.2316612],_
+[0.48,0.2282508,En(3,0.48),En(3,0.48)-0.2282508],_
+[0.49,0.2248990,En(3,0.49),En(3,0.49)-0.2248990],_
+[0.50,0.2216044,En(3,0.50),En(3,0.50)-0.2216044],_
+[0.51,0.2183657,En(3,0.51),En(3,0.51)-0.2183657],_
+[0.52,0.2151818,En(3,0.52),En(3,0.52)-0.2151818],_
+[0.53,0.2120516,En(3,0.53),En(3,0.53)-0.2120516],_
+[0.54,0.2089739,En(3,0.54),En(3,0.54)-0.2089739],_
+[0.55,0.2059475,En(3,0.55),En(3,0.55)-0.2059475],_
+[0.56,0.2029715,En(3,0.56),En(3,0.56)-0.2029715],_
+[0.57,0.2000448,En(3,0.57),En(3,0.57)-0.2000448],_
+[0.58,0.1971664,En(3,0.58),En(3,0.58)-0.1971664],_
+[0.59,0.1943353,En(3,0.59),En(3,0.59)-0.1943353],_
+[0.60,0.1915506,En(3,0.60),En(3,0.60)-0.1915506],_
+[0.61,0.1888114,En(3,0.61),En(3,0.61)-0.1888114],_
+[0.62,0.1861166,En(3,0.62),En(3,0.62)-0.1861166],_
+[0.63,0.1834656,En(3,0.63),En(3,0.63)-0.1834656],_
+[0.64,0.1808573,En(3,0.64),En(3,0.64)-0.1808573],_
+[0.65,0.1782910,En(3,0.65),En(3,0.65)-0.1782910],_
+[0.66,0.1757658,En(3,0.66),En(3,0.66)-0.1757658],_
+[0.67,0.1732810,En(3,0.67),En(3,0.67)-0.1732810],_
+[0.68,0.1708358,En(3,0.68),En(3,0.68)-0.1708358],_
+[0.69,0.1684294,En(3,0.69),En(3,0.69)-0.1684294],_
+[0.70,0.1660612,En(3,0.70),En(3,0.70)-0.1660612],_
+[0.71,0.1637303,En(3,0.71),En(3,0.71)-0.1637303],_
+[0.72,0.1614360,En(3,0.72),En(3,0.72)-0.1614360],_
+[0.73,0.1591778,En(3,0.73),En(3,0.73)-0.1591778],_
+[0.74,0.1569549,En(3,0.74),En(3,0.74)-0.1569549],_
+[0.75,0.1547667,En(3,0.75),En(3,0.75)-0.1547667],_
+[0.76,0.1526125,En(3,0.76),En(3,0.76)-0.1526125],_
+[0.77,0.1504917,En(3,0.77),En(3,0.77)-0.1504917],_
+[0.78,0.1484037,En(3,0.78),En(3,0.78)-0.1484037],_
+[0.79,0.1463479,En(3,0.79),En(3,0.79)-0.1463479],_
+[0.80,0.1443238,En(3,0.80),En(3,0.80)-0.1443238],_
+[0.81,0.1423307,En(3,0.81),En(3,0.81)-0.1423307],_
+[0.82,0.1403681,En(3,0.82),En(3,0.82)-0.1403681],_
+[0.83,0.1384355,En(3,0.83),En(3,0.83)-0.1384355],_
+[0.84,0.1365324,En(3,0.84),En(3,0.84)-0.1365324],_
+[0.85,0.1346581,En(3,0.85),En(3,0.85)-0.1346581],_
+[0.86,0.1328122,En(3,0.86),En(3,0.86)-0.1328122],_
+[0.87,0.1309943,En(3,0.87),En(3,0.87)-0.1309943],_
+[0.88,0.1292037,En(3,0.88),En(3,0.88)-0.1292037],_
+[0.89,0.1274401,En(3,0.89),En(3,0.89)-0.1274401],_
+[0.90,0.1257030,En(3,0.90),En(3,0.90)-0.1257030],_
+[0.91,0.1239919,En(3,0.91),En(3,0.91)-0.1239919],_
+[0.92,0.1223063,En(3,0.92),En(3,0.92)-0.1223063],_
+[0.93,0.1206459,En(3,0.93),En(3,0.93)-0.1206459],_
+[0.94,0.1190102,En(3,0.94),En(3,0.94)-0.1190102],_
+[0.95,0.1173988,En(3,0.95),En(3,0.95)-0.1173988],_
+[0.96,0.1158113,En(3,0.96),En(3,0.96)-0.1158113],_
+[0.97,0.1142472,En(3,0.97),En(3,0.97)-0.1142472],_
+[0.98,0.1127063,En(3,0.98),En(3,0.98)-0.1127063],_
+[0.99,0.1111880,En(3,0.99),En(3,0.99)-0.1111880],_
+[1.00,0.1096920,En(3,1.00),En(3,1.00)-0.1096920],_
+[1.01,0.1082179,En(3,1.01),En(3,1.01)-0.1082179],_
+[1.02,0.1067654,En(3,1.02),En(3,1.02)-0.1067654],_
+[1.03,0.1053342,En(3,1.03),En(3,1.03)-0.1053342],_
+[1.04,0.1039238,En(3,1.04),En(3,1.04)-0.1039238],_
+[1.05,0.1025339,En(3,1.05),En(3,1.05)-0.1025339],_
+[1.06,0.1011643,En(3,1.06),En(3,1.06)-0.1011643],_
+[1.07,0.0998145,En(3,1.07),En(3,1.07)-0.0998145],_
+[1.08,0.0984842,En(3,1.08),En(3,1.08)-0.0984842],_
+[1.09,0.0971731,En(3,1.09),En(3,1.09)-0.0971731],_
+[1.10,0.0958809,En(3,1.10),En(3,1.10)-0.0958809],_
+[1.11,0.0946074,En(3,1.11),En(3,1.11)-0.0946074],_
+[1.12,0.0933521,En(3,1.12),En(3,1.12)-0.0933521],_
+[1.13,0.0921149,En(3,1.13),En(3,1.13)-0.0921149],_
+[1.14,0.0908953,En(3,1.14),En(3,1.14)-0.0908953],_
+[1.15,0.0896932,En(3,1.15),En(3,1.15)-0.0896932],_
+[1.16,0.0885083,En(3,1.16),En(3,1.16)-0.0885083],_
+[1.17,0.0873402,En(3,1.17),En(3,1.17)-0.0873402],_
+[1.18,0.0861888,En(3,1.18),En(3,1.18)-0.0861888],_
+[1.19,0.0850537,En(3,1.19),En(3,1.19)-0.0850537],_
+[1.20,0.0839347,En(3,1.20),En(3,1.20)-0.0839347],_
+[1.21,0.0828315,En(3,1.21),En(3,1.21)-0.0828315],_
+[1.22,0.0817439,En(3,1.22),En(3,1.22)-0.0817439],_
+[1.23,0.0806717,En(3,1.23),En(3,1.23)-0.0806717],_
+[1.24,0.0796146,En(3,1.24),En(3,1.24)-0.0796146],_
+[1.25,0.0785723,En(3,1.25),En(3,1.25)-0.0785723],_
+[1.26,0.0775447,En(3,1.26),En(3,1.26)-0.0775447],_
+[1.27,0.0765316,En(3,1.27),En(3,1.27)-0.0765316],_
+[1.28,0.0755326,En(3,1.28),En(3,1.28)-0.0755326],_
+[1.29,0.0745476,En(3,1.29),En(3,1.29)-0.0745476],_
+[1.30,0.0735763,En(3,1.30),En(3,1.30)-0.0735763],_
+[1.31,0.0726186,En(3,1.31),En(3,1.31)-0.0726186],_
+[1.32,0.0716742,En(3,1.32),En(3,1.32)-0.0716742],_
+[1.33,0.0707429,En(3,1.33),En(3,1.33)-0.0707429],_
+[1.34,0.0698246,En(3,1.34),En(3,1.34)-0.0698246],_
+[1.35,0.0689191,En(3,1.35),En(3,1.35)-0.0689191],_
+[1.36,0.0680260,En(3,1.36),En(3,1.36)-0.0680260],_
+[1.37,0.0671453,En(3,1.37),En(3,1.37)-0.0671453],_
+[1.38,0.0662768,En(3,1.38),En(3,1.38)-0.0662768],_
+[1.39,0.0654203,En(3,1.39),En(3,1.39)-0.0654203],_
+[1.40,0.0645755,En(3,1.40),En(3,1.40)-0.0645755],_
+[1.41,0.0637424,En(3,1.41),En(3,1.41)-0.0637424],_
+[1.42,0.0629207,En(3,1.42),En(3,1.42)-0.0629207],_
+[1.43,0.0621104,En(3,1.43),En(3,1.43)-0.0621104],_
+[1.44,0.0613111,En(3,1.44),En(3,1.44)-0.0613111],_
+[1.45,0.0605227,En(3,1.45),En(3,1.45)-0.0605227],_
+[1.46,0.0597452,En(3,1.46),En(3,1.46)-0.0597452],_
+[1.47,0.0589782,En(3,1.47),En(3,1.47)-0.0589782],_
+[1.48,0.0582217,En(3,1.48),En(3,1.48)-0.0582217],_
+[1.49,0.0574755,En(3,1.49),En(3,1.49)-0.0574755],_
+[1.50,0.0567395,En(3,1.50),En(3,1.50)-0.0567395],_
+[1.51,0.0560135,En(3,1.51),En(3,1.51)-0.0560135],_
+[1.52,0.0552973,En(3,1.52),En(3,1.52)-0.0552973],_
+[1.53,0.0545908,En(3,1.53),En(3,1.53)-0.0545908],_
+[1.54,0.0538939,En(3,1.54),En(3,1.54)-0.0538939],_
+[1.55,0.0532064,En(3,1.55),En(3,1.55)-0.0532064],_
+[1.56,0.0525283,En(3,1.56),En(3,1.56)-0.0525283],_
+[1.57,0.0518592,En(3,1.57),En(3,1.57)-0.0518592],_
+[1.58,0.0511992,En(3,1.58),En(3,1.58)-0.0511992],_
+[1.59,0.0505481,En(3,1.59),En(3,1.59)-0.0505481],_
+[1.60,0.0499057,En(3,1.60),En(3,1.60)-0.0499057],_
+[1.61,0.0492720,En(3,1.61),En(3,1.61)-0.0492720],_
+[1.62,0.0486467,En(3,1.62),En(3,1.62)-0.0486467],_
+[1.63,0.0480299,En(3,1.63),En(3,1.63)-0.0480299],_
+[1.64,0.0474213,En(3,1.64),En(3,1.64)-0.0474213],_
+[1.65,0.0468209,En(3,1.65),En(3,1.65)-0.0468209],_
+[1.66,0.0462284,En(3,1.66),En(3,1.66)-0.0462284],_
+[1.67,0.0456439,En(3,1.67),En(3,1.67)-0.0456439],_
+[1.68,0.0450672,En(3,1.68),En(3,1.68)-0.0450672],_
+[1.69,0.0444982,En(3,1.69),En(3,1.69)-0.0444982],_
+[1.70,0.0439367,En(3,1.70),En(3,1.70)-0.0439367],_
+[1.71,0.0433827,En(3,1.71),En(3,1.71)-0.0433827],_
+[1.72,0.0428361,En(3,1.72),En(3,1.72)-0.0428361],_
+[1.73,0.0422967,En(3,1.73),En(3,1.73)-0.0422967],_
+[1.74,0.0417645,En(3,1.74),En(3,1.74)-0.0417645],_
+[1.75,0.0412393,En(3,1.75),En(3,1.75)-0.0412393],_
+[1.76,0.0407211,En(3,1.76),En(3,1.76)-0.0407211],_
+[1.77,0.0402097,En(3,1.77),En(3,1.77)-0.0402097],_
+[1.78,0.0397051,En(3,1.78),En(3,1.78)-0.0397051],_
+[1.79,0.0392071,En(3,1.79),En(3,1.79)-0.0392071],_
+[1.80,0.0387157,En(3,1.80),En(3,1.80)-0.0387157],_
+[1.81,0.0382308,En(3,1.81),En(3,1.81)-0.0382308],_
+[1.82,0.0377522,En(3,1.82),En(3,1.82)-0.0377522],_
+[1.83,0.0372800,En(3,1.83),En(3,1.83)-0.0372800],_
+[1.84,0.0368139,En(3,1.84),En(3,1.84)-0.0368139],_
+[1.85,0.0363540,En(3,1.85),En(3,1.85)-0.0363540],_
+[1.86,0.0359001,En(3,1.86),En(3,1.86)-0.0359001],_
+[1.87,0.0354521,En(3,1.87),En(3,1.87)-0.0354521],_
+[1.88,0.0350100,En(3,1.88),En(3,1.88)-0.0350100],_
+[1.89,0.0345737,En(3,1.89),En(3,1.89)-0.0345737],_
+[1.90,0.0341430,En(3,1.90),En(3,1.90)-0.0341430],_
+[1.91,0.0337180,En(3,1.91),En(3,1.91)-0.0337180],_
+[1.92,0.0332986,En(3,1.92),En(3,1.92)-0.0332986],_
+[1.93,0.0328846,En(3,1.93),En(3,1.93)-0.0328846],_
+[1.94,0.0324759,En(3,1.94),En(3,1.94)-0.0324759],_
+[1.95,0.0320727,En(3,1.95),En(3,1.95)-0.0320727],_
+[1.96,0.0316746,En(3,1.96),En(3,1.96)-0.0316746],_
+[1.97,0.0312817,En(3,1.97),En(3,1.97)-0.0312817],_
+[1.98,0.0308939,En(3,1.98),En(3,1.98)-0.0308939],_
+[1.99,0.0305112,En(3,1.99),En(3,1.99)-0.0305112],_
+[2.00,0.0301334,En(3,2.00),En(3,2.00)-0.0301334]]
+--R 
+--R
+--R   (4)
+--R   [[1.0E-2,0.49027660000000001,0.49027656418466514,- 3.5815334864519599E-8],
+--R    [2.0E-2,0.48096830000000002,0.48096829147697201,- 8.5230280055803576E-9],
+--R
+--R     [2.9999999999999999E-2, 0.47199770000000002, 0.47199768719683605,
+--R      - 1.2803163973451603E-8]
+--R     ,
+--R
+--R     [4.0000000000000001E-2, 0.46332390000000001, 0.46332394174433533,
+--R      4.1744335321780568E-8]
+--R     ,
+--R
+--R     [5.0000000000000003E-2, 0.45491880000000001, 0.45491884974847663,
+--R      4.9748476615985027E-8]
+--R     ,
+--R
+--R     [5.9999999999999998E-2, 0.44676090000000002, 0.44676088323725571,
+--R      - 1.6762744303733257E-8]
+--R     ,
+--R
+--R     [7.0000000000000007E-2, 0.43883270000000002, 0.43883267979789509,
+--R      - 2.0202104933364495E-8]
+--R     ,
+--R
+--R     [8.0000000000000002E-2, 0.43111969999999999, 0.43111973054612684,
+--R      3.054612685016167E-8]
+--R     ,
+--R
+--R     [8.9999999999999997E-2, 0.42360959999999998, 0.42360960561704109,
+--R      5.6170411100175954E-9]
+--R     ,
+--R
+--R     [0.10000000000000001, 0.41629149999999998, 0.41629145790827876,
+--R      - 4.2091721219605915E-8]
+--R     ,
+--R    [0.11,0.40915570000000001,0.40915568570605237,- 1.4293947636634385E-8],
+--R    [0.12,0.40219369999999999,0.40219369277059286,- 7.2294071284950689E-9],
+--R    [0.13,0.39539770000000002,0.395397711576501,1.1576500980048365E-8],
+--R
+--R     [0.14000000000000001, 0.38876070000000001, 0.38876066929878084,
+--R      - 3.0701219178030925E-8]
+--R     ,
+--R
+--R     [0.14999999999999999, 0.38227610000000001, 0.38227608377400268,
+--R      - 1.6225997323537911E-8]
+--R     ,
+--R    [0.16,0.37593799999999999,0.37593798110903431,- 1.8890965680640193E-8],
+--R
+--R     [0.17000000000000001, 0.36974079999999998, 0.36974082931670565,
+--R      2.9316705674187205E-8]
+--R     ,
+--R
+--R     [0.17999999999999999, 0.36367949999999999, 0.36367948407235051,
+--R      - 1.5927649477109895E-8]
+--R     ,
+--R    [0.19,0.35774909999999999,0.3577491438077095,4.3807709515508719E-8],
+--R
+--R     [0.20000000000000001, 0.35194530000000002, 0.35194531211487057,
+--R      1.2114870551194201E-8]
+--R     ,
+--R
+--R     [0.20999999999999999, 0.34626380000000001, 0.3462637659551443,
+--R      - 3.4044855712345168E-8]
+--R     ,
+--R    [0.22,0.34070050000000002,0.34070052853638638,2.8536386365018984E-8],
+--R
+--R     [0.23000000000000001, 0.33525179999999999, 0.33525184598756436,
+--R      4.5987564367688805E-8]
+--R     ,
+--R
+--R     [0.23999999999999999, 0.32991419999999999, 0.32991416715361832,
+--R      - 3.2846381670115221E-8]
+--R     ,
+--R    [0.25,0.32468409999999998,0.32468412597814367,2.5978143691762767E-8],
+--R
+--R     [0.26000000000000001, 0.31955850000000002, 0.31955852605039498,
+--R      2.6050394952292777E-8]
+--R     ,
+--R
+--R     [0.27000000000000002, 0.31453429999999999, 0.31453432697636552,
+--R      2.6976365530284596E-8]
+--R     ,
+--R
+--R     [0.28000000000000003, 0.30960860000000001, 0.30960863229804725,
+--R      3.2298047236700711E-8]
+--R     ,
+--R
+--R     [0.28999999999999998, 0.30477870000000001, 0.30477867873524889,
+--R      - 2.126475112662618E-8]
+--R     ,
+--R
+--R     [0.29999999999999999, 0.30004180000000003, 0.30004182656401435,
+--R      2.6564014321550644E-8]
+--R     ,
+--R    [0.31,0.29539559999999998,0.29539555097726167,- 4.9022738313198033E-8],
+--R
+--R     [0.32000000000000001, 0.29083740000000002, 0.29083743429861525,
+--R      3.4298615225747398E-8]
+--R     ,
+--R
+--R     [0.33000000000000002, 0.28636519999999999, 0.28636515894092018,
+--R      - 4.1059079802785448E-8]
+--R     ,
+--R
+--R     [0.34000000000000002, 0.28197650000000002, 0.28197650101764582,
+--R      1.0176458009603095E-9]
+--R     ,
+--R
+--R     [0.34999999999999998, 0.27766930000000001, 0.27766932452910809,
+--R      2.4529108078041872E-8]
+--R     ,
+--R
+--R     [0.35999999999999999, 0.27344160000000001, 0.27344157605676817,
+--R      - 2.3943231841627721E-8]
+--R     ,
+--R    [0.37,0.26929130000000001,0.26929127990828022,- 2.0091719787895812E-8],
+--R    [0.38,0.26521650000000002,0.2652165336638212,3.3663821175089481E-8],
+--R
+--R     [0.39000000000000001, 0.26121549999999999, 0.26121550408084138,
+--R      4.0808413870330185E-9]
+--R     ,
+--R
+--R     [0.40000000000000002, 0.25728640000000003, 0.25728642331994478,
+--R      2.3319944753019684E-8]
+--R     ,
+--R
+--R     [0.40999999999999998, 0.25342759999999998, 0.25342758545933575,
+--R      - 1.45406642282353E-8]
+--R     ,
+--R
+--R     [0.41999999999999998, 0.24963730000000001, 0.24963734326929157,
+--R      4.3269291566394585E-8]
+--R     ,
+--R    [0.42999999999999999,0.2459141,0.24591410522156262,5.2215626267226867E-9],
+--R    [0.44,0.24225630000000001,0.24225633271155758,3.2711557573783523E-8],
+--R    [0.45000000000000001,0.2386625,0.23866253747371868,3.7473718683678214E-8],
+--R
+--R     [0.46000000000000002, 0.23513129999999999, 0.23513127917269516,
+--R      - 2.082730482522166E-8]
+--R     ,
+--R
+--R     [0.46999999999999997, 0.23166120000000001, 0.2316611631548362,
+--R      - 3.6845163808862935E-8]
+--R     ,
+--R    [0.47999999999999998,0.2282508,0.22825083834619003,3.8346190028848426E-8],
+--R
+--R     [0.48999999999999999, 0.22489899999999999, 0.22489899528465218,
+--R      - 4.7153478066608301E-9]
+--R     ,
+--R    [0.5,0.22160440000000001,0.22160436427517846,- 3.572482154545753E-8],
+--R    [0.51000000000000001,0.2183657,0.21836571365810192,1.3658101927216393E-8],
+--R
+--R     [0.52000000000000002, 0.21518180000000001, 0.21518184818157665,
+--R      4.8181576645101032E-8]
+--R     ,
+--R
+--R     [0.53000000000000003, 0.21205160000000001, 0.21205160747004631,
+--R      7.4700463037480347E-9]
+--R     ,
+--R
+--R     [0.54000000000000004, 0.20897389999999999, 0.20897386458140554,
+--R      - 3.541859444622375E-8]
+--R     ,
+--R
+--R     [0.55000000000000004, 0.20594750000000001, 0.20594752464620908,
+--R      2.4646209073608816E-8]
+--R     ,
+--R    [0.56000000000000005,0.2029715,0.20297152358289336,2.3582893360352131E-8],
+--R
+--R     [0.56999999999999995, 0.20004479999999999, 0.20004482688351907,
+--R      2.6883519077536278E-8]
+--R     ,
+--R
+--R     [0.57999999999999996, 0.19716639999999999, 0.19716642846502985,
+--R      2.8465029860980096E-8]
+--R     ,
+--R
+--R     [0.58999999999999997, 0.19433529999999999, 0.19433534958145796,
+--R      4.9581457967073916E-8]
+--R     ,
+--R
+--R     [0.59999999999999998, 0.19155059999999999, 0.19155063779289766,
+--R      3.7792897672472847E-8]
+--R     ,
+--R
+--R     [0.60999999999999999, 0.18881139999999999, 0.18881136598742021,
+--R      - 3.4012579780418051E-8]
+--R     ,
+--R    [0.62,0.18611659999999999,0.18611663145241836,3.1452418364219525E-8],
+--R    [0.63,0.18346560000000001,0.18346555499215655,- 4.5007843452182783E-8],
+--R    [0.64000000000000001,0.1808573,0.18085728008855939,- 1.9911440607423003E-8],
+--R
+--R     [0.65000000000000002, 0.17829100000000001, 0.17829097210250539,
+--R      - 2.7897494619955054E-8]
+--R     ,
+--R    [0.66000000000000003,0.1757658,0.17576581751310477,1.7513104771937904E-8],
+--R
+--R     [0.67000000000000004, 0.17328099999999999, 0.17328102319263361,
+--R      2.3192633619162351E-8]
+--R     ,
+--R
+--R     [0.68000000000000005, 0.17083580000000001, 0.17083581571497,
+--R      1.5714969991975636E-8]
+--R     ,
+--R
+--R     [0.68999999999999995, 0.16842940000000001, 0.16842944069553942,
+--R      4.0695539410551262E-8]
+--R     ,
+--R
+--R     [0.69999999999999996, 0.16606119999999999, 0.16606116216092121,
+--R      - 3.7839078786960911E-8]
+--R     ,
+--R    [0.70999999999999996,0.1637303,0.16373026194640114,- 3.8053598855336546E-8],
+--R    [0.71999999999999997,0.161436,0.16143603911987703,3.9119877032200989E-8],
+--R
+--R     [0.72999999999999998, 0.15917780000000001, 0.15917780943063548,
+--R      9.4306354669893011E-9]
+--R     ,
+--R
+--R     [0.73999999999999999, 0.15695490000000001, 0.15695490478162077,
+--R      4.7816207660034138E-9]
+--R     ,
+--R    [0.75,0.15476670000000001,0.15476667272391029,- 2.7276089720196239E-8],
+--R
+--R     [0.76000000000000001, 0.15261250000000001, 0.15261247597219724,
+--R      - 2.4027802775217211E-8]
+--R     ,
+--R
+--R     [0.77000000000000002, 0.15049170000000001, 0.15049169194016149,
+--R      - 8.0598385188146437E-9]
+--R     ,
+--R    [0.78000000000000003,0.1484037,0.14840371229468169,1.2294681689395404E-8],
+--R    [0.79000000000000004,0.1463479,0.14634794252791047,4.2527910470990804E-8],
+--R    [0.80000000000000004,0.1443238,0.14432380154629593,1.5462959246370644E-9],
+--R    [0.81000000000000005,0.1423307,0.14233072127569105,2.1275691042266232E-8],
+--R    [0.81999999999999995,0.1403681,0.14036814628174568,4.628174568588328E-8],
+--R
+--R     [0.82999999999999996, 0.13843549999999999, 0.13843553340482559,
+--R      3.3404825600102939E-8]
+--R     ,
+--R    [0.83999999999999997,0.1365324,0.13653235140874825,- 4.8591251744500852E-8],
+--R    [0.84999999999999998,0.1346581,0.13465808064266868,- 1.935733132119033E-8],
+--R
+--R     [0.85999999999999999, 0.13281219999999999, 0.13281221271548668,
+--R      1.2715486691350364E-8]
+--R     ,
+--R    [0.87,0.13099430000000001,0.1309942501821853,- 4.9817814706809571E-8],
+--R    [0.88,0.1292037,0.12920370624154368,6.2415436707929928E-9],
+--R    [0.89000000000000001,0.1274401,0.12744010444469844,4.4446984426294023E-9],
+--R
+--R     [0.90000000000000002, 0.12570300000000001, 0.12570297841405975,
+--R      - 2.1585940257473624E-8]
+--R     ,
+--R    [0.91000000000000003,0.1239919,0.12399187157211375,- 2.8427886253346024E-8],
+--R
+--R     [0.92000000000000004, 0.12230630000000001, 0.12230633687966941,
+--R      3.6879669401690407E-8]
+--R     ,
+--R    [0.93000000000000005,0.1206459,0.12064593658313376,3.6583133758427699E-8],
+--R    [0.93999999999999995,0.1190102,0.1190102419704193,4.1970419306647244E-8],
+--R    [0.94999999999999996,0.1173988,0.11739883313511031,3.3135110308335491E-8],
+--R
+--R     [0.95999999999999996, 0.11581130000000001, 0.11581129874853491,
+--R      - 1.2514650982176079E-9]
+--R     ,
+--R
+--R     [0.96999999999999997, 0.11424719999999999, 0.11424723583940669,
+--R      3.5839406692383946E-8]
+--R     ,
+--R    [0.97999999999999998,0.1127063,0.11270624958071841,- 5.0419281583113928E-8],
+--R    [0.98999999999999999,0.111188,0.11118795308358656,- 4.6916413434794357E-8],
+--R    [1.,0.109692,0.10969196719776014,- 3.2802239854912152E-8],
+--R    [1.01,0.10821790000000001,0.10821792031852197,2.031852196215933E-8],
+--R    [1.02,0.1067654,0.10676544819972504,4.8199725044550945E-8],
+--R    [1.03,0.1053342,0.10533419377271731,- 6.2272826895082289E-9],
+--R    [1.04,0.1039238,0.1039238069709225,6.9709225059000346E-9],
+--R    [1.05,0.1025339,0.10253394455985448,4.455985448681421E-8],
+--R    [1.0600000000000001,0.1011643,0.10116426997235345,- 3.0027646549801723E-8],
+--R
+--R     [1.0700000000000001, 9.98145E-2, 9.9814453148843241E-2,
+--R      - 4.6851156759730728E-8]
+--R     ,
+--R
+--R     [1.0800000000000001, 9.8484199999999994E-2, 9.8484170382417149E-2,
+--R      - 2.9617582844587709E-8]
+--R     ,
+--R
+--R     [1.0900000000000001, 9.7173099999999998E-2, 9.7173104168569752E-2,
+--R      4.1685697532711785E-9]
+--R     ,
+--R
+--R     [1.1000000000000001, 9.5880900000000005E-2, 9.58809430594003E-2,
+--R      4.305940029536437E-8]
+--R     ,
+--R
+--R     [1.1100000000000001, 9.4607399999999994E-2, 9.4607381522120698E-2,
+--R      - 1.8477879296097122E-8]
+--R     ,
+--R
+--R     [1.1200000000000001, 9.3352099999999993E-2, 9.3352119801709738E-2,
+--R      1.9801709744138307E-8]
+--R     ,
+--R
+--R     [1.1299999999999999, 9.21149E-2, 9.2114863787561826E-2,
+--R      - 3.6212438173088835E-8]
+--R     ,
+--R
+--R     [1.1399999999999999, 9.0895299999999998E-2, 9.0895324883984607E-2,
+--R      2.4883984608981002E-8]
+--R     ,
+--R
+--R     [1.1499999999999999, 8.9693200000000001E-2, 8.969321988440758E-2,
+--R      1.9884407578829588E-8]
+--R     ,
+--R
+--R     [1.1599999999999999, 8.8508299999999998E-2, 8.8508270849168805E-2,
+--R      - 2.9150831193369697E-8]
+--R     ,
+--R
+--R     [1.1699999999999999, 8.7340200000000007E-2, 8.7340204986753447E-2,
+--R      4.9867534407388447E-9]
+--R     ,
+--R
+--R     [1.1799999999999999, 8.6188799999999996E-2, 8.6188754538362E-2,
+--R      - 4.5461637995791726E-8]
+--R     ,
+--R
+--R     [1.1899999999999999, 8.5053699999999996E-2, 8.505365666569388E-2,
+--R      - 4.3334306115694332E-8]
+--R     ,
+--R    [1.2,8.3934700000000001E-2,8.393465334183281E-2,- 4.6658167190960498E-8],
+--R    [1.21,8.2831500000000002E-2,8.283149124512959E-2,- 8.7548704119644327E-9],
+--R    [1.22,8.1743899999999994E-2,8.1743921655978991E-2,2.1655978996171221E-8],
+--R    [1.23,8.0671699999999999E-2,8.067170035639451E-2,3.5639451112512432E-10],
+--R    [1.24,7.9614599999999994E-2,7.9614587532284717E-2,- 1.2467715276853752E-8],
+--R    [1.25,7.8572299999999998E-2,7.857234767834359E-2,4.7678343592649775E-8],
+--R    [1.26,7.7544699999999994E-2,7.7544749505467275E-2,4.950546728110794E-8],
+--R    [1.27,7.6531600000000005E-2,7.6531565850615282E-2,- 3.4149384722792497E-8],
+--R    [1.28,7.5532600000000005E-2,7.5532573589034993E-2,- 2.6410965012213516E-8],
+--R    [1.29,7.4547600000000006E-2,7.454755354877482E-2,- 4.6451225185761835E-8],
+--R    [1.3,7.3576299999999997E-2,7.3576290427412191E-2,- 9.5725878057617564E-9],
+--R
+--R     [1.3100000000000001, 7.2618600000000005E-2, 7.2618572710924589E-2,
+--R      - 2.7289075416048192E-8]
+--R     ,
+--R
+--R     [1.3200000000000001, 7.1674199999999993E-2, 7.1674192594636615E-2,
+--R      - 7.4053633780657435E-9]
+--R     ,
+--R
+--R     [1.3300000000000001, 7.0742899999999997E-2, 7.0742945906179772E-2,
+--R      4.5906179774179989E-8]
+--R     ,
+--R
+--R     [1.3400000000000001, 6.9824600000000001E-2, 6.9824632030397252E-2,
+--R      3.203039725119261E-8]
+--R     ,
+--R
+--R     [1.3500000000000001, 6.8919099999999997E-2, 6.8919053836139416E-2,
+--R      - 4.6163860581427407E-8]
+--R     ,
+--R
+--R     [1.3600000000000001, 6.8026000000000003E-2, 6.8026017604886913E-2,
+--R      1.7604886909383666E-8]
+--R     ,
+--R
+--R     [1.3700000000000001, 6.7145300000000005E-2, 6.7145332961148871E-2,
+--R      3.29611488658621E-8]
+--R     ,
+--R
+--R     [1.3799999999999999, 6.6276799999999997E-2, 6.6276812804578436E-2,
+--R      1.2804578439218339E-8]
+--R     ,
+--R
+--R     [1.3899999999999999, 6.5420300000000001E-2, 6.5420273243759597E-2,
+--R      - 2.6756240403824627E-8]
+--R     ,
+--R
+--R     [1.3999999999999999, 6.4575499999999994E-2, 6.457553353160958E-2,
+--R      3.3531609586190392E-8]
+--R     ,
+--R
+--R     [1.4099999999999999, 6.3742400000000005E-2, 6.3742416002349978E-2,
+--R      1.6002349972898955E-8]
+--R     ,
+--R
+--R     [1.4199999999999999, 6.2920699999999996E-2, 6.2920746010004802E-2,
+--R      4.6010004806085192E-8]
+--R     ,
+--R
+--R     [1.4299999999999999, 6.2110400000000003E-2, 6.211035186837166E-2,
+--R      - 4.8131628342662491E-8]
+--R     ,
+--R
+--R     [1.4399999999999999, 6.13111E-2, 6.1311064792432743E-2,
+--R      - 3.5207567257444783E-8]
+--R     ,
+--R    [1.45,6.0522699999999999E-2,6.0522718841157001E-2,1.8841157002624787E-8],
+--R    [1.46,5.9745199999999998E-2,5.9745150861655022E-2,- 4.9138344976329407E-8],
+--R    [1.47,5.8978200000000001E-2,5.8978200434649602E-2,4.3464960103323236E-10],
+--R    [1.48,5.8221700000000001E-2,5.8221709821222274E-2,9.8212222729432419E-9],
+--R    [1.49,5.7475499999999999E-2,5.7475523910801435E-2,2.3910801436044515E-8],
+--R    [1.5,5.6739499999999998E-2,5.6739490170354172E-2,- 9.8296458264646169E-9],
+--R    [1.51,5.6013500000000001E-2,5.6013458594752839E-2,- 4.140524716145233E-8],
+--R    [1.52,5.5297300000000001E-2,5.52972816582777E-2,- 1.8341722300696528E-8],
+--R    [1.53,5.4590800000000002E-2,5.4590814267229414E-2,1.4267229411779603E-8],
+--R    [1.54,5.3893900000000002E-2,5.3893913713619013E-2,1.3713619011201938E-8],
+--R    [1.55,5.3206400000000001E-2,5.3206439629901937E-2,3.9629901936455436E-8],
+--R
+--R     [1.5600000000000001, 5.25283E-2, 5.2528253944736245E-2,
+--R      - 4.6055263755451215E-8]
+--R     ,
+--R
+--R     [1.5700000000000001, 5.1859200000000001E-2, 5.1859220839728763E-2,
+--R      2.083972876210094E-8]
+--R     ,
+--R    [1.5800000000000001,5.11992E-2,5.1199206707146921E-2,6.7071469211921197E-9],
+--R
+--R     [1.5900000000000001, 5.0548099999999999E-2, 5.0548080108570391E-2,
+--R      - 1.9891429607621358E-8]
+--R     ,
+--R
+--R     [1.6000000000000001, 4.9905699999999997E-2, 4.9905711734454218E-2,
+--R      1.1734454220813095E-8]
+--R     ,
+--R
+--R     [1.6100000000000001, 4.9272000000000003E-2, 4.9271974364586212E-2,
+--R      - 2.5635413791746942E-8]
+--R     ,
+--R
+--R     [1.6200000000000001, 4.8646700000000001E-2, 4.8646742829405641E-2,
+--R      4.2829405640099072E-8]
+--R     ,
+--R
+--R     [1.6299999999999999, 4.80299E-2, 4.8029893972168107E-2,
+--R      - 6.0278318933515429E-9]
+--R     ,
+--R    [1.6399999999999999,4.74213E-2,4.7421306611931634E-2,6.6119316341484158E-9],
+--R
+--R     [1.6499999999999999, 4.6820899999999999E-2, 4.6820861507343305E-2,
+--R      - 3.8492656694033567E-8]
+--R     ,
+--R
+--R     [1.6599999999999999, 4.6228400000000003E-2, 4.622844132120392E-2,
+--R      4.1321203916866445E-8]
+--R     ,
+--R
+--R     [1.6699999999999999, 4.5643900000000001E-2, 4.5643930585794489E-2,
+--R      3.0585794487392182E-8]
+--R     ,
+--R
+--R     [1.6799999999999999, 4.5067200000000002E-2, 4.5067215668940779E-2,
+--R      1.5668940776814022E-8]
+--R     ,
+--R
+--R     [1.6899999999999999, 4.4498200000000002E-2, 4.4498184740800251E-2,
+--R      - 1.5259199750983221E-8]
+--R     ,
+--R    [1.7,4.3936700000000002E-2,4.3936727741353321E-2,2.7741353318677664E-8],
+--R    [1.71,4.3382700000000003E-2,4.3382736348575771E-2,3.6348575767342073E-8],
+--R    [1.72,4.2836100000000002E-2,4.283610394728473E-2,3.9472847282451262E-9],
+--R    [1.73,4.22967E-2,4.229672559863315E-2,2.559863315071409E-8],
+--R    [1.74,4.1764500000000003E-2,4.1764498010238606E-2,- 1.9897613973141048E-9],
+--R    [1.75,4.12393E-2,4.1239319506936156E-2,1.9506936156654664E-8],
+--R    [1.76,4.0721100000000003E-2,4.0721090002130152E-2,- 9.9978698514524567E-9],
+--R    [1.77,4.0209700000000001E-2,4.0209710969742136E-2,1.0969742135491511E-8],
+--R    [1.78,3.97051E-2,3.9705085416725308E-2,- 1.4583274692003823E-8],
+--R    [1.79,3.9207100000000002E-2,3.920711785615049E-2,1.7856150488770872E-8],
+--R    [1.8,3.8715699999999999E-2,3.8715714280832765E-2,1.4280832766333518E-8],
+--R
+--R     [1.8100000000000001, 3.8230800000000002E-2, 3.8230782137495319E-2,
+--R      - 1.7862504683718861E-8]
+--R     ,
+--R    [1.8200000000000001,3.77522E-2,3.7752230301454859E-2,3.0301454859160692E-8],
+--R
+--R     [1.8300000000000001, 3.7280000000000001E-2, 3.7279969051817741E-2,
+--R      - 3.0948182259959989E-8]
+--R     ,
+--R
+--R     [1.8400000000000001, 3.6813899999999997E-2, 3.6813910047171126E-2,
+--R      1.0047171129790033E-8]
+--R     ,
+--R
+--R     [1.8500000000000001, 3.6353999999999997E-2, 3.6353966301762304E-2,
+--R      - 3.3698237693335908E-8]
+--R     ,
+--R
+--R     [1.8600000000000001, 3.5900099999999997E-2, 3.5900052162149899E-2,
+--R      - 4.7837850097876E-8]
+--R     ,
+--R
+--R     [1.8700000000000001, 3.54521E-2, 3.5452083284321229E-2,
+--R      - 1.6715678771705988E-8]
+--R     ,
+--R
+--R     [1.8799999999999999, 3.5009999999999999E-2, 3.5009976611261075E-2,
+--R      - 2.3388738924767782E-8]
+--R     ,
+--R
+--R     [1.8899999999999999, 3.4573699999999999E-2, 3.4573650350956942E-2,
+--R      - 4.9649043057375941E-8]
+--R     ,
+--R    [1.8999999999999999,3.4143E-2,3.414302395484918E-2,2.3954849180662929E-8],
+--R
+--R     [1.9099999999999999, 3.3717999999999998E-2, 3.3718018096686453E-2,
+--R      1.8096686454915911E-8]
+--R     ,
+--R
+--R     [1.9199999999999999, 3.3298599999999998E-2, 3.3298554651807068E-2,
+--R      - 4.5348192929950404E-8]
+--R     ,
+--R
+--R     [1.9299999999999999, 3.28846E-2, 3.2884556676814732E-2,
+--R      - 4.3323185268395736E-8]
+--R     ,
+--R
+--R     [1.9399999999999999, 3.2475900000000002E-2, 3.2475948389653855E-2,
+--R      4.8389653853342374E-8]
+--R     ,
+--R    [1.95,3.2072700000000003E-2,3.2072655150063287E-2,- 4.4849936715884997E-8],
+--R    [1.96,3.1674599999999997E-2,3.167460344041137E-2,3.4404113724573193E-9],
+--R    [1.97,3.1281700000000003E-2,3.1281720846895129E-2,2.0846895126824805E-8],
+--R    [1.98,3.0893899999999998E-2,3.0893936041105241E-2,3.6041105242606841E-8],
+--R    [1.99,3.0511199999999999E-2,3.0511178761929561E-2,- 2.1238070437717971E-8],
+--R    [2.,3.0133400000000001E-2,3.0133379797815663E-2,- 2.0202184338596885E-8]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 4
+
+@
+This table expresses the value of 
+$E_4$ with an argument range of 0.01 to 2.00.
+<<*>>=
+--S 5 of 7
+[[0.01,0.3283824,En(4,0.01),En(4,0.01)-0.3283824],_
+[0.02,0.3235264,En(4,0.02),En(4,0.02)-0.3235264],_
+[0.03,0.3187619,En(4,0.03),En(4,0.03)-0.3187619],_
+[0.04,0.3140855,En(4,0.04),En(4,0.04)-0.3140855],_
+[0.05,0.3094945,En(4,0.05),En(4,0.05)-0.3094945],_
+[0.06,0.3049863,En(4,0.06),En(4,0.06)-0.3049863],_
+[0.07,0.3005585,En(4,0.07),En(4,0.07)-0.3005585],_
+[0.08,0.2962089,En(4,0.08),En(4,0.08)-0.2962089],_
+[0.09,0.2919354,En(4,0.09),En(4,0.09)-0.2919354],_
+[0.10,0.2877361,En(4,0.10),En(4,0.10)-0.2877361],_
+[0.11,0.2836090,En(4,0.11),En(4,0.11)-0.2836090],_
+[0.12,0.2795524,En(4,0.12),En(4,0.12)-0.2795524],_
+[0.13,0.2755646,En(4,0.13),En(4,0.13)-0.2755646],_
+[0.14,0.2716439,En(4,0.14),En(4,0.14)-0.2716439],_
+[0.15,0.2677889,En(4,0.15),En(4,0.15)-0.2677889],_
+[0.16,0.2639979,En(4,0.16),En(4,0.16)-0.2639979],_
+[0.17,0.2602696,En(4,0.17),En(4,0.17)-0.2602696],_
+[0.18,0.2566026,En(4,0.18),En(4,0.18)-0.2566026],_
+[0.19,0.2529956,En(4,0.19),En(4,0.19)-0.2529956],_
+[0.20,0.2494472,En(4,0.20),En(4,0.20)-0.2494472],_
+[0.21,0.2459563,En(4,0.21),En(4,0.21)-0.2459563],_
+[0.22,0.2425216,En(4,0.22),En(4,0.22)-0.2425216],_
+[0.23,0.2391419,En(4,0.23),En(4,0.23)-0.2391419],_
+[0.24,0.2358162,En(4,0.24),En(4,0.24)-0.2358162],_
+[0.25,0.2325432,En(4,0.25),En(4,0.25)-0.2325432],_
+[0.26,0.2293221,En(4,0.26),En(4,0.26)-0.2293221],_
+[0.27,0.2261517,En(4,0.27),En(4,0.27)-0.2261517],_
+[0.28,0.2230311,En(4,0.28),En(4,0.28)-0.2230311],_
+[0.29,0.2199593,En(4,0.29),En(4,0.29)-0.2199593],_
+[0.30,0.2169352,En(4,0.30),En(4,0.30)-0.2169352],_
+[0.31,0.2139581,En(4,0.31),En(4,0.31)-0.2139581],_
+[0.32,0.2110270,En(4,0.32),En(4,0.32)-0.2110270],_
+[0.33,0.2081411,En(4,0.33),En(4,0.33)-0.2081411],_
+[0.34,0.2052994,En(4,0.34),En(4,0.34)-0.2052994],_
+[0.35,0.2025013,En(4,0.35),En(4,0.35)-0.2025013],_
+[0.36,0.1997458,En(4,0.36),En(4,0.36)-0.1997458],_
+[0.37,0.1970322,En(4,0.37),En(4,0.37)-0.1970322],_
+[0.38,0.1943597,En(4,0.38),En(4,0.38)-0.1943597],_
+[0.39,0.1917276,En(4,0.39),En(4,0.39)-0.1917276],_
+[0.40,0.1891352,En(4,0.40),En(4,0.40)-0.1891352],_
+[0.41,0.1865816,En(4,0.41),En(4,0.41)-0.1865816],_
+[0.42,0.1840664,En(4,0.42),En(4,0.42)-0.1840664],_
+[0.43,0.1815887,En(4,0.43),En(4,0.43)-0.1815887],_
+[0.44,0.1791479,En(4,0.44),En(4,0.44)-0.1791479],_
+[0.45,0.1767433,En(4,0.45),En(4,0.45)-0.1767433],_
+[0.46,0.1743744,En(4,0.46),En(4,0.46)-0.1743744],_
+[0.47,0.1720405,En(4,0.47),En(4,0.47)-0.1720405],_
+[0.48,0.1697410,En(4,0.48),En(4,0.48)-0.1697410],_
+[0.49,0.1674753,En(4,0.49),En(4,0.49)-0.1674753],_
+[0.50,0.1652428,En(4,0.50),En(4,0.50)-0.1652428],_
+[0.51,0.1630430,En(4,0.51),En(4,0.51)-0.1630430],_
+[0.52,0.1608753,En(4,0.52),En(4,0.52)-0.1608753],_
+[0.53,0.1587392,En(4,0.53),En(4,0.53)-0.1587392],_
+[0.54,0.1566341,En(4,0.54),En(4,0.54)-0.1566341],_
+[0.55,0.1545596,En(4,0.55),En(4,0.55)-0.1545596],_
+[0.56,0.1525150,En(4,0.56),En(4,0.56)-0.1525150],_
+[0.57,0.1505000,En(4,0.57),En(4,0.57)-0.1505000],_
+[0.58,0.1485139,En(4,0.58),En(4,0.58)-0.1485139],_
+[0.59,0.1465565,En(4,0.59),En(4,0.59)-0.1465565],_
+[0.60,0.1446271,En(4,0.60),En(4,0.60)-0.1446271],_
+[0.61,0.1427253,En(4,0.61),En(4,0.61)-0.1427253],_
+[0.62,0.1408507,En(4,0.62),En(4,0.62)-0.1408507],_
+[0.63,0.1390028,En(4,0.63),En(4,0.63)-0.1390028],_
+[0.64,0.1371813,En(4,0.64),En(4,0.64)-0.1371813],_
+[0.65,0.1353855,En(4,0.65),En(4,0.65)-0.1353855],_
+[0.66,0.1336153,En(4,0.66),En(4,0.66)-0.1336153],_
+[0.67,0.1318701,En(4,0.67),En(4,0.67)-0.1318701],_
+[0.68,0.1301495,En(4,0.68),En(4,0.68)-0.1301495],_
+[0.69,0.1284533,En(4,0.69),En(4,0.69)-0.1284533],_
+[0.70,0.1267808,En(4,0.70),En(4,0.70)-0.1267808],_
+[0.71,0.1251319,En(4,0.71),En(4,0.71)-0.1251319],_
+[0.72,0.1235061,En(4,0.72),En(4,0.72)-0.1235061],_
+[0.73,0.1219031,En(4,0.73),En(4,0.73)-0.1219031],_
+[0.74,0.1203224,En(4,0.74),En(4,0.74)-0.1203224],_
+[0.75,0.1187638,En(4,0.75),En(4,0.75)-0.1187638],_
+[0.76,0.1172270,En(4,0.76),En(4,0.76)-0.1172270],_
+[0.77,0.1157115,En(4,0.77),En(4,0.77)-0.1157115],_
+[0.78,0.1142170,En(4,0.78),En(4,0.78)-0.1142170],_
+[0.79,0.1127433,En(4,0.79),En(4,0.79)-0.1127433],_
+[0.80,0.1112900,En(4,0.80),En(4,0.80)-0.1112900],_
+[0.81,0.1098567,En(4,0.81),En(4,0.81)-0.1098567],_
+[0.82,0.1084433,En(4,0.82),En(4,0.82)-0.1084433],_
+[0.83,0.1070493,En(4,0.83),En(4,0.83)-0.1070493],_
+[0.84,0.1056744,En(4,0.84),En(4,0.84)-0.1056744],_
+[0.85,0.1043185,En(4,0.85),En(4,0.85)-0.1043185],_
+[0.86,0.1029812,En(4,0.86),En(4,0.86)-0.1029812],_
+[0.87,0.1016622,En(4,0.87),En(4,0.87)-0.1016622],_
+[0.88,0.1003612,En(4,0.88),En(4,0.88)-0.1003612],_
+[0.89,0.0990780,En(4,0.89),En(4,0.89)-0.0990780],_
+[0.90,0.0978123,En(4,0.90),En(4,0.90)-0.0978123],_
+[0.91,0.0965639,En(4,0.91),En(4,0.91)-0.0965639],_
+[0.92,0.0953324,En(4,0.92),En(4,0.92)-0.0953324],_
+[0.93,0.0941177,En(4,0.93),En(4,0.93)-0.0941177],_
+[0.94,0.0929194,En(4,0.94),En(4,0.94)-0.0929194],_
+[0.95,0.0917374,En(4,0.95),En(4,0.95)-0.0917374],_
+[0.96,0.0905713,En(4,0.96),En(4,0.96)-0.0905713],_
+[0.97,0.0894211,En(4,0.97),En(4,0.97)-0.0894211],_
+[0.98,0.0882863,En(4,0.98),En(4,0.98)-0.0882863],_
+[0.99,0.0871669,En(4,0.99),En(4,0.99)-0.0871669],_
+[1.00,0.0860625,En(4,1.00),En(4,1.00)-0.0860625],_
+[1.01,0.0849730,En(4,1.01),En(4,1.01)-0.0849730],_
+[1.02,0.0838981,En(4,1.02),En(4,1.02)-0.0838981],_
+[1.03,0.0828376,En(4,1.03),En(4,1.03)-0.0828376],_
+[1.04,0.0817913,En(4,1.04),En(4,1.04)-0.0817913],_
+[1.05,0.0807590,En(4,1.05),En(4,1.05)-0.0807590],_
+[1.06,0.0797406,En(4,1.06),En(4,1.06)-0.0797406],_
+[1.07,0.0787357,En(4,1.07),En(4,1.07)-0.0787357],_
+[1.08,0.0777442,En(4,1.08),En(4,1.08)-0.0777442],_
+[1.09,0.0767659,En(4,1.09),En(4,1.09)-0.0767659],_
+[1.10,0.0758007,En(4,1.10),En(4,1.10)-0.0758007],_
+[1.11,0.0748483,En(4,1.11),En(4,1.11)-0.0748483],_
+[1.12,0.0739085,En(4,1.12),En(4,1.12)-0.0739085],_
+[1.13,0.0729812,En(4,1.13),En(4,1.13)-0.0729812],_
+[1.14,0.0720661,En(4,1.14),En(4,1.14)-0.0720661],_
+[1.15,0.0711632,En(4,1.15),En(4,1.15)-0.0711632],_
+[1.16,0.0702722,En(4,1.16),En(4,1.16)-0.0702722],_
+[1.17,0.0693930,En(4,1.17),En(4,1.17)-0.0693930],_
+[1.18,0.0685253,En(4,1.18),En(4,1.18)-0.0685253],_
+[1.19,0.0676691,En(4,1.19),En(4,1.19)-0.0676691],_
+[1.20,0.0668242,En(4,1.20),En(4,1.20)-0.0668242],_
+[1.21,0.0659904,En(4,1.21),En(4,1.21)-0.0659904],_
+[1.22,0.0651675,En(4,1.22),En(4,1.22)-0.0651675],_
+[1.23,0.0643555,En(4,1.23),En(4,1.23)-0.0643555],_
+[1.24,0.0635540,En(4,1.24),En(4,1.24)-0.0635540],_
+[1.25,0.0627631,En(4,1.25),En(4,1.25)-0.0627631],_
+[1.26,0.0619825,En(4,1.26),En(4,1.26)-0.0619825],_
+[1.27,0.0612122,En(4,1.27),En(4,1.27)-0.0612122],_
+[1.28,0.0604519,En(4,1.28),En(4,1.28)-0.0604519],_
+[1.29,0.0597015,En(4,1.29),En(4,1.29)-0.0597015],_
+[1.30,0.0589609,En(4,1.30),En(4,1.30)-0.0589609],_
+[1.31,0.0582299,En(4,1.31),En(4,1.31)-0.0582299],_
+[1.32,0.0575085,En(4,1.32),En(4,1.32)-0.0575085],_
+[1.33,0.0567964,En(4,1.33),En(4,1.33)-0.0567964],_
+[1.34,0.0560936,En(4,1.34),En(4,1.34)-0.0560936],_
+[1.35,0.0553998,En(4,1.35),En(4,1.35)-0.0553998],_
+[1.36,0.0547151,En(4,1.36),En(4,1.36)-0.0547151],_
+[1.37,0.0540393,En(4,1.37),En(4,1.37)-0.0540393],_
+[1.38,0.0533722,En(4,1.38),En(4,1.38)-0.0533722],_
+[1.39,0.0527137,En(4,1.39),En(4,1.39)-0.0527137],_
+[1.40,0.0520637,En(4,1.40),En(4,1.40)-0.0520637],_
+[1.41,0.0514222,En(4,1.41),En(4,1.41)-0.0514222],_
+[1.42,0.0507889,En(4,1.42),En(4,1.42)-0.0507889],_
+[1.43,0.0501637,En(4,1.43),En(4,1.43)-0.0501637],_
+[1.44,0.0495466,En(4,1.44),En(4,1.44)-0.0495466],_
+[1.45,0.0489374,En(4,1.45),En(4,1.45)-0.0489374],_
+[1.46,0.0483361,En(4,1.46),En(4,1.46)-0.0483361],_
+[1.47,0.0477425,En(4,1.47),En(4,1.47)-0.0477425],_
+[1.48,0.0471565,En(4,1.48),En(4,1.48)-0.0471565],_
+[1.49,0.0465780,En(4,1.49),En(4,1.49)-0.0465780],_
+[1.50,0.0460070,En(4,1.50),En(4,1.50)-0.0460070],_
+[1.51,0.0454432,En(4,1.51),En(4,1.51)-0.0454432],_
+[1.52,0.0448867,En(4,1.52),En(4,1.52)-0.0448867],_
+[1.53,0.0443372,En(4,1.53),En(4,1.53)-0.0443372],_
+[1.54,0.0437948,En(4,1.54),En(4,1.54)-0.0437948],_
+[1.55,0.0432593,En(4,1.55),En(4,1.55)-0.0432593],_
+[1.56,0.0427307,En(4,1.56),En(4,1.56)-0.0427307],_
+[1.57,0.0422087,En(4,1.57),En(4,1.57)-0.0422087],_
+[1.58,0.0416935,En(4,1.58),En(4,1.58)-0.0416935],_
+[1.59,0.0411847,En(4,1.59),En(4,1.59)-0.0411847],_
+[1.60,0.0406825,En(4,1.60),En(4,1.60)-0.0406825],_
+[1.61,0.0401866,En(4,1.61),En(4,1.61)-0.0401866],_
+[1.62,0.0396970,En(4,1.62),En(4,1.62)-0.0396970],_
+[1.63,0.0392136,En(4,1.63),En(4,1.63)-0.0392136],_
+[1.64,0.0387364,En(4,1.64),En(4,1.64)-0.0387364],_
+[1.65,0.0382652,En(4,1.65),En(4,1.65)-0.0382652],_
+[1.66,0.0377999,En(4,1.66),En(4,1.66)-0.0377999],_
+[1.67,0.0373406,En(4,1.67),En(4,1.67)-0.0373406],_
+[1.68,0.0368870,En(4,1.68),En(4,1.68)-0.0368870],_
+[1.69,0.0364392,En(4,1.69),En(4,1.69)-0.0364392],_
+[1.70,0.0359970,En(4,1.70),En(4,1.70)-0.0359970],_
+[1.71,0.0355604,En(4,1.71),En(4,1.71)-0.0355604],_
+[1.72,0.0351293,En(4,1.72),En(4,1.72)-0.0351293],_
+[1.73,0.0347037,En(4,1.73),En(4,1.73)-0.0347037],_
+[1.74,0.0342834,En(4,1.74),En(4,1.74)-0.0342834],_
+[1.75,0.0338684,En(4,1.75),En(4,1.75)-0.0338684],_
+[1.76,0.0334586,En(4,1.76),En(4,1.76)-0.0334586],_
+[1.77,0.0330539,En(4,1.77),En(4,1.77)-0.0330539],_
+[1.78,0.0326544,En(4,1.78),En(4,1.78)-0.0326544],_
+[1.79,0.0322598,En(4,1.79),En(4,1.79)-0.0322598],_
+[1.80,0.0318702,En(4,1.80),En(4,1.80)-0.0318702],_
+[1.81,0.0314855,En(4,1.81),En(4,1.81)-0.0314855],_
+[1.82,0.0311056,En(4,1.82),En(4,1.82)-0.0311056],_
+[1.83,0.0307304,En(4,1.83),En(4,1.83)-0.0307304],_
+[1.84,0.0303599,En(4,1.84),En(4,1.84)-0.0303599],_
+[1.85,0.0299941,En(4,1.85),En(4,1.85)-0.0299941],_
+[1.86,0.0296328,En(4,1.86),En(4,1.86)-0.0296328],_
+[1.87,0.0292761,En(4,1.87),En(4,1.87)-0.0292761],_
+[1.88,0.0289238,En(4,1.88),En(4,1.88)-0.0289238],_
+[1.89,0.0285759,En(4,1.89),En(4,1.89)-0.0285759],_
+[1.90,0.0282323,En(4,1.90),En(4,1.90)-0.0282323],_
+[1.91,0.0278930,En(4,1.91),En(4,1.91)-0.0278930],_
+[1.92,0.0275579,En(4,1.92),En(4,1.92)-0.0275579],_
+[1.93,0.0272270,En(4,1.93),En(4,1.93)-0.0272270],_
+[1.94,0.0269002,En(4,1.94),En(4,1.94)-0.0269002],_
+[1.95,0.0265775,En(4,1.95),En(4,1.95)-0.0265775],_
+[1.96,0.0262587,En(4,1.96),En(4,1.96)-0.0262587],_
+[1.97,0.0259440,En(4,1.97),En(4,1.97)-0.0259440],_
+[1.98,0.0256331,En(4,1.98),En(4,1.98)-0.0256331],_
+[1.99,0.0253261,En(4,1.99),En(4,1.99)-0.0253261],_
+[2.00,0.0250228,En(4,2.00),En(4,2.00)-0.0250228]]
+--R 
+--R
+--R   (5)
+--R   [[1.0E-2,0.32838240000000002,0.32838235603577381,- 4.3964226204007417E-8],
+--R    [2.0E-2,0.32352639999999999,0.32352643582573859,3.5825738597949908E-8],
+--R
+--R     [2.9999999999999999E-2, 0.31876189999999999, 0.318761867644201,
+--R      - 3.2355798984529116E-8]
+--R     ,
+--R
+--R     [4.0000000000000001E-2, 0.31408550000000002, 0.3140854938275166,
+--R      - 6.1724834132803608E-9]
+--R     ,
+--R
+--R     [5.0000000000000003E-2, 0.30949450000000001, 0.30949449400443008,
+--R      - 5.9955699294178544E-9]
+--R     ,
+--R
+--R     [5.9999999999999998E-2, 0.30498629999999999, 0.30498629353000445,
+--R      - 6.4699955393265896E-9]
+--R     ,
+--R
+--R     [7.0000000000000007E-2, 0.30055850000000001, 0.30055851077336515,
+--R      1.0773365144434166E-8]
+--R     ,
+--R    [8.0000000000000002E-2,0.2962089,0.29620892264764853,2.2647648534324105E-8],
+--R
+--R     [8.9999999999999997E-2, 0.29193540000000001, 0.29193544025523149,
+--R      4.0255231481545195E-8]
+--R     ,
+--R
+--R     [0.10000000000000001, 0.28773609999999999, 0.28773609074837719,
+--R      - 9.2516228011874091E-9]
+--R     ,
+--R    [0.11,0.283609,0.2836090032896208,3.2896207979860037E-9],
+--R    [0.12,0.27955239999999998,0.27955239786156211,- 2.1384378712241414E-9],
+--R    [0.13,0.27556459999999999,0.27556457613853869,- 2.3861461306839971E-8],
+--R
+--R     [0.14000000000000001, 0.27164389999999999, 0.27164391389899217,
+--R      1.3898992179406378E-8]
+--R     ,
+--R    [0.14999999999999999,0.2677889,0.26778885461965246,- 4.5380347535317611E-8],
+--R    [0.16,0.26399790000000001,0.26399790399625528,3.9962552711436672E-9],
+--R
+--R     [0.17000000000000001, 0.26026959999999999, 0.26026962520418123,
+--R      2.5204181242077794E-8]
+--R     ,
+--R
+--R     [0.17999999999999999, 0.25660260000000001, 0.25660263475941625,
+--R      3.4759416234209084E-8]
+--R     ,
+--R    [0.19,0.25299559999999999,0.25299559887329914,- 1.1267008437343407E-9],
+--R
+--R     [0.20000000000000001, 0.24944720000000001, 0.24944723021833587,
+--R      3.021833586136502E-8]
+--R     ,
+--R
+--R     [0.20999999999999999, 0.24595629999999999, 0.24595628503986891,
+--R      - 1.496013107837868E-8]
+--R     ,
+--R    [0.22,0.2425216,0.24252156056149116,- 3.9438508847577936E-8],
+--R
+--R     [0.23000000000000001, 0.23914189999999999, 0.23914189264206476,
+--R      - 7.3579352333208448E-9]
+--R     ,
+--R    [0.23999999999999999,0.2358162,0.235816153649895,- 4.6350105004089315E-8],
+--R    [0.25,0.23254320000000001,0.232543250525623,5.0525622991015595E-8],
+--R    [0.26000000000000001,0.2293221,0.22932212301015453,2.3010154531766247E-8],
+--R
+--R     [0.27000000000000002, 0.22615170000000001, 0.22615174201774482,
+--R      4.2017744811273516E-8]
+--R     ,
+--R
+--R     [0.28000000000000003, 0.22303110000000001, 0.22303110813742405,
+--R      8.1374240401554943E-9]
+--R     ,
+--R    [0.28999999999999998,0.2199593,0.21995925024844767,- 4.9751552322341297E-8],
+--R
+--R     [0.29999999999999999, 0.21693519999999999, 0.2169352242375045,
+--R      2.4237504503421547E-8]
+--R     ,
+--R    [0.31,0.21395810000000001,0.21395811180711269,1.1807112676454068E-8],
+--R
+--R     [0.32000000000000001, 0.21102699999999999, 0.21102701936604468,
+--R      1.9366044684554495E-8]
+--R     ,
+--R    [0.33000000000000002,0.2081411,0.20814107699380749,- 2.3006192506613843E-8],
+--R
+--R     [0.34000000000000002, 0.20529939999999999, 0.20529943747220336,
+--R      3.7472203368027479E-8]
+--R     ,
+--R    [0.34999999999999998,0.2025013,0.20250127537784185,- 2.4622158145692907E-8],
+--R    [0.35999999999999999,0.1997458,0.19974578623019815,- 1.3769801854301988E-8],
+--R    [0.37,0.19703219999999999,0.19703218569043029,- 1.4309569695836188E-8],
+--R    [0.38,0.1943597,0.19435970880670123,8.8067012293802094E-9],
+--R    [0.39000000000000001,0.1917276,0.19172760930221217,9.3022121672170499E-9],
+--R    [0.40000000000000002,0.1891352,0.18913515890255378,- 4.1097446223936629E-8],
+--R
+--R     [0.40999999999999998, 0.18658159999999999, 0.18658164669933056,
+--R      4.6699330569222042E-8]
+--R     ,
+--R
+--R     [0.41999999999999998, 0.18406639999999999, 0.18406637854731811,
+--R      - 2.1452681880917979E-8]
+--R     ,
+--R
+--R     [0.42999999999999999, 0.18158869999999999, 0.18158867649268154,
+--R      - 2.3507318452908876E-8]
+--R     ,
+--R    [0.44,0.1791479,0.17914787823001868,- 2.1769981317643072E-8],
+--R
+--R     [0.45000000000000001, 0.17674329999999999, 0.17674333658619995,
+--R      3.6586199958499321E-8]
+--R     ,
+--R
+--R     [0.46000000000000002, 0.17437440000000001, 0.17437441902916206,
+--R      1.9029162051653614E-8]
+--R     ,
+--R
+--R     [0.46999999999999997, 0.17204050000000001, 0.17204050719997591,
+--R      7.1999758932683733E-9]
+--R     ,
+--R    [0.47999999999999998,0.169741,0.16974099646665652,- 3.5333434789119877E-9],
+--R
+--R     [0.48999999999999999, 0.16747529999999999, 0.16747529549831219,
+--R      - 4.5016878003067973E-9]
+--R     ,
+--R    [0.5,0.1652428,0.16524282585834804,2.5858348046137181E-8],
+--R
+--R     [0.51000000000000001, 0.16304299999999999, 0.16304302161554463,
+--R      2.161554463198101E-8]
+--R     ,
+--R    [0.52000000000000002,0.1608753,0.16087532897192483,2.8971924831422413E-8],
+--R    [0.53000000000000003,0.1587392,0.15873920590641022,5.9064102220229842E-9],
+--R    [0.54000000000000004,0.1566341,0.15663412183334352,2.1833343522015269E-8],
+--R
+--R     [0.55000000000000004, 0.15455959999999999, 0.15455955727502388,
+--R      - 4.2724976112573643E-8]
+--R     ,
+--R
+--R     [0.56000000000000005, 0.15251500000000001, 0.15251500354746483,
+--R      3.5474648218958293E-9]
+--R     ,
+--R
+--R     [0.56999999999999995, 0.15049999999999999, 0.15049996245864372,
+--R      - 3.7541356273651516E-8]
+--R     ,
+--R    [0.57999999999999996,0.1485139,0.14851394601856155,4.6018561544736158E-8],
+--R
+--R     [0.58999999999999997, 0.14655650000000001, 0.14655647616048229,
+--R      - 2.3839517721002679E-8]
+--R     ,
+--R
+--R     [0.59999999999999998, 0.14462710000000001, 0.1446270844727626,
+--R      - 1.5527237412493733E-8]
+--R     ,
+--R    [0.60999999999999999,0.1427253,0.14272531194072449,1.1940724492509602E-8],
+--R    [0.62,0.1408507,0.14085070869805835,8.6980583569040704E-9],
+--R    [0.63,0.13900280000000001,0.13900283378727951,3.3787279501984457E-8],
+--R
+--R     [0.64000000000000001, 0.13718130000000001, 0.13718125492879019,
+--R      - 4.5071209819891322E-8]
+--R     ,
+--R
+--R     [0.65000000000000002, 0.13538549999999999, 0.13538554829812915,
+--R      4.8298129162738235E-8]
+--R     ,
+--R
+--R     [0.66000000000000003, 0.13361529999999999, 0.13361529831101665,
+--R      - 1.688983342074124E-9]
+--R     ,
+--R
+--R     [0.67000000000000004, 0.13187009999999999, 0.13187009741582595,
+--R      - 2.5841740414200842E-9]
+--R     ,
+--R    [0.68000000000000005,0.1301495,0.13014954589313665,4.5893136652042088E-8],
+--R
+--R     [0.68999999999999995, 0.12845329999999999, 0.12845325166204447,
+--R      - 4.8337955527166798E-8]
+--R     ,
+--R    [0.69999999999999996,0.1267808,0.12678083009292157,3.0092921571300835E-8],
+--R
+--R     [0.70999999999999996, 0.12513189999999999, 0.1251319038263401,
+--R      3.8263401058724611E-9]
+--R     ,
+--R
+--R     [0.71999999999999997, 0.12350609999999999, 0.12350610259788675,
+--R      2.5978867529419247E-9]
+--R     ,
+--R    [0.72999999999999998,0.1219031,0.12190306306861284,- 3.6931387156768913E-8],
+--R    [0.73999999999999999,0.1203224,0.12032242866087833,2.8660878337305462E-8],
+--R    [0.75,0.1187638,0.11876384939936065,4.9399360643742618E-8],
+--R    [0.76000000000000001,0.117227,0.1172269817570131,- 1.8242986898897584E-8],
+--R
+--R     [0.77000000000000002, 0.11571149999999999, 0.1157114885057679,
+--R      - 1.1494232093900436E-8]
+--R     ,
+--R    [0.78000000000000003,0.114217,0.11421703857179058,3.8571790580976817E-8],
+--R    [0.79000000000000004,0.1127433,0.11274330689510217,6.895102169779399E-9],
+--R    [0.80000000000000004,0.11129,0.11128997429339493,- 2.570660506684419E-8],
+--R    [0.81000000000000005,0.1098567,0.10985672732987711,2.7329877111403889E-8],
+--R
+--R     [0.81999999999999995, 0.10844330000000001, 0.10844325818498929,
+--R      - 4.1815010717871637E-8]
+--R     ,
+--R    [0.82999999999999996,0.1070493,0.10704926453184345,- 3.5468156550644281E-8],
+--R    [0.83999999999999997,0.1056744,0.10567444941524373,4.9415243730255298E-8],
+--R
+--R     [0.84999999999999998, 0.10431849999999999, 0.10431852113415277,
+--R      2.1134152775048598E-8]
+--R     ,
+--R
+--R     [0.85999999999999999, 0.10298119999999999, 0.10298119312747675,
+--R      - 6.8725232449384777E-9]
+--R     ,
+--R    [0.87,0.10166219999999999,0.10166218386304593,- 1.6136954064660536E-8],
+--R    [0.88,0.1003612,0.10036121672967431,1.6729674312543352E-8],
+--R
+--R     [0.89000000000000001, 9.9077999999999999E-2, 9.9078019932187969E-2,
+--R      1.9932187969584625E-8]
+--R     ,
+--R
+--R     [0.90000000000000002, 9.7812300000000005E-2, 9.7812326389315099E-2,
+--R      2.6389315094244026E-8]
+--R     ,
+--R
+--R     [0.91000000000000003, 9.6563899999999994E-2, 9.6563873634337494E-2,
+--R      - 2.6365662500471565E-8]
+--R     ,
+--R
+--R     [0.92000000000000004, 9.5332399999999998E-2, 9.5332403718406081E-2,
+--R      3.7184060830419696E-9]
+--R     ,
+--R
+--R     [0.93000000000000005, 9.4117699999999999E-2, 9.4117663116428885E-2,
+--R      - 3.6883571113976998E-8]
+--R     ,
+--R
+--R     [0.93999999999999995, 9.2919399999999999E-2, 9.2919402635442322E-2,
+--R      2.6354423227070711E-9]
+--R     ,
+--R
+--R     [0.94999999999999996, 9.1737399999999997E-2, 9.1737377325382147E-2,
+--R      - 2.2674617849993695E-8]
+--R     ,
+--R
+--R     [0.95999999999999996, 9.0571299999999993E-2, 9.0571346392172847E-2,
+--R      4.6392172853293445E-8]
+--R     ,
+--R
+--R     [0.96999999999999997, 8.9421100000000003E-2, 8.9421073113058108E-2,
+--R      - 2.6886941895121197E-8]
+--R     ,
+--R
+--R     [0.97999999999999998, 8.8286299999999998E-2, 8.8286324754098505E-2,
+--R      2.4754098507107081E-8]
+--R     ,
+--R
+--R     [0.98999999999999999, 8.7166900000000005E-2, 8.7166872489765013E-2,
+--R      - 2.7510234992811355E-8]
+--R     ,
+--R    [1.,8.60625E-2,8.606249132456073E-2,- 8.6754392697452332E-9],
+--R    [1.01,8.4973000000000007E-2,8.497296001660537E-2,- 3.9983394636844416E-8],
+--R    [1.02,8.3898100000000003E-2,8.3898061003119584E-2,- 3.8996880419306379E-8],
+--R    [1.03,8.2837599999999997E-2,8.2837580327749516E-2,- 1.967225048138399E-8],
+--R    [1.04,8.1791299999999997E-2,8.179130756967358E-2,7.5696735823971295E-9],
+--R    [1.05,8.0758999999999997E-2,8.0759035774436022E-2,3.5774436024649248E-8],
+--R
+--R     [1.0600000000000001, 7.9740599999999995E-2, 7.9740561386454245E-2,
+--R      - 3.8613545749743849E-8]
+--R     ,
+--R
+--R     [1.0700000000000001, 7.8735700000000006E-2, 7.8735684183148108E-2,
+--R      - 1.5816851897665352E-8]
+--R     ,
+--R
+--R     [1.0800000000000001, 7.7744199999999999E-2, 7.774420721064286E-2,
+--R      7.2106428605778206E-9]
+--R     ,
+--R
+--R     [1.0900000000000001, 7.6765899999999998E-2, 7.6765936720997435E-2,
+--R      3.6720997437234004E-8]
+--R     ,
+--R
+--R     [1.1000000000000001, 7.5800699999999999E-2, 7.5800682110913065E-2,
+--R      - 1.7889086934008702E-8]
+--R     ,
+--R
+--R     [1.1100000000000001, 7.4848300000000006E-2, 7.4848255861878349E-2,
+--R      - 4.4138121657599783E-8]
+--R     ,
+--R
+--R     [1.1200000000000001, 7.3908500000000002E-2, 7.3908473481708176E-2,
+--R      - 2.6518291826316265E-8]
+--R     ,
+--R
+--R     [1.1299999999999999, 7.2981199999999996E-2, 7.2981153447436026E-2,
+--R      - 4.6552563970347904E-8]
+--R     ,
+--R
+--R     [1.1399999999999999, 7.2066099999999994E-2, 7.2066117149520492E-2,
+--R      1.7149520498138671E-8]
+--R     ,
+--R
+--R     [1.1499999999999999, 7.1163199999999996E-2, 7.1163188837328195E-2,
+--R      - 1.1162671800790669E-8]
+--R     ,
+--R
+--R     [1.1599999999999999, 7.0272200000000007E-2, 7.027219556585651E-2,
+--R      - 4.4341434968231397E-9]
+--R     ,
+--R
+--R     [1.1699999999999999, 6.9392999999999996E-2, 6.9392967143661166E-2,
+--R      - 3.2856338830211662E-8]
+--R     ,
+--R
+--R     [1.1799999999999999, 6.8525299999999997E-2, 6.8525336081954685E-2,
+--R      3.6081954687960938E-8]
+--R     ,
+--R
+--R     [1.1899999999999999, 6.7669099999999996E-2, 6.7669137544842786E-2,
+--R      3.7544842790282473E-8]
+--R     ,
+--R    [1.2,6.68242E-2,6.6824209300667584E-2,9.300667583311828E-9],
+--R    [1.21,6.5990400000000005E-2,6.5990391674426857E-2,- 8.3255731475606609E-9],
+--R    [1.22,6.5167500000000003E-2,6.5167527501239941E-2,2.7501239938110267E-8],
+--R    [1.23,6.4355499999999996E-2,6.4355462080831385E-2,- 3.7919168610622833E-8],
+--R    [1.24,6.3553999999999999E-2,6.3554043133005847E-2,4.313300584757318E-8],
+--R    [1.25,6.2763100000000002E-2,6.2763120754086868E-2,2.0754086865748533E-8],
+--R    [1.26,6.1982500000000003E-2,6.1982547374293873E-2,4.7374293869750783E-8],
+--R    [1.27,6.1212200000000001E-2,6.1212177716032781E-2,- 2.2283967220093626E-8],
+--R    [1.28,6.0451900000000003E-2,6.0451868753076454E-2,- 3.124692354861125E-8],
+--R    [1.29,5.9701499999999998E-2,5.9701479670610937E-2,- 2.0329389061046754E-8],
+--R    [1.3,5.8960899999999997E-2,5.8960871826125583E-2,- 2.8173874414005873E-8],
+--R
+--R     [1.3100000000000001, 5.8229900000000001E-2, 5.822990871112519E-2,
+--R      8.7111251892424413E-9]
+--R     ,
+--R
+--R     [1.3200000000000001, 5.7508499999999997E-2, 5.7508455913643328E-2,
+--R      - 4.4086356669470472E-8]
+--R     ,
+--R
+--R     [1.3300000000000001, 5.6796399999999997E-2, 5.6796381081534952E-2,
+--R      - 1.8918465044537314E-8]
+--R     ,
+--R
+--R     [1.3400000000000001, 5.60936E-2, 5.6093553886531214E-2,
+--R      - 4.6113468786523892E-8]
+--R     ,
+--R
+--R     [1.3500000000000001, 5.5399799999999999E-2, 5.5399845989034427E-2,
+--R      4.5989034427873499E-8]
+--R     ,
+--R
+--R     [1.3600000000000001, 5.4715100000000003E-2, 5.4715131003636563E-2,
+--R      3.1003636560833492E-8]
+--R     ,
+--R
+--R     [1.3700000000000001, 5.4039299999999998E-2, 5.4039284465342098E-2,
+--R      - 1.5534657900329041E-8]
+--R     ,
+--R
+--R     [1.3799999999999999, 5.3372200000000002E-2, 5.3372183796479428E-2,
+--R      - 1.6203520573909724E-8]
+--R     ,
+--R
+--R     [1.3899999999999999, 5.2713700000000002E-2, 5.2713708274280791E-2,
+--R      8.2742807883118275E-9]
+--R     ,
+--R
+--R     [1.3999999999999999, 5.2063699999999997E-2, 5.206373899911769E-2,
+--R      3.8999117692173346E-8]
+--R     ,
+--R
+--R     [1.4099999999999999, 5.1422200000000001E-2, 5.1422158863374548E-2,
+--R      - 4.1136625453241926E-8]
+--R     ,
+--R
+--R     [1.4199999999999999, 5.0788899999999998E-2, 5.0788852520943208E-2,
+--R      - 4.7479056790311613E-8]
+--R     ,
+--R
+--R     [1.4299999999999999, 5.0163699999999999E-2, 5.0163706357327698E-2,
+--R      6.357327698991444E-9]
+--R     ,
+--R
+--R     [1.4399999999999999, 4.9546600000000003E-2, 4.9546608460339536E-2,
+--R      8.4603395331117603E-9]
+--R     ,
+--R    [1.45,4.8937399999999999E-2,4.8937448591373334E-2,4.859137333473873E-8],
+--R    [1.46,4.83361E-2,4.833611815724749E-2,1.8157247490269146E-8],
+--R    [1.47,4.77425E-2,4.7742510182596301E-2,1.0182596300867264E-8],
+--R    [1.48,4.7156499999999997E-2,4.7156519282801249E-2,1.9282801251574444E-8],
+--R    [1.49,4.6578000000000001E-2,4.6578041637448199E-2,4.1637448197628935E-8],
+--R    [1.5,4.6006999999999999E-2,4.6006974964299521E-2,- 2.503570047801551E-8],
+--R    [1.51,4.5443200000000003E-2,4.5443218493767132E-2,1.8493767128813232E-8],
+--R    [1.52,4.4886700000000002E-2,4.4886672943877548E-2,- 2.7056122453572584E-8],
+--R    [1.53,4.43372E-2,4.4337240495715358E-2,4.0495715357413697E-8],
+--R    [1.54,4.3794800000000002E-2,4.3794824769334884E-2,2.4769334881769023E-8],
+--R    [1.55,4.3259300000000001E-2,4.3259330800131675E-2,3.0800131674280085E-8],
+--R
+--R     [1.5600000000000001, 4.2730700000000003E-2, 4.2730665015658725E-2,
+--R      - 3.4984341278077036E-8]
+--R     ,
+--R
+--R     [1.5700000000000001, 4.2208700000000002E-2, 4.2208735212882098E-2,
+--R      3.5212882096291587E-8]
+--R     ,
+--R
+--R     [1.5800000000000001, 4.1693500000000001E-2, 4.1693450535863763E-2,
+--R      - 4.9464136238352996E-8]
+--R     ,
+--R
+--R     [1.5900000000000001, 4.1184699999999998E-2, 4.1184721453862164E-2,
+--R      2.1453862165954352E-8]
+--R     ,
+--R
+--R     [1.6000000000000001, 4.0682500000000003E-2, 4.0682459739842872E-2,
+--R      - 4.0260157131710717E-8]
+--R     ,
+--R
+--R     [1.6100000000000001, 4.0186600000000003E-2, 4.0186578449386889E-2,
+--R      - 2.1550613113485717E-8]
+--R     ,
+--R
+--R     [1.6200000000000001, 3.9697000000000003E-2, 3.9696991899992498E-2,
+--R      - 8.1000075050075004E-9]
+--R     ,
+--R
+--R     [1.6299999999999999, 3.9213600000000001E-2, 3.9213615650758454E-2,
+--R      1.5650758453111813E-8]
+--R     ,
+--R
+--R     [1.6399999999999999, 3.8736399999999997E-2, 3.8736366482441345E-2,
+--R      - 3.3517558652162993E-8]
+--R     ,
+--R
+--R     [1.6499999999999999, 3.8265199999999999E-2, 3.8265162377879226E-2,
+--R      - 3.7622120772906609E-8]
+--R     ,
+--R
+--R     [1.6599999999999999, 3.7799899999999997E-2, 3.7799922502774017E-2,
+--R      2.2502774019161897E-8]
+--R     ,
+--R
+--R     [1.6699999999999999, 3.7340600000000002E-2, 3.7340567186823333E-2,
+--R      - 3.2813176668866628E-8]
+--R     ,
+--R
+--R     [1.6799999999999999, 3.6887000000000003E-2, 3.6887017905196488E-2,
+--R      1.7905196485201724E-8]
+--R     ,
+--R
+--R     [1.6899999999999999, 3.6439199999999998E-2, 3.6439197260345613E-2,
+--R      - 2.7396543852975519E-9]
+--R     ,
+--R    [1.7,3.5997000000000001E-2,3.5997028964144667E-2,2.8964144666021596E-8],
+--R    [1.71,3.5560399999999999E-2,3.5560437820352508E-2,3.7820352509487787E-8],
+--R    [1.72,3.5129300000000002E-2,3.5129349707387826E-2,4.9707387823894056E-8],
+--R    [1.73,3.4703699999999997E-2,3.4703691561414153E-2,- 8.4385858439839367E-9],
+--R    [1.74,3.4283399999999999E-2,3.4283391359727231E-2,- 8.6402727680900959E-9],
+--R    [1.75,3.38684E-2,3.3868378104435617E-2,- 2.1895564382423682E-8],
+--R    [1.76,3.3458599999999998E-2,3.3458581806433824E-2,- 1.8193566174440345E-8],
+--R    [1.77,3.3053899999999997E-2,3.3053933469655281E-2,3.3469655283391297E-8],
+--R    [1.78,3.26544E-2,3.2654365075608149E-2,- 3.4924391850710279E-8],
+--R    [1.79,3.2259799999999998E-2,3.2259809568177103E-2,9.5681771047906317E-9],
+--R    [1.8,3.1870200000000001E-2,3.187020083869585E-2,8.3869584888152104E-10],
+--R    [1.8100000000000001,3.14855E-2,3.1485473711279174E-2,- 2.628872082521827E-8]
+--R     ,
+--R
+--R     [1.8200000000000001, 3.1105600000000001E-2, 3.1105563928410965E-2,
+--R      - 3.6071589035180374E-8]
+--R     ,
+--R
+--R     [1.8300000000000001, 3.0730400000000001E-2, 3.073040813678209E-2,
+--R      8.1367820883859743E-9]
+--R     ,
+--R
+--R     [1.8400000000000001, 3.0359899999999999E-2, 3.0359943873375265E-2,
+--R      4.387337526612356E-8]
+--R     ,
+--R
+--R     [1.8500000000000001, 2.9994099999999999E-2, 2.9994109551789112E-2,
+--R      9.5517891131324806E-9]
+--R     ,
+--R
+--R     [1.8600000000000001, 2.9632800000000001E-2, 2.9632844448799496E-2,
+--R      4.4448799495222513E-8]
+--R     ,
+--R
+--R     [1.8700000000000001, 2.9276099999999999E-2, 2.9276088691150233E-2,
+--R      - 1.1308849766356044E-8]
+--R     ,
+--R
+--R     [1.8799999999999999, 2.89238E-2, 2.8923783242571027E-2,
+--R      - 1.6757428972224986E-8]
+--R     ,
+--R
+--R     [1.8899999999999999, 2.8575900000000001E-2, 2.8575869891020747E-2,
+--R      - 3.0108979254261925E-8]
+--R     ,
+--R
+--R     [1.8999999999999999, 2.8232299999999998E-2, 2.8232291236140537E-2,
+--R      - 8.7638594610528475E-9]
+--R     ,
+--R
+--R     [1.9099999999999999, 2.7893000000000001E-2, 2.7892990676930449E-2,
+--R      - 9.3230695524804119E-9]
+--R     ,
+--R    [1.9199999999999999,2.75579E-2,2.7557912399626863E-2,1.2399626863474067E-8],
+--R
+--R     [1.9299999999999999, 2.7227000000000001E-2, 2.7227001365790433E-2,
+--R      1.3657904321395797E-9]
+--R     ,
+--R
+--R     [1.9399999999999999, 2.6900199999999999E-2, 2.6900203300591483E-2,
+--R      3.300591484151072E-9]
+--R     ,
+--R    [1.95,2.65775E-2,2.6577464681296726E-2,- 3.5318703274500596E-8],
+--R    [1.96,2.6258699999999999E-2,2.6258732725946241E-2,3.2725946241124459E-8],
+--R    [1.97,2.5943999999999998E-2,2.5943955382222512E-2,- 4.4617777486544163E-8],
+--R    [1.98,2.5633099999999999E-2,2.5633081316501483E-2,- 1.8683498515664754E-8],
+--R    [1.99,2.5326100000000001E-2,2.5326059903094673E-2,- 4.0096905327274834E-8],
+--R    [2.,2.5022800000000001E-2,2.5022841213660458E-2,4.1213660456618229E-8]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 5
+
+@
+This table expresses the value of 
+$E_10$ with an argument range of 0.01 to 2.00.
+<<*>>=
+--S 6 of 7
+[[0.01,0.1098682,En(10,0.01),En(10,0.01)-0.1098682],_
+[0.02,0.1086395,En(10,0.02),En(10,0.02)-0.1086395],_
+[0.03,0.1074246,En(10,0.03),En(10,0.03)-0.1074246],_
+[0.04,0.1062236,En(10,0.04),En(10,0.04)-0.1062236],_
+[0.05,0.1050363,En(10,0.05),En(10,0.05)-0.1050363],_
+[0.06,0.1038624,En(10,0.06),En(10,0.06)-0.1038624],_
+[0.07,0.1027018,En(10,0.07),En(10,0.07)-0.1027018],_
+[0.08,0.1015544,En(10,0.08),En(10,0.08)-0.1015544],_
+[0.09,0.1004200,En(10,0.09),En(10,0.09)-0.1004200],_
+[0.10,0.0992984,En(10,0.10),En(10,0.10)-0.0992984],_
+[0.11,0.0981896,En(10,0.11),En(10,0.11)-0.0981896],_
+[0.12,0.0970934,En(10,0.12),En(10,0.12)-0.0970934],_
+[0.13,0.0960095,En(10,0.13),En(10,0.13)-0.0960095],_
+[0.14,0.0949380,En(10,0.14),En(10,0.14)-0.0949380],_
+[0.15,0.0938786,En(10,0.15),En(10,0.15)-0.0938786],_
+[0.16,0.0928312,En(10,0.16),En(10,0.16)-0.0928312],_
+[0.17,0.0917956,En(10,0.17),En(10,0.17)-0.0917956],_
+[0.18,0.0907718,En(10,0.18),En(10,0.18)-0.0907718],_
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+[0.48,0.0648907,En(10,0.48),En(10,0.48)-0.0648907],_
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+[0.50,0.0634583,En(10,0.50),En(10,0.50)-0.0634583],_
+[0.51,0.0627542,En(10,0.51),En(10,0.51)-0.0627542],_
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+[0.57,0.0586925,En(10,0.57),En(10,0.57)-0.0586925],_
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+[0.59,0.0573986,En(10,0.59),En(10,0.59)-0.0573986],_
+[0.60,0.0567626,En(10,0.60),En(10,0.60)-0.0567626],_
+[0.61,0.0561336,En(10,0.61),En(10,0.61)-0.0561336],_
+[0.62,0.0555118,En(10,0.62),En(10,0.62)-0.0555118],_
+[0.63,0.0548969,En(10,0.63),En(10,0.63)-0.0548969],_
+[0.64,0.0542889,En(10,0.64),En(10,0.64)-0.0542889],_
+[0.65,0.0536877,En(10,0.65),En(10,0.65)-0.0536877],_
+[0.66,0.0530933,En(10,0.66),En(10,0.66)-0.0530933],_
+[0.67,0.0525055,En(10,0.67),En(10,0.67)-0.0525055],_
+[0.68,0.0519243,En(10,0.68),En(10,0.68)-0.0519243],_
+[0.69,0.0513497,En(10,0.69),En(10,0.69)-0.0513497],_
+[0.70,0.0507815,En(10,0.70),En(10,0.70)-0.0507815],_
+[0.71,0.0502196,En(10,0.71),En(10,0.71)-0.0502196],_
+[0.72,0.0496640,En(10,0.72),En(10,0.72)-0.0496640],_
+[0.73,0.0491147,En(10,0.73),En(10,0.73)-0.0491147],_
+[0.74,0.0485715,En(10,0.74),En(10,0.74)-0.0485715],_
+[0.75,0.0480344,En(10,0.75),En(10,0.75)-0.0480344],_
+[0.76,0.0475033,En(10,0.76),En(10,0.76)-0.0475033],_
+[0.77,0.0469781,En(10,0.77),En(10,0.77)-0.0469781],_
+[0.78,0.0464588,En(10,0.78),En(10,0.78)-0.0464588],_
+[0.79,0.0459453,En(10,0.79),En(10,0.79)-0.0459453],_
+[0.80,0.0454376,En(10,0.80),En(10,0.80)-0.0454376],_
+[0.81,0.0449356,En(10,0.81),En(10,0.81)-0.0449356],_
+[0.82,0.0444391,En(10,0.82),En(10,0.82)-0.0444391],_
+[0.83,0.0439482,En(10,0.83),En(10,0.83)-0.0439482],_
+[0.84,0.0434628,En(10,0.84),En(10,0.84)-0.0434628],_
+[0.85,0.0429829,En(10,0.85),En(10,0.85)-0.0429829],_
+[0.86,0.0425082,En(10,0.86),En(10,0.86)-0.0425082],_
+[0.87,0.0420389,En(10,0.87),En(10,0.87)-0.0420389],_
+[0.88,0.0415749,En(10,0.88),En(10,0.88)-0.0415749],_
+[0.89,0.0411160,En(10,0.89),En(10,0.89)-0.0411160],_
+[0.90,0.0406622,En(10,0.90),En(10,0.90)-0.0406622],_
+[0.91,0.0402135,En(10,0.91),En(10,0.91)-0.0402135],_
+[0.92,0.0397698,En(10,0.92),En(10,0.92)-0.0397698],_
+[0.93,0.0393311,En(10,0.93),En(10,0.93)-0.0393311],_
+[0.94,0.0388973,En(10,0.94),En(10,0.94)-0.0388973],_
+[0.95,0.0384683,En(10,0.95),En(10,0.95)-0.0384683],_
+[0.96,0.0380441,En(10,0.96),En(10,0.96)-0.0380441],_
+[0.97,0.0376246,En(10,0.97),En(10,0.97)-0.0376246],_
+[0.98,0.0372098,En(10,0.98),En(10,0.98)-0.0372098],_
+[0.99,0.0367996,En(10,0.99),En(10,0.99)-0.0367996],_
+[1.00,0.0363940,En(10,1.00),En(10,1.00)-0.0363940],_
+[1.01,0.0359929,En(10,1.01),En(10,1.01)-0.0359929],_
+[1.02,0.0355963,En(10,1.02),En(10,1.02)-0.0355963],_
+[1.03,0.0352041,En(10,1.03),En(10,1.03)-0.0352041],_
+[1.04,0.0348163,En(10,1.04),En(10,1.04)-0.0348163],_
+[1.05,0.0344328,En(10,1.05),En(10,1.05)-0.0344328],_
+[1.06,0.0340535,En(10,1.06),En(10,1.06)-0.0340535],_
+[1.07,0.0336785,En(10,1.07),En(10,1.07)-0.0336785],_
+[1.08,0.0333077,En(10,1.08),En(10,1.08)-0.0333077],_
+[1.09,0.0329410,En(10,1.09),En(10,1.09)-0.0329410],_
+[1.10,0.0325784,En(10,1.10),En(10,1.10)-0.0325784],_
+[1.11,0.0322198,En(10,1.11),En(10,1.11)-0.0322198],_
+[1.12,0.0318652,En(10,1.12),En(10,1.12)-0.0318652],_
+[1.13,0.0315145,En(10,1.13),En(10,1.13)-0.0315145],_
+[1.14,0.0311678,En(10,1.14),En(10,1.14)-0.0311678],_
+[1.15,0.0308249,En(10,1.15),En(10,1.15)-0.0308249],_
+[1.16,0.0304858,En(10,1.16),En(10,1.16)-0.0304858],_
+[1.17,0.0301505,En(10,1.17),En(10,1.17)-0.0301505],_
+[1.18,0.0298189,En(10,1.18),En(10,1.18)-0.0298189],_
+[1.19,0.0294910,En(10,1.19),En(10,1.19)-0.0294910],_
+[1.20,0.0291668,En(10,1.20),En(10,1.20)-0.0291668],_
+[1.21,0.0288461,En(10,1.21),En(10,1.21)-0.0288461],_
+[1.22,0.0285290,En(10,1.22),En(10,1.22)-0.0285290],_
+[1.23,0.0282155,En(10,1.23),En(10,1.23)-0.0282155],_
+[1.24,0.0279054,En(10,1.24),En(10,1.24)-0.0279054],_
+[1.25,0.0275988,En(10,1.25),En(10,1.25)-0.0275988],_
+[1.26,0.0272955,En(10,1.26),En(10,1.26)-0.0272955],_
+[1.27,0.0269957,En(10,1.27),En(10,1.27)-0.0269957],_
+[1.28,0.0266991,En(10,1.28),En(10,1.28)-0.0266991],_
+[1.29,0.0264059,En(10,1.29),En(10,1.29)-0.0264059],_
+[1.30,0.0261159,En(10,1.30),En(10,1.30)-0.0261159],_
+[1.31,0.0258291,En(10,1.31),En(10,1.31)-0.0258291],_
+[1.32,0.0255455,En(10,1.32),En(10,1.32)-0.0255455],_
+[1.33,0.0252651,En(10,1.33),En(10,1.33)-0.0252651],_
+[1.34,0.0249878,En(10,1.34),En(10,1.34)-0.0249878],_
+[1.35,0.0247135,En(10,1.35),En(10,1.35)-0.0247135],_
+[1.36,0.0244423,En(10,1.36),En(10,1.36)-0.0244423],_
+[1.37,0.0241741,En(10,1.37),En(10,1.37)-0.0241741],_
+[1.38,0.0239088,En(10,1.38),En(10,1.38)-0.0239088],_
+[1.39,0.0236465,En(10,1.39),En(10,1.39)-0.0236465],_
+[1.40,0.0233872,En(10,1.40),En(10,1.40)-0.0233872],_
+[1.41,0.0231306,En(10,1.41),En(10,1.41)-0.0231306],_
+[1.42,0.0228770,En(10,1.42),En(10,1.42)-0.0228770],_
+[1.43,0.0226261,En(10,1.43),En(10,1.43)-0.0226261],_
+[1.44,0.0223780,En(10,1.44),En(10,1.44)-0.0223780],_
+[1.45,0.0221327,En(10,1.45),En(10,1.45)-0.0221327],_
+[1.46,0.0218901,En(10,1.46),En(10,1.46)-0.0218901],_
+[1.47,0.0216501,En(10,1.47),En(10,1.47)-0.0216501],_
+[1.48,0.0214128,En(10,1.48),En(10,1.48)-0.0214128],_
+[1.49,0.0211782,En(10,1.49),En(10,1.49)-0.0211782],_
+[1.50,0.0209461,En(10,1.50),En(10,1.50)-0.0209461],_
+[1.51,0.0207167,En(10,1.51),En(10,1.51)-0.0207167],_
+[1.52,0.0204897,En(10,1.52),En(10,1.52)-0.0204897],_
+[1.53,0.0202653,En(10,1.53),En(10,1.53)-0.0202653],_
+[1.54,0.0200433,En(10,1.54),En(10,1.54)-0.0200433],_
+[1.55,0.0198238,En(10,1.55),En(10,1.55)-0.0198238],_
+[1.56,0.0196067,En(10,1.56),En(10,1.56)-0.0196067],_
+[1.57,0.0193921,En(10,1.57),En(10,1.57)-0.0193921],_
+[1.58,0.0191798,En(10,1.58),En(10,1.58)-0.0191798],_
+[1.59,0.0189698,En(10,1.59),En(10,1.59)-0.0189698],_
+[1.60,0.0187622,En(10,1.60),En(10,1.60)-0.0187622],_
+[1.61,0.0185568,En(10,1.61),En(10,1.61)-0.0185568],_
+[1.62,0.0183538,En(10,1.62),En(10,1.62)-0.0183538],_
+[1.63,0.0181530,En(10,1.63),En(10,1.63)-0.0181530],_
+[1.64,0.0179543,En(10,1.64),En(10,1.64)-0.0179543],_
+[1.65,0.0177579,En(10,1.65),En(10,1.65)-0.0177579],_
+[1.66,0.0175637,En(10,1.66),En(10,1.66)-0.0175637],_
+[1.67,0.0173716,En(10,1.67),En(10,1.67)-0.0173716],_
+[1.68,0.0171816,En(10,1.68),En(10,1.68)-0.0171816],_
+[1.69,0.0169937,En(10,1.69),En(10,1.69)-0.0169937],_
+[1.70,0.0168079,En(10,1.70),En(10,1.70)-0.0168079],_
+[1.71,0.0166242,En(10,1.71),En(10,1.71)-0.0166242],_
+[1.72,0.0164424,En(10,1.72),En(10,1.72)-0.0164424],_
+[1.73,0.0162627,En(10,1.73),En(10,1.73)-0.0162627],_
+[1.74,0.0160850,En(10,1.74),En(10,1.74)-0.0160850],_
+[1.75,0.0159092,En(10,1.75),En(10,1.75)-0.0159092],_
+[1.76,0.0157354,En(10,1.76),En(10,1.76)-0.0157354],_
+[1.77,0.0155634,En(10,1.77),En(10,1.77)-0.0155634],_
+[1.78,0.0153934,En(10,1.78),En(10,1.78)-0.0153934],_
+[1.79,0.0152253,En(10,1.79),En(10,1.79)-0.0152253],_
+[1.80,0.0150590,En(10,1.80),En(10,1.80)-0.0150590],_
+[1.81,0.0148945,En(10,1.81),En(10,1.81)-0.0148945],_
+[1.82,0.0147318,En(10,1.82),En(10,1.82)-0.0147318],_
+[1.83,0.0145710,En(10,1.83),En(10,1.83)-0.0145710],_
+[1.84,0.0144119,En(10,1.84),En(10,1.84)-0.0144119],_
+[1.85,0.0142546,En(10,1.85),En(10,1.85)-0.0142546],_
+[1.86,0.0140990,En(10,1.86),En(10,1.86)-0.0140990],_
+[1.87,0.0139451,En(10,1.87),En(10,1.87)-0.0139451],_
+[1.88,0.0137929,En(10,1.88),En(10,1.88)-0.0137929],_
+[1.89,0.0136424,En(10,1.89),En(10,1.89)-0.0136424],_
+[1.90,0.0134935,En(10,1.90),En(10,1.90)-0.0134935],_
+[1.91,0.0133463,En(10,1.91),En(10,1.91)-0.0133463],_
+[1.92,0.0132007,En(10,1.92),En(10,1.92)-0.0132007],_
+[1.93,0.0130567,En(10,1.93),En(10,1.93)-0.0130567],_
+[1.94,0.0129143,En(10,1.94),En(10,1.94)-0.0129143],_
+[1.95,0.0127734,En(10,1.95),En(10,1.95)-0.0127734],_
+[1.96,0.0126341,En(10,1.96),En(10,1.96)-0.0126341],_
+[1.97,0.0124964,En(10,1.97),En(10,1.97)-0.0124964],_
+[1.98,0.0123601,En(10,1.98),En(10,1.98)-0.0123601],_
+[1.99,0.0122254,En(10,1.99),En(10,1.99)-0.0122254],_
+[2.00,0.0120921,En(10,2.00),En(10,2.00)-0.0120921]]
+--R 
+--R
+--R   (6)
+--R   [[1.0E-2,0.1098682,0.10986822627360165,2.6273601655413259E-8],
+--R    [2.0E-2,0.1086395,0.10863946164415648,- 3.8355843515192056E-8],
+--R    [2.9999999999999999E-2,0.1074246,0.10742465352510716,5.3525107165941499E-8],
+--R    [4.0000000000000001E-2,0.1062236,0.10622364016949924,4.016949924079416E-8],
+--R
+--R     [5.0000000000000003E-2, 0.1050363, 0.10503626175690921,
+--R      - 3.8243090791367784E-8]
+--R     ,
+--R
+--R     [5.9999999999999998E-2, 0.10386239999999999, 0.10386236036958028,
+--R      - 3.9630419709779652E-8]
+--R     ,
+--R    [7.0000000000000007E-2,0.1027018,0.1027017799688722,- 2.0031127798136872E-8]
+--R     ,
+--R    [8.0000000000000002E-2,0.1015544,0.1015543663720207,- 3.3627979303951783E-8]
+--R     ,
+--R    [8.9999999999999997E-2,0.10042,0.1004199672292018,- 3.2770798200076889E-8],
+--R
+--R     [0.10000000000000001, 9.9298399999999995E-2, 9.9298432000896802E-2,
+--R      3.2000896807438117E-8]
+--R     ,
+--R    [0.11,9.8189600000000002E-2,9.8189611935553478E-2,1.1935553476116745E-8],
+--R    [0.12,9.7093399999999996E-2,9.7093360047539198E-2,- 3.9952460798020617E-8],
+--R    [0.13,9.6009499999999998E-2,9.6009531095381809E-2,3.109538181111926E-8],
+--R
+--R     [0.14000000000000001, 9.4937999999999995E-2, 9.4937981560294218E-2,
+--R      - 1.8439705776196469E-8]
+--R     ,
+--R
+--R     [0.14999999999999999, 9.3878600000000006E-2, 9.3878569624978384E-2,
+--R      - 3.037502162295258E-8]
+--R     ,
+--R    [0.16,9.2831200000000003E-2,9.2831155152705111E-2,- 4.4847294891625644E-8],
+--R
+--R     [0.17000000000000001, 9.1795600000000005E-2, 9.1795599666665256E-2,
+--R      - 3.3333474869223778E-10]
+--R     ,
+--R
+--R     [0.17999999999999999, 9.07718E-2, 9.0771766329588957E-2,
+--R      - 3.3670411042630022E-8]
+--R     ,
+--R    [0.19,8.9759500000000006E-2,8.9759519923628739E-2,1.9923628732931853E-8],
+--R
+--R     [0.20000000000000001, 8.8758699999999996E-2, 8.8758726830502982E-2,
+--R      2.6830502986019411E-8]
+--R     ,
+--R
+--R     [0.20999999999999999, 8.7769299999999995E-2, 8.7769255011895919E-2,
+--R      - 4.4988104075383006E-8]
+--R     ,
+--R    [0.22,8.6790999999999993E-2,8.6790973990110751E-2,- 2.6009889242395445E-8],
+--R
+--R     [0.23000000000000001, 8.5823800000000006E-2, 8.5823754828972157E-2,
+--R      - 4.5171027848733836E-8]
+--R     ,
+--R
+--R     [0.23999999999999999, 8.4867499999999998E-2, 8.4867470114974794E-2,
+--R      - 2.9885025204512417E-8]
+--R     ,
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+--R     [1.6000000000000001, 1.87622E-2, 1.8762186249609149E-2,
+--R      - 1.3750390850247873E-8]
+--R     ,
+--R
+--R     [1.6100000000000001, 1.8556799999999998E-2, 1.8556848960566173E-2,
+--R      4.8960566174233167E-8]
+--R     ,
+--R    [1.6200000000000001,1.83538E-2,1.835378056160069E-2,- 1.9438399310317545E-8]
+--R     ,
+--R
+--R     [1.6299999999999999, 1.8152999999999999E-2, 1.8152955689888815E-2,
+--R      - 4.431011118438688E-8]
+--R     ,
+--R
+--R     [1.6399999999999999, 1.7954299999999999E-2, 1.7954349270203122E-2,
+--R      4.927020312225916E-8]
+--R     ,
+--R    [1.6499999999999999,1.77579E-2,1.77579365115926E-2,3.6511592600013687E-8],
+--R
+--R     [1.6599999999999999, 1.7563700000000002E-2, 1.7563692904101796E-2,
+--R      - 7.0958982058277886E-9]
+--R     ,
+--R
+--R     [1.6699999999999999, 1.7371600000000001E-2, 1.7371594215528752E-2,
+--R      - 5.7844712492149952E-9]
+--R     ,
+--R
+--R     [1.6799999999999999, 1.7181600000000002E-2, 1.718161648822112E-2,
+--R      1.6488221118299284E-8]
+--R     ,
+--R    [1.6899999999999999,1.69937E-2,1.6993736035910194E-2,3.6035910193354947E-8],
+--R    [1.7,1.6807900000000001E-2,1.6807929440582167E-2,2.9440582166584406E-8],
+--R    [1.71,1.6624199999999999E-2,1.6624173549386351E-2,- 2.6450613647283072E-8],
+--R    [1.72,1.6442399999999999E-2,1.6442445471579824E-2,4.5471579824402086E-8],
+--R    [1.73,1.6262700000000001E-2,1.6262722575508065E-2,2.2575508063643612E-8],
+--R    [1.74,1.6084999999999999E-2,1.6084982485621131E-2,- 1.7514378867350411E-8],
+--R    [1.75,1.5909199999999998E-2,1.590920307952504E-2,3.0795250412218866E-9],
+--R    [1.76,1.57354E-2,1.5735362485067742E-2,- 3.7514932257898259E-8],
+--R    [1.77,1.55634E-2,1.5563439077459485E-2,3.9077459485295507E-8],
+--R    [1.78,1.53934E-2,1.5393411476427005E-2,1.1476427004900036E-8],
+--R    [1.79,1.5225300000000001E-2,1.5225258543401145E-2,- 4.1456598855182936E-8],
+--R    [1.8,1.5058999999999999E-2,1.5058959378737585E-2,- 4.0621262414539117E-8],
+--R
+--R     [1.8100000000000001, 1.48945E-2, 1.4894493318970201E-2,
+--R      - 6.6810297988384448E-9]
+--R     ,
+--R    [1.8200000000000001,1.47318E-2,1.4731839934096682E-2,3.9934096682237019E-8],
+--R
+--R     [1.8300000000000001, 1.4571000000000001E-2, 1.4570979024896022E-2,
+--R      - 2.0975103978346232E-8]
+--R     ,
+--R
+--R     [1.8400000000000001, 1.44119E-2, 1.4411890620277503E-2,
+--R      - 9.3797224969688342E-9]
+--R     ,
+--R
+--R     [1.8500000000000001, 1.4254599999999999E-2, 1.4254554974660784E-2,
+--R      - 4.5025339215701288E-8]
+--R     ,
+--R    [1.8600000000000001,1.4099E-2,1.4098952565386716E-2,- 4.7434613284144667E-8]
+--R     ,
+--R    [1.8700000000000001,1.39451E-2,1.3945064090158516E-2,- 3.590984148406362E-8]
+--R     ,
+--R
+--R     [1.8799999999999999, 1.37929E-2, 1.3792870464512974E-2,
+--R      - 2.9535487026596807E-8]
+--R     ,
+--R
+--R     [1.8899999999999999, 1.3642400000000001E-2, 1.3642352819321199E-2,
+--R      - 4.7180678801328479E-8]
+--R     ,
+--R
+--R     [1.8999999999999999, 1.34935E-2, 1.3493492498318767E-2,
+--R      - 7.5016812327993732E-9]
+--R     ,
+--R
+--R     [1.9099999999999999, 1.33463E-2, 1.3346271055664704E-2,
+--R      - 2.8944335296599011E-8]
+--R     ,
+--R
+--R     [1.9199999999999999, 1.3200699999999999E-2, 1.3200670253529076E-2,
+--R      - 2.9746470923616708E-8]
+--R     ,
+--R
+--R     [1.9299999999999999, 1.3056699999999999E-2, 1.3056672059708809E-2,
+--R      - 2.794029119006225E-8]
+--R     ,
+--R
+--R     [1.9399999999999999, 1.29143E-2, 1.2914258645271409E-2,
+--R      - 4.1354728591222467E-8]
+--R     ,
+--R    [1.95,1.2773400000000001E-2,1.2773412382226235E-2,1.2382226233925708E-8],
+--R    [1.96,1.2634100000000001E-2,1.2634115841222995E-2,1.5841222994819604E-8],
+--R    [1.97,1.24964E-2,1.2496351789277165E-2,- 4.8210722834035602E-8],
+--R    [1.98,1.2360100000000001E-2,1.236010318752195E-2,3.1875219495131057E-9],
+--R    [1.99,1.2225400000000001E-2,1.222535318898654E-2,- 4.6811013461323103E-8],
+--R    [2.,1.20921E-2,1.2092085136400298E-2,- 1.4863599701736563E-8]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 6
+
+@
+This table expresses the value of 
+$E_20$ with an argument range of 0.01 to 2.00.
+<<*>>=
+--S 7 of 7
+[[0.01,0.0520790,En(20,0.01),En(20,0.01)-0.0520790],_
+[0.02,0.0515321,En(20,0.02),En(20,0.02)-0.0515321],_
+[0.03,0.0509911,En(20,0.03),En(20,0.03)-0.0509911],_
+[0.04,0.0504558,En(20,0.04),En(20,0.04)-0.0504558],_
+[0.05,0.0499260,En(20,0.05),En(20,0.05)-0.0499260],_
+[0.06,0.0494019,En(20,0.06),En(20,0.06)-0.0494019],_
+[0.07,0.0488833,En(20,0.07),En(20,0.07)-0.0488833],_
+[0.08,0.0483702,En(20,0.08),En(20,0.08)-0.0483702],_
+[0.09,0.0478624,En(20,0.09),En(20,0.09)-0.0478624],_
+[0.10,0.0473600,En(20,0.10),En(20,0.10)-0.0473600],_
+[0.11,0.0468629,En(20,0.11),En(20,0.11)-0.0468629],_
+[0.12,0.0463710,En(20,0.12),En(20,0.12)-0.0463710],_
+[0.13,0.0458843,En(20,0.13),En(20,0.13)-0.0458843],_
+[0.14,0.0454027,En(20,0.14),En(20,0.14)-0.0454027],_
+[0.15,0.0449262,En(20,0.15),En(20,0.15)-0.0449262],_
+[0.16,0.0444547,En(20,0.16),En(20,0.16)-0.0444547],_
+[0.17,0.0439882,En(20,0.17),En(20,0.17)-0.0439882],_
+[0.18,0.0435266,En(20,0.18),En(20,0.18)-0.0435266],_
+[0.19,0.0430698,En(20,0.19),En(20,0.19)-0.0430698],_
+[0.20,0.0426179,En(20,0.20),En(20,0.20)-0.0426179],_
+[0.21,0.0421707,En(20,0.21),En(20,0.21)-0.0421707],_
+[0.22,0.0417282,En(20,0.22),En(20,0.22)-0.0417282],_
+[0.23,0.0412903,En(20,0.23),En(20,0.23)-0.0412903],_
+[0.24,0.0408571,En(20,0.24),En(20,0.24)-0.0408571],_
+[0.25,0.0404285,En(20,0.25),En(20,0.25)-0.0404285],_
+[0.26,0.0400043,En(20,0.26),En(20,0.26)-0.0400043],_
+[0.27,0.0395846,En(20,0.27),En(20,0.27)-0.0395846],_
+[0.28,0.0391693,En(20,0.28),En(20,0.28)-0.0391693],_
+[0.29,0.0387584,En(20,0.29),En(20,0.29)-0.0387584],_
+[0.30,0.0383518,En(20,0.30),En(20,0.30)-0.0383518],_
+[0.31,0.0379495,En(20,0.31),En(20,0.31)-0.0379495],_
+[0.32,0.0375515,En(20,0.32),En(20,0.32)-0.0375515],_
+[0.33,0.0371576,En(20,0.33),En(20,0.33)-0.0371576],_
+[0.34,0.0367678,En(20,0.34),En(20,0.34)-0.0367678],_
+[0.35,0.0363822,En(20,0.35),En(20,0.35)-0.0363822],_
+[0.36,0.0360006,En(20,0.36),En(20,0.36)-0.0360006],_
+[0.37,0.0356231,En(20,0.37),En(20,0.37)-0.0356231],_
+[0.38,0.0352495,En(20,0.38),En(20,0.38)-0.0352495],_
+[0.39,0.0348798,En(20,0.39),En(20,0.39)-0.0348798],_
+[0.40,0.0345140,En(20,0.40),En(20,0.40)-0.0345140],_
+[0.41,0.0341521,En(20,0.41),En(20,0.41)-0.0341521],_
+[0.42,0.0337939,En(20,0.42),En(20,0.42)-0.0337939],_
+[0.43,0.0334396,En(20,0.43),En(20,0.43)-0.0334396],_
+[0.44,0.0330889,En(20,0.44),En(20,0.44)-0.0330889],_
+[0.45,0.0327420,En(20,0.45),En(20,0.45)-0.0327420],_
+[0.46,0.0323987,En(20,0.46),En(20,0.46)-0.0323987],_
+[0.47,0.0320590,En(20,0.47),En(20,0.47)-0.0320590],_
+[0.48,0.0317229,En(20,0.48),En(20,0.48)-0.0317229],_
+[0.49,0.0313903,En(20,0.49),En(20,0.49)-0.0313903],_
+[0.50,0.0310612,En(20,0.50),En(20,0.50)-0.0310612],_
+[0.51,0.0307356,En(20,0.51),En(20,0.51)-0.0307356],_
+[0.52,0.0304134,En(20,0.52),En(20,0.52)-0.0304134],_
+[0.53,0.0300946,En(20,0.53),En(20,0.53)-0.0300946],_
+[0.54,0.0297791,En(20,0.54),En(20,0.54)-0.0297791],_
+[0.55,0.0294670,En(20,0.55),En(20,0.55)-0.0294670],_
+[0.56,0.0291581,En(20,0.56),En(20,0.56)-0.0291581],_
+[0.57,0.0288525,En(20,0.57),En(20,0.57)-0.0288525],_
+[0.58,0.0285501,En(20,0.58),En(20,0.58)-0.0285501],_
+[0.59,0.0282508,En(20,0.59),En(20,0.59)-0.0282508],_
+[0.60,0.0279548,En(20,0.60),En(20,0.60)-0.0279548],_
+[0.61,0.0276618,En(20,0.61),En(20,0.61)-0.0276618],_
+[0.62,0.0273719,En(20,0.62),En(20,0.62)-0.0273719],_
+[0.63,0.0270850,En(20,0.63),En(20,0.63)-0.0270850],_
+[0.64,0.0268012,En(20,0.64),En(20,0.64)-0.0268012],_
+[0.65,0.0265204,En(20,0.65),En(20,0.65)-0.0265204],_
+[0.66,0.0262425,En(20,0.66),En(20,0.66)-0.0262425],_
+[0.67,0.0259675,En(20,0.67),En(20,0.67)-0.0259675],_
+[0.68,0.0256954,En(20,0.68),En(20,0.68)-0.0256954],_
+[0.69,0.0254262,En(20,0.69),En(20,0.69)-0.0254262],_
+[0.70,0.0251598,En(20,0.70),En(20,0.70)-0.0251598],_
+[0.71,0.0248962,En(20,0.71),En(20,0.71)-0.0248962],_
+[0.72,0.0246353,En(20,0.72),En(20,0.72)-0.0246353],_
+[0.73,0.0243772,En(20,0.73),En(20,0.73)-0.0243772],_
+[0.74,0.0241219,En(20,0.74),En(20,0.74)-0.0241219],_
+[0.75,0.0238692,En(20,0.75),En(20,0.75)-0.0238692],_
+[0.76,0.0236191,En(20,0.76),En(20,0.76)-0.0236191],_
+[0.77,0.0233717,En(20,0.77),En(20,0.77)-0.0233717],_
+[0.78,0.0231269,En(20,0.78),En(20,0.78)-0.0231269],_
+[0.79,0.0228846,En(20,0.79),En(20,0.79)-0.0228846],_
+[0.80,0.0226449,En(20,0.80),En(20,0.80)-0.0226449],_
+[0.81,0.0224078,En(20,0.81),En(20,0.81)-0.0224078],_
+[0.82,0.0221731,En(20,0.82),En(20,0.82)-0.0221731],_
+[0.83,0.0219408,En(20,0.83),En(20,0.83)-0.0219408],_
+[0.84,0.0217111,En(20,0.84),En(20,0.84)-0.0217111],_
+[0.85,0.0214837,En(20,0.85),En(20,0.85)-0.0214837],_
+[0.86,0.0212587,En(20,0.86),En(20,0.86)-0.0212587],_
+[0.87,0.0210361,En(20,0.87),En(20,0.87)-0.0210361],_
+[0.88,0.0208158,En(20,0.88),En(20,0.88)-0.0208158],_
+[0.89,0.0205978,En(20,0.89),En(20,0.89)-0.0205978],_
+[0.90,0.0203821,En(20,0.90),En(20,0.90)-0.0203821],_
+[0.91,0.0201687,En(20,0.91),En(20,0.91)-0.0201687],_
+[0.92,0.0199575,En(20,0.92),En(20,0.92)-0.0199575],_
+[0.93,0.0197486,En(20,0.93),En(20,0.93)-0.0197486],_
+[0.94,0.0195418,En(20,0.94),En(20,0.94)-0.0195418],_
+[0.95,0.0193372,En(20,0.95),En(20,0.95)-0.0193372],_
+[0.96,0.0191347,En(20,0.96),En(20,0.96)-0.0191347],_
+[0.97,0.0189344,En(20,0.97),En(20,0.97)-0.0189344],_
+[0.98,0.0187362,En(20,0.98),En(20,0.98)-0.0187362],_
+[0.99,0.0185401,En(20,0.99),En(20,0.99)-0.0185401],_
+[1.00,0.0183460,En(20,1.00),En(20,1.00)-0.0183460],_
+[1.01,0.0181539,En(20,1.01),En(20,1.01)-0.0181539],_
+[1.02,0.0179639,En(20,1.02),En(20,1.02)-0.0179639],_
+[1.03,0.0177759,En(20,1.03),En(20,1.03)-0.0177759],_
+[1.04,0.0175898,En(20,1.04),En(20,1.04)-0.0175898],_
+[1.05,0.0174057,En(20,1.05),En(20,1.05)-0.0174057],_
+[1.06,0.0172235,En(20,1.06),En(20,1.06)-0.0172235],_
+[1.07,0.0170433,En(20,1.07),En(20,1.07)-0.0170433],_
+[1.08,0.0168649,En(20,1.08),En(20,1.08)-0.0168649],_
+[1.09,0.0166884,En(20,1.09),En(20,1.09)-0.0166884],_
+[1.10,0.0165137,En(20,1.10),En(20,1.10)-0.0165137],_
+[1.11,0.0163409,En(20,1.11),En(20,1.11)-0.0163409],_
+[1.12,0.0161699,En(20,1.12),En(20,1.12)-0.0161699],_
+[1.13,0.0160007,En(20,1.13),En(20,1.13)-0.0160007],_
+[1.14,0.0158333,En(20,1.14),En(20,1.14)-0.0158333],_
+[1.15,0.0156676,En(20,1.15),En(20,1.15)-0.0156676],_
+[1.16,0.0155037,En(20,1.16),En(20,1.16)-0.0155037],_
+[1.17,0.0153414,En(20,1.17),En(20,1.17)-0.0153414],_
+[1.18,0.0151809,En(20,1.18),En(20,1.18)-0.0151809],_
+[1.19,0.0150221,En(20,1.19),En(20,1.19)-0.0150221],_
+[1.20,0.0148649,En(20,1.20),En(20,1.20)-0.0148649],_
+[1.21,0.0147094,En(20,1.21),En(20,1.21)-0.0147094],_
+[1.22,0.0145555,En(20,1.22),En(20,1.22)-0.0145555],_
+[1.23,0.0144032,En(20,1.23),En(20,1.23)-0.0144032],_
+[1.24,0.0142526,En(20,1.24),En(20,1.24)-0.0142526],_
+[1.25,0.0141035,En(20,1.25),En(20,1.25)-0.0141035],_
+[1.26,0.0139559,En(20,1.26),En(20,1.26)-0.0139559],_
+[1.27,0.0138100,En(20,1.27),En(20,1.27)-0.0138100],_
+[1.28,0.0136655,En(20,1.28),En(20,1.28)-0.0136655],_
+[1.29,0.0135226,En(20,1.29),En(20,1.29)-0.0135226],_
+[1.30,0.0133811,En(20,1.30),En(20,1.30)-0.0133811],_
+[1.31,0.0132412,En(20,1.31),En(20,1.31)-0.0132412],_
+[1.32,0.0131027,En(20,1.32),En(20,1.32)-0.0131027],_
+[1.33,0.0129657,En(20,1.33),En(20,1.33)-0.0129657],_
+[1.34,0.0128301,En(20,1.34),En(20,1.34)-0.0128301],_
+[1.35,0.0126959,En(20,1.35),En(20,1.35)-0.0126959],_
+[1.36,0.0125631,En(20,1.36),En(20,1.36)-0.0125631],_
+[1.37,0.0124317,En(20,1.37),En(20,1.37)-0.0124317],_
+[1.38,0.0123017,En(20,1.38),En(20,1.38)-0.0123017],_
+[1.39,0.0121731,En(20,1.39),En(20,1.39)-0.0121731],_
+[1.40,0.0120458,En(20,1.40),En(20,1.40)-0.0120458],_
+[1.41,0.0119198,En(20,1.41),En(20,1.41)-0.0119198],_
+[1.42,0.0117952,En(20,1.42),En(20,1.42)-0.0117952],_
+[1.43,0.0116719,En(20,1.43),En(20,1.43)-0.0116719],_
+[1.44,0.0115499,En(20,1.44),En(20,1.44)-0.0115499],_
+[1.45,0.0114291,En(20,1.45),En(20,1.45)-0.0114291],_
+[1.46,0.0113096,En(20,1.46),En(20,1.46)-0.0113096],_
+[1.47,0.0111914,En(20,1.47),En(20,1.47)-0.0111914],_
+[1.48,0.0110744,En(20,1.48),En(20,1.48)-0.0110744],_
+[1.49,0.0109586,En(20,1.49),En(20,1.49)-0.0109586],_
+[1.50,0.0108440,En(20,1.50),En(20,1.50)-0.0108440],_
+[1.51,0.0107307,En(20,1.51),En(20,1.51)-0.0107307],_
+[1.52,0.0106185,En(20,1.52),En(20,1.52)-0.0106185],_
+[1.53,0.0105075,En(20,1.53),En(20,1.53)-0.0105075],_
+[1.54,0.0103977,En(20,1.54),En(20,1.54)-0.0103977],_
+[1.55,0.0102890,En(20,1.55),En(20,1.55)-0.0102890],_
+[1.56,0.0101815,En(20,1.56),En(20,1.56)-0.0101815],_
+[1.57,0.0100750,En(20,1.57),En(20,1.57)-0.0100750],_
+[1.58,0.0099697,En(20,1.58),En(20,1.58)-0.0099697],_
+[1.59,0.0098656,En(20,1.59),En(20,1.59)-0.0098656],_
+[1.60,0.0097624,En(20,1.60),En(20,1.60)-0.0097624],_
+[1.61,0.0096604,En(20,1.61),En(20,1.61)-0.0096604],_
+[1.62,0.0095595,En(20,1.62),En(20,1.62)-0.0095595],_
+[1.63,0.0094596,En(20,1.63),En(20,1.63)-0.0094596],_
+[1.64,0.0093607,En(20,1.64),En(20,1.64)-0.0093607],_
+[1.65,0.0092629,En(20,1.65),En(20,1.65)-0.0092629],_
+[1.66,0.0091661,En(20,1.66),En(20,1.66)-0.0091661],_
+[1.67,0.0090703,En(20,1.67),En(20,1.67)-0.0090703],_
+[1.68,0.0089756,En(20,1.68),En(20,1.68)-0.0089756],_
+[1.69,0.0088818,En(20,1.69),En(20,1.69)-0.0088818],_
+[1.70,0.0087890,En(20,1.70),En(20,1.70)-0.0087890],_
+[1.71,0.0086972,En(20,1.71),En(20,1.71)-0.0086972],_
+[1.72,0.0086063,En(20,1.72),En(20,1.72)-0.0086063],_
+[1.73,0.0085164,En(20,1.73),En(20,1.73)-0.0085164],_
+[1.74,0.0084274,En(20,1.74),En(20,1.74)-0.0084274],_
+[1.75,0.0083394,En(20,1.75),En(20,1.75)-0.0083394],_
+[1.76,0.0082522,En(20,1.76),En(20,1.76)-0.0082522],_
+[1.77,0.0081660,En(20,1.77),En(20,1.77)-0.0081660],_
+[1.78,0.0080807,En(20,1.78),En(20,1.78)-0.0080807],_
+[1.79,0.0079963,En(20,1.79),En(20,1.79)-0.0079963],_
+[1.80,0.0079128,En(20,1.80),En(20,1.80)-0.0079128],_
+[1.81,0.0078302,En(20,1.81),En(20,1.81)-0.0078302],_
+[1.82,0.0077484,En(20,1.82),En(20,1.82)-0.0077484],_
+[1.83,0.0076674,En(20,1.83),En(20,1.83)-0.0076674],_
+[1.84,0.0075874,En(20,1.84),En(20,1.84)-0.0075874],_
+[1.85,0.0075081,En(20,1.85),En(20,1.85)-0.0075081],_
+[1.86,0.0074297,En(20,1.86),En(20,1.86)-0.0074297],_
+[1.87,0.0073521,En(20,1.87),En(20,1.87)-0.0073521],_
+[1.88,0.0072753,En(20,1.88),En(20,1.88)-0.0072753],_
+[1.89,0.0071993,En(20,1.89),En(20,1.89)-0.0071993],_
+[1.90,0.0071242,En(20,1.90),En(20,1.90)-0.0071242],_
+[1.91,0.0070498,En(20,1.91),En(20,1.91)-0.0070498],_
+[1.92,0.0069762,En(20,1.92),En(20,1.92)-0.0069762],_
+[1.93,0.0069033,En(20,1.93),En(20,1.93)-0.0069033],_
+[1.94,0.0068312,En(20,1.94),En(20,1.94)-0.0068312],_
+[1.95,0.0067599,En(20,1.95),En(20,1.95)-0.0067599],_
+[1.96,0.0066893,En(20,1.96),En(20,1.96)-0.0066893],_
+[1.97,0.0066195,En(20,1.97),En(20,1.97)-0.0066195],_
+[1.98,0.0065504,En(20,1.98),En(20,1.98)-0.0065504],_
+[1.99,0.0064820,En(20,1.99),En(20,1.99)-0.0064820],_
+[2.00,0.0064143,En(20,2.00),En(20,2.00)-0.0064143]]
+--R 
+--R
+--R   (7)
+--R   [[1.0E-2,5.2079E-2,5.2078954179335148E-2,- 4.5820664852647131E-8],
+--R    [2.0E-2,5.1532099999999997E-2,5.1532149651352818E-2,4.9651352820867523E-8],
+--R
+--R     [2.9999999999999999E-2, 5.0991099999999998E-2, 5.0991103854550281E-2,
+--R      3.8545502831222045E-9]
+--R     ,
+--R
+--R     [4.0000000000000001E-2, 5.0455800000000002E-2, 5.0455755932602576E-2,
+--R      - 4.4067397425573418E-8]
+--R     ,
+--R
+--R     [5.0000000000000003E-2, 4.9925999999999998E-2, 4.9926045674777729E-2,
+--R      4.5674777730819738E-8]
+--R     ,
+--R
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+--R      1.3509057830707327E-8]
+--R     ,
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+--R      4.9533392665335185E-10]
+--R     ,
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+--R      - 5.1681326301844521E-8]
+--R     ,
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+--R      - 7.1733870926626864E-10]
+--R     ,
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+--R      - 3.6971917125039333E-9]
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+--R    [0.13,4.5884300000000003E-2,4.5884301897836918E-2,1.8978369153987984E-9],
+--R
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+--R      2.430960564792084E-8]
+--R     ,
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+--R      1.6308288566801998E-8]
+--R     ,
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+--R
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+--R      - 4.5554353483856502E-9]
+--R     ,
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+--R      - 2.2815457884073354E-8]
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+--R     [0.20000000000000001, 4.26179E-2, 4.2617865691013848E-2,
+--R      - 3.4308986152087328E-8]
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+--R      - 2.962345675011635E-8]
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+--R
+--R     [0.23000000000000001, 4.1290300000000002E-2, 4.1290349454660515E-2,
+--R      4.9454660512593396E-8]
+--R     ,
+--R    [0.23999999999999999,4.08571E-2,4.0857124978128601E-2,2.4978128600194882E-8]
+--R     ,
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+--R    [0.26000000000000001,4.00043E-2,4.0004304738165339E-2,4.7381653392464251E-9]
+--R     ,
+--R
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+--R      1.3215981260750187E-8]
+--R     ,
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+--R     [0.28000000000000003, 3.9169299999999997E-2, 3.916933776458735E-2,
+--R      3.7764587353106283E-8]
+--R     ,
+--R
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+--R      3.1763662325379194E-8]
+--R     ,
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+--R      4.908688519544846E-8]
+--R     ,
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+--R      - 2.8360991423392878E-8]
+--R     ,
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+--R      - 1.2961760643970255E-8]
+--R     ,
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+--R      4.6092043924639281E-8]
+--R     ,
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+--R     ,
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+--R
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+--R      - 1.8400129926654518E-8]
+--R     ,
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+--R     ,
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+--R      - 3.410559876876329E-8]
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+--R     ,
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+--R     [0.42999999999999999, 3.34396E-2, 3.3439576546725337E-2,
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+--R     [0.45000000000000001, 3.2742E-2, 3.2741993946556171E-2,
+--R      - 6.0534438295345439E-9]
+--R     ,
+--R
+--R     [0.46000000000000002, 3.2398700000000002E-2, 3.2398694643532566E-2,
+--R      - 5.3564674362682752E-9]
+--R     ,
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+--R      5.216573520372414E-9]
+--R     ,
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+--R      - 1.2415324886216883E-8]
+--R     ,
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+--R     [0.48999999999999999, 3.1390300000000003E-2, 3.1390304070007917E-2,
+--R      4.070007914025453E-9]
+--R     ,
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+--R     [0.51000000000000001, 3.0735599999999998E-2, 3.0735590671251607E-2,
+--R      - 9.3287483918236269E-9]
+--R     ,
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+--R     [0.52000000000000002, 3.04134E-2, 3.0413387409030435E-2,
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+--R     ,
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+--R     [0.53000000000000003, 3.0094599999999999E-2, 3.009457149943125E-2,
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+--R     ,
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+--R     [0.54000000000000004, 2.9779099999999999E-2, 2.9779107217115259E-2,
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+--R     ,
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+--R     [0.55000000000000004, 2.9467E-2, 2.9466959214879015E-2,
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+--R     ,
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+--R     [0.56000000000000005, 2.9158099999999999E-2, 2.9158092519635862E-2,
+--R      - 7.4803641371412599E-9]
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+--R     [0.56999999999999995, 2.88525E-2, 2.8852472528440212E-2,
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+--R     ,
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+--R      - 4.7780506436806425E-8]
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+--R     [0.60999999999999999, 2.76618E-2, 2.7661780281068072E-2,
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+--R
+--R     [0.64000000000000001, 2.6801200000000001E-2, 2.6801209587681844E-2,
+--R      9.5876818428930477E-9]
+--R     ,
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+--R     [0.65000000000000002, 2.65204E-2, 2.6520360660970379E-2,
+--R      - 3.9339029620638621E-8]
+--R     ,
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+--R     [0.66000000000000003, 2.6242499999999998E-2, 2.6242463032806879E-2,
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+--R     ,
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+--R     [0.67000000000000004, 2.5967500000000001E-2, 2.5967485592108871E-2,
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+--R     ,
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+--R     [0.68000000000000005, 2.56954E-2, 2.5695397556905687E-2,
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+--R     ,
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+--R     [0.68999999999999995, 2.5426199999999999E-2, 2.5426168470843124E-2,
+--R      - 3.1529156875598208E-8]
+--R     ,
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+--R     [0.69999999999999996, 2.51598E-2, 2.5159768199725301E-2,
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+--R     ,
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+--R     [0.70999999999999996, 2.48962E-2, 2.489616692809353E-2,
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+--R     ,
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+--R     [0.71999999999999997, 2.4635299999999999E-2, 2.4635335155841546E-2,
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+--R     ,
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+--R     [0.72999999999999998, 2.4377200000000002E-2, 2.4377243694866949E-2,
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+--R     ,
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+--R     [0.73999999999999999, 2.4121900000000002E-2, 2.4121863665758296E-2,
+--R      - 3.6334241705626713E-8]
+--R     ,
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+--R
+--R     [0.76000000000000001, 2.3619100000000001E-2, 2.3619123909317606E-2,
+--R      2.3909317605624869E-8]
+--R     ,
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+--R     [0.77000000000000002, 2.3371699999999999E-2, 2.3371707937294008E-2,
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+--R     ,
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+--R     [0.78000000000000003, 2.3126899999999999E-2, 2.3126890901371419E-2,
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+--R     ,
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+--R     [0.79000000000000004, 2.2884600000000001E-2, 2.2884645417123543E-2,
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+--R     ,
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+--R     [0.80000000000000004, 2.2644899999999999E-2, 2.2644944389666643E-2,
+--R      4.4389666643829262E-8]
+--R     ,
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+--R     [0.81000000000000005, 2.2407799999999999E-2, 2.2407761010586214E-2,
+--R      - 3.8989413784357163E-8]
+--R     ,
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+--R     [0.81999999999999995, 2.2173100000000001E-2, 2.2173068754896382E-2,
+--R      - 3.124510361898758E-8]
+--R     ,
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+--R     ,
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+--R     [0.83999999999999997, 2.1711100000000001E-2, 2.1711052912871148E-2,
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+--R     ,
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+--R     [0.84999999999999998, 2.1483700000000001E-2, 2.1483677666793578E-2,
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+--R     ,
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+--R     [0.85999999999999999, 2.1258699999999998E-2, 2.1258690218764878E-2,
+--R      - 9.7812351204651371E-9]
+--R     ,
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+--R    [0.88,2.0815799999999999E-2,2.0815778373391495E-2,- 2.162660850402176E-8],
+--R
+--R     [0.89000000000000001, 2.0597799999999999E-2, 2.0597804466129153E-2,
+--R      4.4661291540015746E-9]
+--R     ,
+--R    [0.90000000000000002,2.03821E-2,2.0382119331468756E-2,1.9331468755628967E-8]
+--R     ,
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+--R     [0.91000000000000003, 2.0168700000000001E-2, 2.0168698863691235E-2,
+--R      - 1.136308765831151E-9]
+--R     ,
+--R    [0.92000000000000004,1.99575E-2,1.9957519211827056E-2,1.9211827056303532E-8]
+--R     ,
+--R
+--R     [0.93000000000000005, 1.9748600000000002E-2, 1.9748556776953658E-2,
+--R      - 4.3223046343465743E-8]
+--R     ,
+--R
+--R     [0.93999999999999995, 1.9541800000000002E-2, 1.9541788209521748E-2,
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+--R     ,
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+--R     [0.94999999999999996, 1.9337199999999999E-2, 1.9337190406710072E-2,
+--R      - 9.5932899263351246E-9]
+--R     ,
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+--R     [0.95999999999999996, 1.9134700000000001E-2, 1.9134740509808349E-2,
+--R      4.0509808348077048E-8]
+--R     ,
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+--R     [0.96999999999999997, 1.8934400000000001E-2, 1.8934415901628112E-2,
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+--R     ,
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+--R     [0.97999999999999998, 1.8736200000000001E-2, 1.8736194203941098E-2,
+--R      - 5.7960589036620824E-9]
+--R     ,
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+--R     [0.98999999999999999, 1.85401E-2, 1.8540053274944939E-2,
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+--R     ,
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+--R    [1.03,1.7775900000000001E-2,1.7775862545064098E-2,- 3.745493590279203E-8],
+--R    [1.04,1.7589799999999999E-2,1.7589801433368843E-2,1.433368843756222E-9],
+--R    [1.05,1.74057E-2,1.7405693065937998E-2,- 6.9340620016677956E-9],
+--R
+--R     [1.0600000000000001, 1.7223499999999999E-2, 1.7223516887221315E-2,
+--R      1.6887221315764478E-8]
+--R     ,
+--R
+--R     [1.0700000000000001, 1.7043300000000001E-2, 1.7043252558767878E-2,
+--R      - 4.744123212330198E-8]
+--R     ,
+--R
+--R     [1.0800000000000001, 1.6864899999999999E-2, 1.6864879956924574E-2,
+--R      - 2.0043075425052459E-8]
+--R     ,
+--R
+--R     [1.0900000000000001, 1.6688399999999999E-2, 1.6688379170559084E-2,
+--R      - 2.082944091513772E-8]
+--R     ,
+--R
+--R     [1.1000000000000001, 1.6513699999999999E-2, 1.6513730498807128E-2,
+--R      3.0498807129103467E-8]
+--R     ,
+--R
+--R     [1.1100000000000001, 1.6340899999999998E-2, 1.6340914448843669E-2,
+--R      1.4448843670133993E-8]
+--R     ,
+--R
+--R     [1.1200000000000001, 1.6169900000000001E-2, 1.6169911733677846E-2,
+--R      1.1733677845321422E-8]
+--R     ,
+--R    [1.1299999999999999,1.60007E-2,1.6000703269971401E-2,3.269971401292926E-9],
+--R
+--R     [1.1399999999999999, 1.5833300000000002E-2, 1.5833270175880269E-2,
+--R      - 2.9824119732874133E-8]
+--R     ,
+--R    [1.1499999999999999,1.56676E-2,1.566759376891921E-2,- 6.2310807902310472E-9]
+--R     ,
+--R
+--R     [1.1599999999999999, 1.55037E-2, 1.5503655563849129E-2,
+--R      - 4.4436150871754454E-8]
+--R     ,
+--R    [1.1699999999999999,1.53414E-2,1.5341437270586872E-2,3.7270586871959721E-8],
+--R
+--R     [1.1799999999999999, 1.5180900000000001E-2, 1.5180920792137275E-2,
+--R      2.079213727405882E-8]
+--R     ,
+--R
+--R     [1.1899999999999999, 1.50221E-2, 1.5022088222547223E-2,
+--R      - 1.1777452776917663E-8]
+--R     ,
+--R    [1.2,1.48649E-2,1.4864921844881475E-2,2.1844881474900046E-8],
+--R    [1.21,1.4709399999999999E-2,1.4709404129220015E-2,4.129220016260704E-9],
+--R    [1.22,1.4555500000000001E-2,1.4555517730676726E-2,1.7730676725447037E-8],
+--R    [1.23,1.44032E-2,1.4403245487439129E-2,4.5487439129590634E-8],
+--R    [1.24,1.4252600000000001E-2,1.4252570418828979E-2,- 2.9581171021378361E-8],
+--R    [1.25,1.41035E-2,1.4103475723383493E-2,- 2.427661650698798E-8],
+--R    [1.26,1.39559E-2,1.395594477695699E-2,4.4776956989456274E-8],
+--R    [1.27,1.3809999999999999E-2,1.3809961130842716E-2,- 3.8869157283419331E-8],
+--R    [1.28,1.3665500000000001E-2,1.3665508509914652E-2,8.5099146514278079E-9],
+--R    [1.29,1.3522599999999999E-2,1.3522570810789076E-2,- 2.9189210923319386E-8],
+--R    [1.3,1.33811E-2,1.338113210000569E-2,3.2100005689997269E-8],
+--R
+--R     [1.3100000000000001, 1.32412E-2, 1.3241176612228069E-2,
+--R      - 2.3387771930921675E-8]
+--R     ,
+--R    [1.3200000000000001,1.31027E-2,1.3102688748463267E-2,- 1.125153673356416E-8]
+--R     ,
+--R
+--R     [1.3300000000000001, 1.29657E-2, 1.2965653074300341E-2,
+--R      - 4.6925699659361442E-8]
+--R     ,
+--R
+--R     [1.3400000000000001, 1.2830100000000001E-2, 1.2830054318167606E-2,
+--R      - 4.5681832394964261E-8]
+--R     ,
+--R    [1.3500000000000001,1.26959E-2,1.2695877369608423E-2,- 2.263039157630875E-8]
+--R     ,
+--R
+--R     [1.3600000000000001, 1.2563100000000001E-2, 1.2563107277575319E-2,
+--R      7.2775753184178837E-9]
+--R     ,
+--R    [1.3700000000000001,1.24317E-2,1.2431729248742227E-2,2.9248742226817281E-8],
+--R    [1.3799999999999999,1.23017E-2,1.2301728645834703E-2,2.8645834702564765E-8],
+--R
+--R     [1.3899999999999999, 1.2173099999999999E-2, 1.2173090985977835E-2,
+--R      - 9.0140221641055174E-9]
+--R     ,
+--R
+--R     [1.3999999999999999, 1.2045800000000001E-2, 1.2045801939061785E-2,
+--R      1.9390617844033953E-9]
+--R     ,
+--R
+--R     [1.4099999999999999, 1.1919799999999999E-2, 1.1919847326124635E-2,
+--R      4.7326124635524436E-8]
+--R     ,
+--R    [1.4199999999999999,1.17952E-2,1.1795213117752473E-2,1.3117752472799338E-8],
+--R
+--R     [1.4299999999999999, 1.1671900000000001E-2, 1.1671885432496448E-2,
+--R      - 1.4567503552401218E-8]
+--R     ,
+--R
+--R     [1.4399999999999999, 1.15499E-2, 1.1549850535306669E-2,
+--R      - 4.9464693330919518E-8]
+--R     ,
+--R    [1.45,1.1429099999999999E-2,1.1429094835982728E-2,- 5.1640172710892829E-9],
+--R    [1.46,1.13096E-2,1.1309604887640717E-2,4.8876407175019176E-9],
+--R    [1.47,1.1191400000000001E-2,1.1191367385196493E-2,- 3.2614803507219348E-8],
+--R    [1.48,1.10744E-2,1.1074369163865097E-2,- 3.0836134902456624E-8],
+--R    [1.49,1.0958600000000001E-2,1.0958597197676098E-2,- 2.8023239024427271E-9],
+--R    [1.5,1.0843999999999999E-2,1.0844038598004712E-2,3.8598004712536715E-8],
+--R    [1.51,1.0730699999999999E-2,1.0730680612118525E-2,- 1.9387881473875712E-8],
+--R    [1.52,1.0618499999999999E-2,1.0618510621739656E-2,1.062173965622748E-8],
+--R    [1.53,1.0507499999999999E-2,1.05075161416222E-2,1.6141622200330219E-8],
+--R    [1.54,1.0397699999999999E-2,1.0397684818144773E-2,- 1.5181855226101271E-8],
+--R    [1.55,1.0289E-2,1.0289004427918007E-2,4.4279180079487279E-9],
+--R
+--R     [1.5600000000000001, 1.01815E-2, 1.0181462876406836E-2,
+--R      - 3.7123593163088109E-8]
+--R     ,
+--R
+--R     [1.5700000000000001, 1.0075000000000001E-2, 1.0075048196567427E-2,
+--R      4.8196567426286863E-8]
+--R     ,
+--R
+--R     [1.5800000000000001, 9.9696999999999997E-3, 9.9697485474985574E-3,
+--R      4.8547498557663382E-8]
+--R     ,
+--R
+--R     [1.5900000000000001, 9.8656000000000004E-3, 9.8655522131073474E-3,
+--R      - 4.7786892653015056E-8]
+--R     ,
+--R
+--R     [1.6000000000000001, 9.7623999999999992E-3, 9.7624476007891394E-3,
+--R      4.7600789140206379E-8]
+--R     ,
+--R
+--R     [1.6100000000000001, 9.6603999999999995E-3, 9.660423240121405E-3,
+--R      2.3240121405435077E-8]
+--R     ,
+--R
+--R     [1.6200000000000001, 9.5595000000000003E-3, 9.5594677815715092E-3,
+--R      - 3.2218428491076478E-8]
+--R     ,
+--R
+--R     [1.6299999999999999, 9.4596000000000003E-3, 9.4595699952182213E-3,
+--R      - 3.0004781778961154E-8]
+--R     ,
+--R
+--R     [1.6399999999999999, 9.3606999999999996E-3, 9.3607187694867526E-3,
+--R      1.8769486753028586E-8]
+--R     ,
+--R
+--R     [1.6499999999999999, 9.2628999999999993E-3, 9.2629031098972782E-3,
+--R      3.1098972789328494E-9]
+--R     ,
+--R
+--R     [1.6599999999999999, 9.1660999999999999E-3, 9.1661121378267089E-3,
+--R      1.2137826709002209E-8]
+--R     ,
+--R
+--R     [1.6699999999999999, 9.0702999999999999E-3, 9.0703350892836175E-3,
+--R      3.5089283617617539E-8]
+--R     ,
+--R
+--R     [1.6799999999999999, 8.9756000000000002E-3, 8.9755613136961645E-3,
+--R      - 3.8686303835758218E-8]
+--R     ,
+--R
+--R     [1.6899999999999999, 8.8818000000000005E-3, 8.8817802727129049E-3,
+--R      - 1.9727287095538948E-8]
+--R     ,
+--R    [1.7,8.7889999999999999E-3,8.7889815390163051E-3,- 1.8460983694859601E-8],
+--R    [1.71,8.6972000000000004E-3,8.6971547951488551E-3,- 4.5204851145327907E-8],
+--R    [1.72,8.6063000000000008E-3,8.6062898323516594E-3,- 1.0167648341330437E-8],
+--R    [1.73,8.5164000000000004E-3,8.5163765494153266E-3,- 2.345058467377592E-8],
+--R    [1.74,8.4273999999999998E-3,8.4274049515430643E-3,4.9515430644575531E-9],
+--R    [1.75,8.3394000000000003E-3,8.3393651492258501E-3,- 3.4850774150232966E-8],
+--R    [1.76,8.2521999999999995E-3,8.2522473571295107E-3,4.7357129511274576E-8],
+--R    [1.77,8.1659999999999996E-3,8.1660418929936432E-3,4.1892993643544152E-8],
+--R    [1.78,8.0806999999999997E-3,8.0807391765421923E-3,3.9176542192612129E-8],
+--R    [1.79,7.9962999999999996E-3,7.9963297284055979E-3,2.9728405598339336E-8],
+--R    [1.8,7.9127999999999993E-3,7.9128041690543771E-3,4.1690543777644917E-9],
+--R
+--R     [1.8100000000000001, 7.8302000000000007E-3, 7.8301532177440097E-3,
+--R      - 4.6782255990959754E-8]
+--R     ,
+--R
+--R     [1.8200000000000001, 7.7483999999999999E-3, 7.7483676914710427E-3,
+--R      - 3.2308528957226967E-8]
+--R     ,
+--R
+--R     [1.8300000000000001, 7.6674000000000004E-3, 7.6674385039402267E-3,
+--R      3.8503940226301825E-8]
+--R     ,
+--R
+--R     [1.8400000000000001, 7.5874000000000002E-3, 7.5873566645426526E-3,
+--R      - 4.3335457347581929E-8]
+--R     ,
+--R
+--R     [1.8500000000000001, 7.5081000000000002E-3, 7.50811327734469E-3,
+--R      1.327734468984515E-8]
+--R     ,
+--R    [1.8600000000000001,7.4297E-3,7.429699540087673E-3,- 4.5991232697217832E-10]
+--R     ,
+--R
+--R     [1.8700000000000001, 7.3521000000000003E-3, 7.3521067431981824E-3,
+--R      6.743198182126986E-9]
+--R     ,
+--R
+--R     [1.8799999999999999, 7.2753000000000002E-3, 7.2753262688088375E-3,
+--R      2.6268808837361102E-8]
+--R     ,
+--R
+--R     [1.8899999999999999, 7.1992999999999996E-3, 7.1993495897894446E-3,
+--R      4.9589789444942634E-8]
+--R     ,
+--R
+--R     [1.8999999999999999, 7.1241999999999998E-3, 7.1241682687884587E-3,
+--R      - 3.1731211541131954E-8]
+--R     ,
+--R
+--R     [1.9099999999999999, 7.0498000000000002E-3, 7.049773957284575E-3,
+--R      - 2.6042715425139695E-8]
+--R     ,
+--R
+--R     [1.9199999999999999, 6.9762000000000001E-3, 6.9761583946483935E-3,
+--R      - 4.1605351606618934E-8]
+--R     ,
+--R
+--R     [1.9299999999999999, 6.9033000000000002E-3, 6.9033134072140311E-3,
+--R      1.3407214030820847E-8]
+--R     ,
+--R
+--R     [1.9399999999999999, 6.8311999999999999E-3, 6.8312309073605745E-3,
+--R      3.0907360574518317E-8]
+--R     ,
+--R    [1.95,6.7599000000000001E-3,6.759902892603269E-3,2.8926032688905701E-9],
+--R    [1.96,6.6892999999999996E-3,6.6893214446943419E-3,2.1444694342336035E-8],
+--R    [1.97,6.6195000000000004E-3,6.6194787287333721E-3,- 2.1271266628306029E-8],
+--R    [1.98,6.5503999999999996E-3,6.5503669922870687E-3,- 3.3007712930965827E-8],
+--R    [1.99,6.4819999999999999E-3,6.4819785645183897E-3,- 2.1435481610196372E-8],
+--R    [2.,6.4143000000000004E-3,6.4143058553248998E-3,5.8553248994189233E-9]]
+--R                               Type: List List OnePointCompletion DoubleFloat
+--E 7
+)spool 
+)lisp (bye)
+ 
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} Abramowitz and Stegun,``Handbook of Mathematical Functions'',
+Dover Publications, Inc. New York 1965. pp245-248
+\end{thebibliography}
+\end{document}
+



From MAILER-DAEMON Fri Jan 25 13:39:08 2008
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	(ondrej@certik.cz)
Subject: Re: [Axiom-developer] Nice Fixes! Another go on IssueTracker
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Ondrej,

>I think this should be written on the front page of the axiom project,

It is all over the project. What would be the point of literate
documentation, indeed, of using noweb at all, if you didn't plan
to do literate programming? You'll find very few files in Axiom
that are not literate documents at least in form.

If you look at the actual progression of the project you'll
see that it is slowly re-organizing into books. The tutorial
book (vol 1) has been published, the interpreter (vol 5) is being
worked on, the firefox hyperdoc (vol 11) is checked in, and other
books exist.

The ultimate planned form will be similar to Knuth's Tex book or
the Lisp in Small Pieces book rather than a traditional "bag of
files" structure.

This has been debated endlessly on the mailing list so it cannot
come as a surprise.


Tim



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Subject: Re: [Axiom-developer] Nice Fixes! Another go on IssueTracker
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Ondrej,

>because my personal opinion is, that many people want to know the
>strategy behind the project. Also, I think, but maybe I am wrong, that
>most people prefer the "fix it now, instead of tomorrow" strategy. 

Yes, they do prefer that strategy. But they don't support it.

How many diff-Naur patches do you see for the various "fix proposed"
quick fixes? Saying that "it is fixed in FriCAS" is trivial. You might
as well say it works in Mathematica. It takes a while to decide what
the problem is, what the correct fix is, what the regression test is,
whether the newly fixed code builds correctly, and then to create a
proper diff-Naur patch that includes all of that work.

Creating a diff-Naur patch is hard work. Complaining is trivial.
So if your personal opinion is that these things ought to be fixed
please post diff-Naur patches to the Axiom mailing list.




You have the incorrect impression that fixes are not being done
in Axiom. In fact, since the November release there are 15 bug fixes,
roughly one every 4 days. Some, but not all, of these bugs are the
same as the "published" ones. If you check the posted diff-Naur
patches on the mailing list you can see the fixes.


Tim




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Ondrej,




>So one should then ask - do I want to create a project that most of the
>people would agree with the course of development (strategy of fixing
>bugs)? Or do I want to create a project that most people disagree with
>the strategy, but maybe in 30 years there is some chance that axiom
>will become very popular. I, personally, prefer the former, because I
>(but that's just my opinion) prefer my project to be joined by a lot
>of people now, not in 30 years.

I have had many projects disappear. Some were commercial products,
some were corporate internal projects, and two were open source
projects (Pinger and RentRoll). I have seen how and why projects die.

How do you keep a project alive after every original developer dies?

I spent a considerable amount of time pondering this question while
I was building Axiom from the code NAG gave me. The answer to the
question, in my opinion, rests with literate programming. Computer
algebra systems are based on research work. In order to understand,
maintain, and modify such systems you need to merge the research
work with the source code. And you need to explain the details of
the algorithm to someone you will never meet.

Unlike other programs a computer algebra system will get the same
answers 30 years from now. It is worthwhile investing in the literate
documentation to ensure that the code is fully and deeply explained.
"Literate", in this context (see Knuth), means that you can sit down
and read the program like a book. You can understand what people
thought before you decide to change things.

The problem is that programmers don't know how, and do not want, to do
literate programs. They like hack-and-slash, little-C-file, 1970s
technology include-file thinking. Hack-make-publish is ok if you want
to build toys for today. Why use noweb at all? It is just overhead for
any project that is not trying to be literate and should be removed.

But if you want a quality piece of polished software backed by
research publications and literate documentation you need to
fundamentally change your mindset.

When Axiom is fully literate people can teach from the source code.
Researchers can read an algorithm and understand all the little
practical issues. Programmers can understand the research that
supports the algorithms. New research can be based on well-documented
sources and easily added to the system. The goal is to have a literate
Axiom be the basis for Computational Mathematics teaching and research.

Tim



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Ondrej,

>And with this perspective, I think (again just my opinion) that even
>if you want to optimize the project over 30 years, the strategy is
>wrong too, because it's imho essential that people join the
>development. I think the best strategy is to optimize for the needs
>and opinions of potential developers in 2008, not 2038.

The needs and opinions of potential developers are based in the last
30 years of code development. C programs live in tiny files because
I only had 4K of memory on my PDP 11/40. Include files exist so I
can tell the compiler what it can't figure out because it won't all
fit in memory. Makefiles exist because these tiny sand-grain files
exist. Libraries exist and have to be included (in order) because my
paper tape reader needed to make one pass in order to copy all of
the required libraries directly to the output punch. /* comments
exist because I cannot write real documentation in source code.
Autoconf exists because unix systems people can't agree where to
keep the piles of sand (include files). Documentation is a burden
and an afterthought. Good documentation involves clever variable 
naming. And all the great research in computer science is published
in "Numerical Recipes in C".

You look at all of this and think this is the best way; clearly
the standard way; absolutely the only way to build systems.

I look at all of this and search for a better way.
I believe literate programming is fundamentally better.


Tim






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Following Axiom's example :-), the Sage developers now have a tutorial
available on Lulu:

http://www.lulu.com/content/1932117

Tim, considering the keywords used for Sage tutorial:

    * Computer Algebra
    * number theory
    * sage
    * math

  Listed in:
  Computers & Internet

you might want to review the keywords you assigned to:

http://www.lulu.com/content/190827

    * math
    * science
    * mathematics
    * Statistics
    * General

  Listed in:
  Medicine & Science

to make sure that these tutorials are equally accessible. (math is too general).

Regards,
Bill Page.



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I have just added a new bug status "fixed somewhere".

Please use "fix proposed" and "fixed somewhere" according to the following
rules:

> 1) fix proposed
> 
>    would be used if a patch is attached, but it hasn't been fixed in one of the
>    systems
> 
> 2) fixed somewhere  
> 
>    would be used if the issue is fixed somewhere, regardless whether a patch is
>    included or not.
> -------------------------------------------------------------------------------

I will upload patches to those issues I marked "fix proposed" with comment
"fixed in FriCAS" upon request.  I hope that's OK for everybody.  Otherwise,
please complain and I'll try to find a solution.

I'd be also very grateful for a better name, replacing "fixed somewhere", it
shouldn't be longer though.

Martin




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Update the release date information for the January 2008 release

=======================================================================
diff --git a/Makefile b/Makefile
index 7458a7b..829a773 100644
--- a/Makefile
+++ b/Makefile
@@ -1,4 +1,4 @@
-VERSION="Axiom (Sept 2007)"
+VERSION="Axiom (January 2008)"
 SPD=$(shell pwd)
 SYS=$(notdir $(AXIOM))
 SPAD=${SPD}/mnt/${SYS}
diff --git a/Makefile.pamphlet b/Makefile.pamphlet
index 663f9f3..2775bb5 100644
--- a/Makefile.pamphlet
+++ b/Makefile.pamphlet
@@ -292,7 +292,7 @@ The DOCUMENT variable is now set to replace the direct call
 to the document command. This will allow it to be
 changed on the command line.
 <<environment>>=
-VERSION="Axiom (Sept 2007)"
+VERSION="Axiom (January 2008)"
 SPD=$(shell pwd)
 SYS=$(notdir $(AXIOM))
 SPAD=${SPD}/mnt/${SYS}
diff --git a/changelog b/changelog
index 9cfc6cf..052e649 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080125 tpd --patch-55 (January 2008) release
 20080125 tpd src/input/Makefile add En regression test 
 20080125 tpd src/input/en.input regression test En
 20080125 tpd src/algebra/special.spad add special function En 



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Summary: January 2008 release

There have been two major concentrations of effort in this release.

The first concentration was on the new Firefox Hyperdoc and the
second concentration was the verification of Axiom against
publish standards.




Firefox Hyperdoc

The Firefox Hyperdoc has been integrated with the rest of the
interpreter. The new )browse command causes Axiom to listen and
serve hyperdoc pages on port 8085. 

The interpreter was changed to add the )browse command. As a 
side-effect new documentation was added to the interpreter
volume (bookvol5) to explain top-level command handling. In
addition, lisp and boot code was rewritten as part of the
literate change.

New sections were added to cover the beginning of the Computer
Algebra Test Suite (CATS) subsection which brings a focus on
compliance with published standards.

Arthur Ralf's mathml-enabled version of the Jenks book is fully
integrated into the Firefox Hyperdoc. Arthur also fixed some
rendering and ambiguity issues.

  axbook.tgz              fix the user/group settings
  axserver.spad           fix lastType output re: errors
  bookvol5                browse and top-level command handling 
  bookvol11               add standards compliance for gamma
  gammacomplexinverse.png added
  gammacomplex.png        added
  gammareal3.png          added
  loggamma.png            added
  mathml.spad             fix ambiguity bug in mathml output
  mathml.spad             fix hex(10) mathml rendering 
  mathml.spad             fix F,3 mathml rendering 
  mathml.spad             remove code to eat %% 
  psi.png                 added 




Standards Verification

The Computer Algebra Test Suite (CATS) effort checks the results
that Axiom generates against published results. Axiom has an
extensive set of regression tests (the KAMKE suite) for ordinary
differential equations, for integration (the SCHAUM suite), and
for numeric special functions (the ABRAMOWITZ suite). In addition,
results have been checked against Mathematica, Maple, and Maxima.

  asinatan.input   regression for the functions asin and atan
  asinhatanh.input regression for the functions asinh and atanh
  besselk.input    regression for the function besselK
  e1.input         regression for the function E1
  en.input         regression for the function En
  exp.input        regression for the function exp
  gamma.input      regression for the function gamma
  log.input        regression for the functions log
  pfaffian.input   regression for the function pfaffian
  seccsc.input     regression for the functions sec and csc
  sincos.input     regression for the functions sin and cos
  sinhcosh.input   regression for the functions sinh and cosh
  tancot.input     regression for the functions tan and cot
  tanhcoth.input   regression for the functions tanh and coth




New Functions Added

Axiom is missing various special functions found in other computer
algebra systems. This release adds two new ones, the Exponential
Integral E1 and the higher order Exponential Integral En.  These have
been tested against the publish results.

  special.spad E1 added
  special.spad En added




Bugs Fixed

There are 15 bug fixes in this release:

  bug 7015: fix hex(10) mathml rendering
  bug 7016: remove code to eat %% 
  bug 7019: fix F,3 mathml rendering 
  bug 7023: discardGraph free corrected 
  bug 7042: ignore regression test gensym
  bug 7045: wrong Makefile for Xpm fix 
  bug 7052: spurious remake of axbook
  bug 7054: /home/silver path in bookvol11
  bug 7057: ambiguity in mathml
  bug 7089/343: FreeAbelianGroup order 
  bug 7090/355 handle besselK
  bug 7093: Function name fix
  bug 7100/149: numlock in hyperdoc
  bug 7101/204: MoreSystemCommand unnecessary loading 
  bug 7102/412: Equality testing in TableAggregate 




Regression test fixes

As changes happen in the system the regression tests are updated
to reflect the new conditions. Changing the category of PositiveInteger
caused (a + -bi) to print as (a - bi). New builds raised gensym faults
which were fixed. And new builds change random numbers so the tests
that depend on them are marked "ok" despite failures due to randomness.
The FreeAbelianGroup bug is tested.

  acplot.spad     fix output form of negative numbers
  calculus2.input fix function names 
  classtalk.input ignore gensyms
  collect.input   fix function names 
  dfloat.input    handle negative number output
  easter.input    fix function names 
  elemnum.input   handle negative number output
  exlap.input     fix function names 
  exsum.input     fix function names 
  free.input      added to test bug
  grpthry.input   mark random generation failures ok
  grpthry.input   fix function names 
  ico.input       mark random generation failures ok
  intg0.input     ignore gensyms
  is.input        type declare function
  kamke3.input    mark random generation failures ok
  knot2.input     fix function names 
  lodo.spad       ignore regression test gensym 
  mapleok.input   ignore gensyms
  mathml.input    handle new mathml sub/sup change
  ndftip.input    fix missing blank lines
  pmint.input     rewritten
  repa6.input     fix function names 
  r20bugs.input   change spacing
  sf.spad         fix output form of negative numbers
  tbagg.input     regression for equality testing in TableAggregate




Algebra file changes

The fundamental change was a supposedly transparent move of
the category for PositiveInteger. This had the effect of changing
the output form and broke several regression tests. Some mathml
issues were fixed. The new functions E1 and En were added. The
FreeAbelianGroup bug was fixed.

  axserver.spad  fix lastType output re: errors
  acplot.spad    fix output form of negative numbers
  combfunc.spad  fix bold font handling
  integer.spad   category change for PositiveInteger
  sf.spad fix    output form of negative numbers
  sf.spad        handle besselK
  op.spad        handle besselK 
  combfunc.spad  handle besselK 
  free.spad      fix FreeAbelianGroup bug 
  special.spad   add E1
  special.spad   add En
  mathml.spad    fix ambiguity bug in mathml output
  mathml.spad    fix hex(10) mathml rendering 
  mathml.spad    fix F,3 mathml rendering 
  mathml.spad    remove code to eat %% 



Interpreter changes

The primary changes are the addition of bookvol11 for the Firefox
Hyperdoc and the literate documentation of the top level command
handling in bookvol5 (the interpreter) along with rewrites of the
lisp/boot code.

  bootfuns.lisp move $systemCommands to bookvol5
  bookvol5      browse and top-level command handling
  bookvol11     added
  http.lisp     mathObject2String for hex(10) 
  incl.boot     move incBiteOff to bookvol5
  intint.lisp   move setCurrentLine to bookvol5
  int-top.boot  move ncloopCommand, etc. to bookvol5
  i-syscmd.boot move $SYSCOMMANDS to bookvol5
  Makefile      wrong Makefile for Xpm fix 
  makegraph.c   discardGraph free corrected 
  nci.lisp      move ncloopInclude to bookvol5
  setq.lisp     move command initialization to bookvol5



Documentation changes

  bookvol5      explain top level input handling (lisp/boot rewrite)
  combfunc.spad fix bold font handling
  axiom.sty     add binom 



Patches released

  20071129.01.tpd.patch
  20071129.02.tpd.patch
  20071205.01.tpd.patch
  20071206.01.tpd.patch
  20071208.01.tpd.patch
  20071215.01.tpd.patch
  20071215.02.tpd.patch
  20071215.03.gxv.patch
  20071216.01.tpd.patch
  20071216.02.tpd.patch
  20071216.03.acr.patch
  20071217.01.acr.patch
  20071217.02.tpd.patch
  20071218.01.acr.patch
  20071225.01.sxw.patch
  20071228.01.tpd.patch
  20071229.01.jap.patch
  20071230.01.acr.patch
  20071230.02.tpd.patch
  20071230.03.tpd.patch
  20080102.01.tpd.patch
  20080103.01.tpd.patch
  20080104.01.tpd.patch
  20080104.02.tpd.patch
  20080106.01.tpd.patch
  20080107.01.tpd.patch
  20080107.02.tpd.patch
  20080107.03.tpd.patch
  20080116.01.tpd.patch
  20080119.01.tpd.patch
  20080119.02.tpd.patch
  20080119.03.tpd.patch
  20080120.01.gxv.patch
  20080120.02.tpd.patch
  20080120.03.tpd.patch
  20080125.01.tpd.patch
  20080125.02.tpd.patch





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From: Martin Rubey <martin.rubey@univie.ac.at>
Date: 26 Jan 2008 08:11:37 +0100
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Yes, I like this proposal equally well.  Comments?

"William Sit" <wyscc@sci.ccny.cuny.edu> writes:

> Dear Martin:
> 
> May I suggest using a bit vector to notate the fixes? I propose that the bits
> of the bit vector be assigned according to chronological order of axiom
> branches, so that bit 0 is for Tim's Axiom, bit 1 for FriCAS and bit 2 for
> OpenAxiom, etc. So we need only two status: fixed xxx and fixed proposed xxx
> for now. The bit vector is extensible if more flavors come along.
> 
> Example: "fixed 101" means patches exist and implemented for OpenAxiom and for
> Tim's Axiom, each patch according to the requirements of the flavor.
> 
> "fixed proposed 010" means a patch exist (proposed) but not implemented (for
> whatever reasons) on FriCAS.
> 
> "fixed 000" means no patch implemented
> 
> "fixed proposed 000" means no patch proposed (implies of course "fixed 000").
> 
> You may forward this to the groups or disregard as you please.
> 
> William
> 
> 
> On 25 Jan 2008 22:11:55 +0100
>   Martin Rubey <martin.rubey@univie.ac.at> wrote:
> >I have just added a new bug status "fixed somewhere".
> >
> > Please use "fix proposed" and "fixed somewhere" according to the following
> >rules:
> >
> >> 1) fix proposed
> >>    would be used if a patch is attached, but it hasn't been fixed in one of
> >> the
> >>    systems
> >> 2) fixed somewhere     would be used if the issue is fixed somewhere,
> >> regardless whether a patch is
> >>    included or not.
> >> -------------------------------------------------------------------------------
> >
> > I will upload patches to those issues I marked "fix proposed" with comment
> > "fixed in FriCAS" upon request.  I hope that's OK for everybody.  Otherwise,
> >please complain and I'll try to find a solution.
> >
> > I'd be also very grateful for a better name, replacing "fixed somewhere", it
> >shouldn't be longer though.
> >
> >Martin
> >
> >
> >
> >_______________________________________________
> >Axiom-developer mailing list
> >Axiom-developer@nongnu.org
> >http://lists.nongnu.org/mailman/listinfo/axiom-developer
> 
> William Sit, Professor of Mathematics, City College of New York
> Office: R6/202C Tel: 212-650-5179, Fax: 212-862-0004
> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/




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Martin Rubey <martin.rubey@univie.ac.at> writes:

> Yes, I like this proposal equally well.  Comments?

Oops, sorry, I don't understand it:  it would imply having 7 additional
statuses too.  I don't think it's possible to have a "variable" status?

The point is being able to filter out all issues that are fixed somewhere.

Martin
> 
> "William Sit" <wyscc@sci.ccny.cuny.edu> writes:
> 
> > Dear Martin:
> > 
> > May I suggest using a bit vector to notate the fixes? I propose that the bits
> > of the bit vector be assigned according to chronological order of axiom
> > branches, so that bit 0 is for Tim's Axiom, bit 1 for FriCAS and bit 2 for
> > OpenAxiom, etc. So we need only two status: fixed xxx and fixed proposed xxx
> > for now. The bit vector is extensible if more flavors come along.
> > 
> > Example: "fixed 101" means patches exist and implemented for OpenAxiom and for
> > Tim's Axiom, each patch according to the requirements of the flavor.
> > 
> > "fixed proposed 010" means a patch exist (proposed) but not implemented (for
> > whatever reasons) on FriCAS.
> > 
> > "fixed 000" means no patch implemented
> > 
> > "fixed proposed 000" means no patch proposed (implies of course "fixed 000").
> > 
> > You may forward this to the groups or disregard as you please.
> > 
> > William
> > 
> > 
> > On 25 Jan 2008 22:11:55 +0100
> >   Martin Rubey <martin.rubey@univie.ac.at> wrote:
> > >I have just added a new bug status "fixed somewhere".
> > >
> > > Please use "fix proposed" and "fixed somewhere" according to the following
> > >rules:
> > >
> > >> 1) fix proposed
> > >>    would be used if a patch is attached, but it hasn't been fixed in one of
> > >> the
> > >>    systems
> > >> 2) fixed somewhere     would be used if the issue is fixed somewhere,
> > >> regardless whether a patch is
> > >>    included or not.
> > >> -------------------------------------------------------------------------------
> > >
> > > I will upload patches to those issues I marked "fix proposed" with comment
> > > "fixed in FriCAS" upon request.  I hope that's OK for everybody.  Otherwise,
> > >please complain and I'll try to find a solution.
> > >
> > > I'd be also very grateful for a better name, replacing "fixed somewhere", it
> > >shouldn't be longer though.
> > >
> > >Martin
> > >
> > >
> > >
> > >_______________________________________________
> > >Axiom-developer mailing list
> > >Axiom-developer@nongnu.org
> > >http://lists.nongnu.org/mailman/listinfo/axiom-developer
> > 
> > William Sit, Professor of Mathematics, City College of New York
> > Office: R6/202C Tel: 212-650-5179, Fax: 212-862-0004
> > Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
> 
> 
> 
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer




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From: "Bill Page" <bill.page@newsynthesis.org>
To: "Martin Rubey" <martin.rubey@univie.ac.at>
Subject: Re: [Axiom-developer] Re: Axiom: fixed and fixed proposed bit vectors
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I must say that I do not like the sound of "fixed somewhere" - it
sounds too vague - but I understand the intent. Maybe it would be
better to write:

  "fixed as noted ..."

And then expect to find a comment saying in which versions/forks the
fix occurs. This is almost like you are doing now except for the
distinction between "as noted" and "proposed" (patch attached to the
bug report).

I don't think William intended 7 different statuses. The parameter can
be included as a comment. But I do not see any point in being so
cryptic.

Regards,
Bill Page.

On 26 Jan 2008 09:47:43 +0100, Martin Rubey wrote:
> Martin Rubey <martin.rubey@univie.ac.at> writes:
>
> > Yes, I like this proposal equally well.  Comments?
>
> Oops, sorry, I don't understand it:  it would imply having 7
> additional statuses too.  I don't think it's possible to have
> a "variable" status?
>
> The point is being able to filter out all issues that are
> 'fixed somewhere.'
>
> Martin
> >
> > "William Sit" <wyscc@sci.ccny.cuny.edu> writes:
> >
> > > Dear Martin:
> > >
> > > May I suggest using a bit vector to notate the fixes? I propose
> > > that the bits of the bit vector be assigned according to chronological
> > > order of axiom branches, so that bit 0 is for Tim's Axiom, bit 1 for
> > > FriCAS and bit 2 for OpenAxiom, etc. So we need only two status:
> > > fixed xxx and fixed proposed xxx for now. The bit vector is extensible
> > > if more flavors come along.
> > >
> > > Example: "fixed 101" means patches exist and implemented for
> > > OpenAxiom and for Tim's Axiom, each patch according to the
> > > requirements of the flavor.
> > >
> > > "fixed proposed 010" means a patch exist (proposed) but not
> > > implemented (for whatever reasons) on FriCAS.
> > >
> > > "fixed 000" means no patch implemented
> > >
> > > "fixed proposed 000" means no patch proposed (implies of
> > > course "fixed 000").
> > >
> > > You may forward this to the groups or disregard as you
> > > please.
> > >
> > > William
> > >
> > >
> > > On 25 Jan 2008 22:11:55 +0100
> > >   Martin Rubey <martin.rubey@univie.ac.at> wrote:
> > > >I have just added a new bug status "fixed somewhere".
> > > >
> > > > Please use "fix proposed" and "fixed somewhere" according
> > > > to the following rules:
> > > >
> > > >> 1) fix proposed
> > > >>    would be used if a patch is attached, but it hasn't been
> > > >> fixed in one of the systems
> > > >> 2) fixed somewhere would be used if the issue is fixed
> > > >> somewhere, regardless whether a patch is included or
> > > >> not.
> > > >> ...



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"Bill Page" <bill.page@newsynthesis.org> writes:

> I must say that I do not like the sound of "fixed somewhere" - it
> sounds too vague - but I understand the intent. Maybe it would be
> better to write:
> 
>   "fixed as noted ..."

fine with me!  As I said already, I dislike the sound of "fixed somewhere",
too, but couldn't think of anything better.  If you like, please change the
name.  (lightgreen is good, I think)

> And then expect to find a comment saying in which versions/forks the fix
> occurs. This is almost like you are doing now except for the distinction
> between "as noted" and "proposed" (patch attached to the bug report).

Exactly, that was the idea.  Fortunately, I mainly modified the status of
algebra bugs, where a patch is reasonably easy to upload.  As I said already,
I'll do so on request.

I'd like to return to the algebra, now.

Martin




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Dear all,

I'd like to understand the following behaviour.  Consider

-------------------------------------------------------------------------------
)abb package TEST Test
Test(): with

     foo: Fraction Integer -> Fraction Integer
     foo: PrimeField 5 -> PrimeField 5
     foo: Expression Integer -> Expression Integer

  == add

    foo(n: Fraction Integer) == n
    foo(n: PrimeField 5) == n
    foo(n: Expression Integer) == n
-------------------------------------------------------------------------------

Then we have:

(4) -> foo 3

 Function Selection for foo
      Arguments: PI 
   -> no appropriate foo found in PositiveInteger 
   -> no appropriate foo found in Integer 
   -> no appropriate foo found in PositiveInteger 
   -> no appropriate foo found in Integer 

 Modemaps from Associated Packages 
   no modemaps

 Remaining General Modemaps 
   [1] Expression Integer -> Expression Integer from Test
   [2] PrimeField 5 -> PrimeField 5 from Test
   [3] Fraction Integer -> Fraction Integer from Test
 
 [1]  signature:   PF 5 -> PF 5
      implemented: slot (PrimeField 5)(PrimeField 5) from TEST
 [2]  signature:   FRAC INT -> FRAC INT
      implemented: slot (Fraction (Integer))(Fraction (Integer)) from TEST
 [3]  signature:   EXPR INT -> EXPR INT
      implemented: slot (Expression (Integer))(Expression (Integer)) from TEST
 

   (4)  3
                                                           Type: PrimeField 5
-------------------------------------------------------------------------------

Note that the interpreter seems to prefer PF 5 over FRAC INT over EXPR INT.
Now consider:

p := z^10; s:=z=0..1; 

(6) -> integrate(p, s)

 Function Selection for integrate
      Arguments: (POLY INT,SEGBIND NNI) 
   -> no appropriate integrate found in Polynomial Integer 
   -> no appropriate integrate found in SegmentBinding NonNegativeInteger 
   -> no appropriate integrate found in Polynomial Integer 
   -> no appropriate integrate found in SegmentBinding NonNegativeInteger 

 Modemaps from Associated Packages 
   [1] (Polynomial D2,Symbol) -> Polynomial D2 from Polynomial D2
            if D2 has ALGEBRA FRAC INT and D2 has RING

 Remaining General Modemaps 
   [1] (D,D1) -> D from D
            if D1 = SYMBOL and D has UTSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has UTSCAT D2 and
            D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
            integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
   [2] (D,D1) -> D from D
            if D1 = SYMBOL and D has UPXSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has UPXSCAT D2 
            and D2 has RING and D2 has variables: D2 -> List D1 and D2 
            has integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
         
   [3] (D,D1) -> D from D
            if D1 = SYMBOL and D has ULSCAT D2 and D2 has RING and D2 
            has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
            has ALGEBRA FRAC INT or D1 = SYMBOL and D has ULSCAT D2 and
            D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
            integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
   [4] (D,D1) -> D from D
            if D has MTSCAT(D2,D1) and D2 has RING and D1 has ORDSET 
            and D2 has ALGEBRA FRAC INT
   [5] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List 
            Expression D4)
            from IntegrationResultRFToFunction D4
            if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo 
            Integer,OrderedSet,LinearlyExplicitRingOver Integer)
   [6] (GeneralUnivariatePowerSeries(D2,D3,D4),Variable D3) -> 
            GeneralUnivariatePowerSeries(D2,D3,D4)
            from GeneralUnivariatePowerSeries(D2,D3,D4)
            if D3: SYMBOL and D2 has ALGEBRA FRAC INT and D2 has RING 
            and D4: D2
   [7] (D2,Symbol) -> Union(D2,List D2) from FunctionSpaceIntegration(
            D4,D2)
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer) and D2 has Join(
            TranscendentalFunctionCategory,PrimitiveFunctionCategory,
            AlgebraicallyClosedFunctionSpace D4)
   [8] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
            Fraction Polynomial D4) -> Union(f1: OrderedCompletion Expression
            D4,f2: List OrderedCompletion Expression D4,fail: failed,
            pole: potentialPole)
            from RationalFunctionDefiniteIntegration D4
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)
   [9] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
            Expression D4) -> Union(f1: OrderedCompletion Expression D4,f2: 
            List OrderedCompletion Expression D4,fail: failed,pole: 
            potentialPole)
            from RationalFunctionDefiniteIntegration D4
            if D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)
   [10] (D2,SegmentBinding OrderedCompletion D2) -> Union(f1: 
            OrderedCompletion D2,f2: List OrderedCompletion D2,fail: failed,
            pole: potentialPole)
            from ElementaryFunctionDefiniteIntegration(D4,D2)
            if D2 has Join(TranscendentalFunctionCategory,
            PrimitiveFunctionCategory,AlgebraicallyClosedFunctionSpace 
            D4) and D4 has Join(EuclideanDomain,OrderedSet,
            CharacteristicZero,RetractableTo Integer,
            LinearlyExplicitRingOver Integer)
 
 [1]  signature:   (EXPR INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Expression (Integer))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTEF(INT,EXPR INT)
 [2]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTRF INT
 [3]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Fraction (Polynomial (Integer))))) from DEFINTRF INT
 

 Function Selection for map by coercion facility (map) 
      Arguments: ((NNI -> ORDCOMP EXPR INT),SEGBIND NNI) 
      Target type: SEGBIND ORDCOMP EXPR INT 
   -> no appropriate map found in SegmentBinding NonNegativeInteger 
   -> no appropriate map found in SegmentBinding OrderedCompletion Expression Integer 
   -> no appropriate map found in OrderedCompletion Expression Integer 
   -> no appropriate map found in NonNegativeInteger 
   -> no appropriate map found in OrderedCompletion Expression Integer 

 Modemaps from Associated Packages 
   [1] ((D4 -> D5),SegmentBinding D4) -> SegmentBinding D5
            from SegmentBindingFunctions2(D4,D5)
            if D4 has TYPE and D5 has TYPE
 
 [1]  signature:   ((NNI -> ORDCOMP EXPR INT),SEGBIND NNI) -> SEGBIND ORDCOMP EXPR INT
      implemented: slot (SegmentBinding (OrderedCompletion (Expression (Integer))))(Mapping (OrderedCompletion (Expression (Integer))) (NonNegativeInteger))(SegmentBinding (NonNegativeInteger)) from SEGBIND2(NNI,ORDCOMP EXPR INT)
 

         1
   (6)  --
        11
                    Type: Union(f1: OrderedCompletion Expression Integer,...)

There are many things I do not understand:

1) why doesn't the interpreter choose the "General Modemap"

   [8] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
            Fraction Polynomial D4)

  ?

2) why is its preference different now:


 [1]  signature:   (EXPR INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Expression (Integer))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTEF(INT,EXPR INT)
 [2]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTRF INT
 [3]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
      implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Fraction (Polynomial (Integer))))) from DEFINTRF INT
 
I.e., in package DEFINTRF, it seems to prefer (FRAC POLY INT,SEGBIND ORDCOMP
EXPR INT) over (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT).

3) why does it prefer DEFINTEF(INT,EXPR INT) over DEFINTRF INT ?


ANY hints?

Martin




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From: "William Sit" <wyscc@sci.ccny.cuny.edu>
Subject: Re: [Axiom-developer] Re: Axiom: fixed and fixed proposed bit vectors
To: "Bill Page" <bill.page@newsynthesis.org>,"Martin Rubey"
	<martin.rubey@univie.ac.at>
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I consider "fixed xxx" and "fixed proposed" as two 
statuses, not 16 (but that is up to your interpretation 
what counts as a status). One reason for proposing this 
"cryptic" version is to enable automatic sorting. One can 
write a program to search "fixed" and "fixed proposed" and 
then parse xxx to filter out those that are fixed or not 
fixed, or proposed or not proposed for any of the flavors. 
This program will allow users to do searches and be 
presented only with a particular xxx view.

As long as the bit vector is documented, it is no more 
cryptic than other code. A bit vector is the most 
efficient of data representation. Since each axiom has its 
own bit, it is easy to even spot the status visually.

If you use comments after "fixed" and "fixed proposed", 
someone will have to read the comments manually and sort 
the bug reports manually. It would be better to know ahead 
of reading the comments what one is about to read.

In the best scenario, the code xxx would only go through 4 
values (starting with 000 and increasing by changing a 0 
to a 1, at most three times).


William

On Sat, 26 Jan 2008 08:54:55 -0500
  "Bill Page" <bill.page@newsynthesis.org> wrote:
>I must say that I do not like the sound of "fixed 
>somewhere" - it
>sounds too vague - but I understand the intent. Maybe it 
>would be
>better to write:
>
>   "fixed as noted ..."
>
>And then expect to find a comment saying in which 
>versions/forks the
>fix occurs. This is almost like you are doing now except 
>for the
>distinction between "as noted" and "proposed" (patch 
>attached to the
>bug report).
>
>I don't think William intended 7 different statuses. The 
>parameter can
>be included as a comment. But I do not see any point in 
>being so
>cryptic.
>
>Regards,
>Bill Page.
>
>On 26 Jan 2008 09:47:43 +0100, Martin Rubey wrote:
>> Martin Rubey <martin.rubey@univie.ac.at> writes:
>>
>> > Yes, I like this proposal equally well.  Comments?
>>
>> Oops, sorry, I don't understand it:  it would imply 
>>having 7
>> additional statuses too.  I don't think it's possible to 
>>have
>> a "variable" status?
>>
>> The point is being able to filter out all issues that 
>>are
>> 'fixed somewhere.'
>>
>> Martin
>> >
>> > "William Sit" <wyscc@sci.ccny.cuny.edu> writes:
>> >
>> > > Dear Martin:
>> > >
>> > > May I suggest using a bit vector to notate the 
>>fixes? I propose
>> > > that the bits of the bit vector be assigned 
>>according to chronological
>> > > order of axiom branches, so that bit 0 is for Tim's 
>>Axiom, bit 1 for
>> > > FriCAS and bit 2 for OpenAxiom, etc. So we need only 
>>two status:
>> > > fixed xxx and fixed proposed xxx for now. The bit 
>>vector is extensible
>> > > if more flavors come along.
>> > >
>> > > Example: "fixed 101" means patches exist and 
>>implemented for
>> > > OpenAxiom and for Tim's Axiom, each patch according 
>>to the
>> > > requirements of the flavor.
>> > >
>> > > "fixed proposed 010" means a patch exist (proposed) 
>>but not
>> > > implemented (for whatever reasons) on FriCAS.
>> > >
>> > > "fixed 000" means no patch implemented
>> > >
>> > > "fixed proposed 000" means no patch proposed 
>>(implies of
>> > > course "fixed 000").
>> > >
>> > > You may forward this to the groups or disregard as 
>>you
>> > > please.
>> > >
>> > > William
>> > >
>> > >
>> > > On 25 Jan 2008 22:11:55 +0100
>> > >   Martin Rubey <martin.rubey@univie.ac.at> wrote:
>> > > >I have just added a new bug status "fixed 
>>somewhere".
>> > > >
>> > > > Please use "fix proposed" and "fixed somewhere" 
>>according
>> > > > to the following rules:
>> > > >
>> > > >> 1) fix proposed
>> > > >>    would be used if a patch is attached, but it 
>>hasn't been
>> > > >> fixed in one of the systems
>> > > >> 2) fixed somewhere would be used if the issue is 
>>fixed
>> > > >> somewhere, regardless whether a patch is included 
>>or
>> > > >> not.
>> > > >> ...
>
>
>_______________________________________________
>Axiom-developer mailing list
>Axiom-developer@nongnu.org
>http://lists.nongnu.org/mailman/listinfo/axiom-developer

William Sit, 
Professor of Mathematics, City College of New York 
                 
Office: R6/202C Tel: 212-650-5179, Fax: 212-862-0004
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/



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This patch create a one-page, 2-sided reference card for Axiom.

Tim

=========================================================================
diff --git a/changelog b/changelog
index 052e649..d486296 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20080127 tpd src/doc/Makefile add refcard
+20080127 tpd src/doc/refcard added
 20080125 tpd --patch-55 (January 2008) release
 20080125 tpd src/input/Makefile add En regression test 
 20080125 tpd src/input/en.input regression test En
diff --git a/src/doc/Makefile.pamphlet b/src/doc/Makefile.pamphlet
index cd17c11..6cbc985 100644
--- a/src/doc/Makefile.pamphlet
+++ b/src/doc/Makefile.pamphlet
@@ -81,6 +81,17 @@ ${DVI}/bookvol1.dvi: ${IN}/bookvol1.pamphlet
 	cp ${IN}/ps/*ps* ${DVI}/ps )
 
 @
+\section{The Reference Card}
+This is the one-page, 2 sided reference card
+<<refcard>>=
+${DVI}/refcard.dvi: ${IN}/refcard.pamphlet
+	@echo 4 making ${DVI}/refcard.dvi from ${IN}/refcard.pamphlet
+	@(cd ${MID} ; \
+	cp ${IN}/refcard.pamphlet ${MID} ;\
+	${DOCUMENT} ${NOISE} refcard ; \
+	cp refcard.dvi ${DVI} )
+
+@
 \section{The End Papers}
 This document reproduces the diagrams on the inside covers of the
 original Jenks Axiom book but adds hyperlinks.
@@ -164,8 +175,8 @@ DVI=${MNT}/${SYS}/doc
 DOC=${INT}/doc
 
 FILES= ${MID}/axiom.bib ${STY}/axiom.sty ${DVI}/bookvol4.dvi \
-       ${DVI}/book.dvi ${DVI}/bookvol1.dvi ${DVI}/endpaper.dvi \
-       ${DVI}/rosetta.dvi ${DVI}/spadhelp/spadhelp.files
+       ${DVI}/book.dvi ${DVI}/bookvol1.dvi ${DVI}/refcard.dvi \
+       ${DVI}/endpaper.dvi ${DVI}/rosetta.dvi ${DVI}/spadhelp/spadhelp.files
 
 CMDS=${OUT}/booklet
 
@@ -180,6 +191,7 @@ all: ${FILES} ${CMDS}
 <<bookvol4>>
 <<Book>>
 <<bookvol1>>
+<<refcard>>
 <<Endpapers>>
 <<rosetta>>
 <<spadhelp.files>>
diff --git a/src/doc/refcard.pamphlet b/src/doc/refcard.pamphlet
new file mode 100644
index 0000000..ef6afbe
--- /dev/null
+++ b/src/doc/refcard.pamphlet
@@ -0,0 +1,514 @@
+\documentclass{article}
+\usepackage[landscape]{geometry}
+\usepackage{multicol}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\advance\topmargin-.8in
+\advance\textheight2in
+\advance\textwidth3in
+\advance\oddsidemargin-1.5in
+\advance\evensidemargin-1.5in
+\parindent0pt
+\parskip2pt
+\newcommand{\hr}{\centerline{\rule{3.5in}{1pt}}}
+\begin{document}
+\begin{multicols*}{3}
+\begin{center}
+\textbf{Axiom Quick Reference (January 2008)}\\
+\end{center}
+
+\textbf{Command Line}
+
+)cd $\langle$pathname$\rangle$
+
+)clear all -- clear workspace
+
+)display op $\langle$function$\rangle$ -- function arguments
+
+)set message autoload off -- quietly load algebra
+
+)set message bottom on -- show selection process
+
+)set stream calculate 20 -- number of terms to calculate
+
+)show $\langle$domain$\rangle$ -- list all functions
+
+)spool $\langle${\sl filename}$\rangle$ -- start save session
+
+)spool -- close spool file
+
+)trace $\langle$domain$\rangle$ )math -- trace execution
+
+)quit -- exit Axiom
+
+)read $\langle$filename$\rangle$[.input] -- evaluate a file
+
+)sys $\langle$command line$\rangle$ -- execute command
+
+\_ continues input lines or escapes chars \verb|a_ b| = ``a b''
+
+\% is last value
+
+\%\%(n) is $n$th value
+
+-- and ++ start comment lines
+
+%*********************************************
+\hr\textbf{Programming}
+
+assignment: var := value\\
+\hbox{\hskip 2cm}x:=3
+
+conditional: if $\langle$pred$\rangle$ then $\langle$truecase$\rangle$
+else $\langle$falsecase$\rangle$\\
+\hbox{\hskip 2cm}\verb|if (2 > 4) then 4 else 5|
+
+loop: for $\langle$pred$\rangle$ repeat $\langle$block$\rangle$\\
+\hbox{\hskip 2cm}\verb|for i in 1..5 repeat print i|
+\hbox{\hskip 2cm}\verb|while i < 3 repeat (print i ; i:=i+1)|
+
+function: $f(x)=x^2$ \\
+\hbox{\hskip 2cm}\verb|f(x)==x^2|
+
+anon. function: \verb|g:=x +-> x+1| \quad g(3) $\rightarrow$ 4
+
+Indentation is significant:\\
+\hbox{\hskip 2cm}\verb|f(x)==(x > 3 => x ; 0)|\\
+\hbox{\hskip 2cm}\verb|f(x)==|\\
+\hbox{\hskip 2.4cm}\verb|x > 3 => x|\\
+\hbox{\hskip 2.4cm}\verb|0|
+
+%*********************************************
+\hr\textbf{Basic constants and functions}
+
+$\pi=$ \verb|%pi| \quad $e=$ \verb|%e| \quad $i=$ \verb|%i| 
+\quad $\infty=$ \verb|%infinity|
+
+$+\infty$=\verb|%plusInfinity|\quad $-\infty$=\verb|%minusInfinity|
+
+\verb|numeric(%pi)| $=3.1415926535\ 897932385$
+
+%Binary operations: \verb|+  -  *  /  ^|
+
+Functions: \verb|sin cos tan sec csc cot sinh cosh tanh| \verb|sech csch coth log ln exp|
+
+$ab=$ \verb|a*b| \quad $\frac a b=$ \verb|a/b| 
+\quad 
+$a^b=$ \verb|a^b| \quad $\sqrt{x}=$ \verb|sqrt(x)|
+
+$\sqrt[n]{x}=$\verb|x^(1/n)|
+$|x|=$\verb|abs(x)|
+$\log_b(x)=$\verb|log(x)/log(b)|
+
+%*********************************************
+\hr\textbf{Operations on expressions}
+
+\verb|factor(...)|\qquad \verb|expand(...)|\qquad \verb|simplify(...)|
+
+Symbolic equations: \verb|f(x)=g(x)|
+
+Solve $f(x)=g(x)$: \verb| solve(f(x)=g(x),x)|
+
+\verb|solve([x^2*y-1,x*y^2-2],.01)|\\
+\hbox{\hskip 2.0cm} $\rightarrow$ $[[y=1.5859375,x=0.79296875]]$
+
+\verb|complexSolve([x^2*y-1,x*y^2-2],1/1000)|
+
+\verb|radicalSolve([x^2/a+a+y^3-1,a*y+a+1],[x,y])|
+
+$\displaystyle\sum_{i=k}^n f(i)=$ \verb|reduce(+,[f(i) for i in k..n])|
+
+$\displaystyle\prod_{i=k}^n f(i)=$ \verb|reduce(*,[f(i) for i in k..n])|
+
+%*********************************************
+\hr\textbf{Pattern Matching}
+
+logrule:=rule log(x)+log(y) == log(x*y) $\rightarrow$\\
+\hbox{\hskip 2.1cm}\verb|log(y)+log(x)+%B==log(x y)+%B|
+
+f:=log sin x + log x $\rightarrow$ log(sin(x))+log(x)
+
+logrule f $\rightarrow$ log(x sin(x))
+
+%*********************************************
+\hr\textbf{Calculus}
+
+$\displaystyle\lim_{x\to a} f(x)=$ \verb|limit(f(x), x=a)|
+
+$\displaystyle\lim_{x\to a^-} f(x)=$ \verb|limit(f(x), x=a, "left")|
+
+$\displaystyle\lim_{x\to a^+} f(x)=$ \verb|limit(f(x), x=a, "right")|
+
+$\displaystyle\lim_{x\to \infty} f(x)=$ \verb|limit(f(x), x=%plusInfinity)|
+
+\verb|limit(sin(x)/x,x=%plusInfinity)| $\rightarrow$ 0
+
+\verb|complexLimit(sin(x)/x,x=%infinity)| $\rightarrow$ "failed"
+
+$\frac{d}{dx}(f(x))=$ \verb|D(f(x),x)|
+
+$\frac{\partial}{\partial x}(f(x,y))=$ \verb|D(f(x,y),x)|
+
+$\int f(x)dx=$ \verb|integrate(f(x),x)|
+
+$\int_a^b f(x)dx=$ \verb|integrate(f(x),x=a..b)|
+
+%*********************************************
+\hr\textbf{Series}
+
+x:=series 'x
+
+y:=sin(x) $\rightarrow$ 
+$x-\frac{1}{6}x^3+\frac{1}{120}x^5-\frac{1}{5040}x^7+O(x^9)$
+
+coefficient(y,3) $\rightarrow$ $-\frac{1}{6}$
+
+taylor(f(x),x=a)
+
+laurent(x/log(x),x=1)
+
+\verb|puiseux(sqrt(sec(x)),x=3*%pi/2)|
+
+%*********************************************
+\hr\textbf{2D graphics}
+
+\verb|draw(cos(5*t/8),t=0..16*%pi,coordinates==polar)|
+
+\verb|f(t:SF):SF == sin(3*t/5)|
+
+\verb|g(t:SF):SF == sin(t)|
+
+\verb|draw(curve(f,g),0..%pi)|
+
+\verb|draw(x^2+y^3-1=0,x,y,range==[-1..1,-1..1])|
+
+v1:=draw(Gamma(i),i=-4.2..4,adaptive==true)
+
+v2:=draw(1/Gamma(i),i=-4.2..4,adaptive==true)
+
+putGraph(v2,getGraph(v1,1),2)
+
+makeViewport2D(v2)
+
+options: adaptive clip toScale curveColor pointColor\\
+unit range coordinates
+
+%*********************************************
+\hr\textbf{3D graphics}
+
+m(u:SF,v:SF):SF == 1
+
+\verb|draw(m,0..2*%pi,0..%pi,coordinates==spherical)|
+
+options: title style colorFunction coordinates tubeRadius
+tubePoints var1Steps var2Steps space
+
+%*********************************************
+\hr\textbf{Discrete math}
+
+$\lfloor x\rfloor=$ \verb|floor(x)| 
+\quad 
+$\lceil x\rceil=$ \verb|ceiling(x)|
+
+Remainder of $n$ divided by $k=$ \verb|rem(n,k) |, $k|n$ iff \verb| n%k==0|
+
+$n!=$ \verb|factorial(n)| \qquad
+${x\choose m}=$ \verb|binomial(x,m)|
+
+$\phi(n)=$ \texttt{eulerPhi($n$)}\quad Tuples: \ \verb|(1,'Hello,x)| 
+
+%*********************************************
+\hr\textbf{Type Conversions}
+
+\verb|r:=(2/3)*x^2-y+4/5| $\rightarrow$ $-y+\frac{2}{3}x^2+\frac{4}{5}$\\
+\hbox{\hskip 2.0cm} Type: Polynomial Fraction Integer
+
+r::FRAC POLY INT $\rightarrow$ $\frac{-15y+10x^2+12}{15}$\\
+\hbox{\hskip 2.0cm} Type: Fraction Polynomial Integer
+
+\verb|s:=(3+4*%i)/(7+3*%i)| $\rightarrow$ $\frac{33}{58}+\frac{19}{58}\%i$
+
+s::FRAC COMPLEX INT $\rightarrow$ $\frac{3+4\%i}{7+3\%i}$
+
+%*********************************************
+\hr\textbf{Equation}
+
+eq1:=3*x+4*y=5 $\rightarrow$ $4y+3x=5$
+
+eq2:=2*x+2*y=3 $\rightarrow$ $2y+2x=3$
+
+lhs eq1 $\rightarrow$ $4y+3x$
+
+rhs eq1 $\rightarrow$ 5
+
+eq1+eq2 $\rightarrow$ $6y+5x=8$
+
+%*********************************************
+\hr\textbf{Factored}
+
+g:=factor(4312) $\rightarrow$ $2^3 7^2 11$
+
+unit g $\rightarrow$ 1
+
+numberOfFactors g $\rightarrow$ 3
+
+nthFactor(g,2) $\rightarrow$ 7
+
+nthExponent(g,2) $\rightarrow$ 2
+
+nthFlag(g,2) $\rightarrow$ "prime"
+
+map(factor,55739/2520) $\rightarrow$ $\frac{139\ 401}{2^3\ 3^2\ 5\ 7}$
+
+%*********************************************
+\hr\textbf{List}
+
+a:=[1,2,3,4] $\rightarrow$ $[1,2,3,4]$
+
+b:=[3,4,5,6] $\rightarrow$ $[3,4,5,6]$
+
+append(a,b) $\rightarrow$ $[1,2,3,4,3,4,5,6]$
+
+cons(10,a) $\rightarrow$ $[10,1,2,3,4]$
+
+empty? a $\rightarrow$ false
+
+a.2 $\rightarrow$ 2
+
+a.2 := 99 $\rightarrow$ $[1,99,3,4]$
+
+reverse b $\rightarrow$ $[6,5,4,3]$
+
+%*********************************************
+\hr\textbf{MakeFunction}
+
+\verb|expr:=(x+a)^3| $\rightarrow$ $x^3+ 3ax^2 + 3a^2x + a^3$
+
+function(expr,f,x) $\rightarrow$ f
+
+f(2) $\rightarrow$ $a^3 + 6a^2 + 12a + 8$
+
+function(expr,g,a) $\rightarrow$ g
+
+g(2) $\rightarrow$ $x^3 + 6x^2 + 12x + 8$
+
+%*********************************************
+\hr\textbf{Matrix}
+
+A:=matrix([[1,2],[3,4]]) $\rightarrow$
+$
+\left[
+\begin{array}{cc}
+1 & 2\\
+3 & 4\\
+\end{array}
+\right]
+$
+
+determinant A $\rightarrow$ -2
+
+v:=vector([1,2]) $\rightarrow$ $[1,2]$
+
+A*v $\rightarrow$ $[5,11]$
+
+\verb|A^-1| $\rightarrow$
+$
+\left[
+\begin{array}{cc}
+2 & 1\\
+\frac{3}{2} & \frac{1}{2}\\
+\end{array}
+\right]
+$
+
+transpose(A) $\rightarrow$ 
+$
+\left[
+\begin{array}{cc}
+1 & 3\\
+2 & 4\\
+\end{array}
+\right]
+$
+
+nrows A $\rightarrow$ 2
+
+ncols A $\rightarrow$ 2
+
+nullity A $\rightarrow$ 0
+
+rank A $\rightarrow$ 2
+
+trace A $\rightarrow$ 5
+
+%*********************************************
+\hr\textbf{Polynomial}
+
+x+1 yields Type {\bf Polynomial Integer}
+
+z-2.3 yields Type {\bf Polynomial Float}
+
+\verb|y^2-z+3/4| yields Type {\bf Polynomial Fraction Integer}
+
+\verb|p:=(y-1)^2*x*z| $\rightarrow$ $(xy^2-2xy+x)z$
+
+\verb|q:=(y-1)*x*(z+5)| $\rightarrow$ $(xy - x)z + 5xy - 5x$
+
+gcd(p,q) $\rightarrow$ $x y - x$
+
+mainVariable p $\rightarrow$ $z$
+
+variables p $\rightarrow$ $[z,y,x]$
+
+degree(p,y) $\rightarrow$ 2
+
+totaldegree p $\rightarrow$ 4
+
+eval(p,x,w) $\rightarrow$ $(wy^2  - 2wy + w)z$
+
+D(p,x) $\rightarrow$  $(y^2- 2y + 1)z$
+
+integrate(p,x) $\rightarrow$ $(\frac{1}{2}x^2y^2-x^2y+\frac{1}{2}x^2)z$
+
+%*********************************************
+\hr\textbf{PrimeField}
+
+x:PrimeField(7):=5 $\rightarrow$ 5
+
+\verb|x^3| $\rightarrow$ 6
+
+1/x $\rightarrow$ 3
+
+%*********************************************
+\hr\textbf{Set}
+
+\verb|s:=brace([1,2,3,4,5])| $\rightarrow$ $\{1,2,3,4,5\}$
+
+\verb;t:=brace([2,3,5,7]); $\rightarrow$ $\{2,3,5,7\}$
+
+intersect(s,t) $\rightarrow$ $\{2,3,5\}$
+
+union(s,t) $\rightarrow$ $\{1,2,3,4,5,7\}$
+
+difference(s,t) $\rightarrow$ $\{1,4\}$
+
+insert!(7,s) $\rightarrow$ $\{1,2,3,4,5,7\}$
+
+remove!(7,s) $\rightarrow$ $\{1,2,3,4,5\}$
+
+$\{1,2,1,a\}=$ \verb|brace([1,2,1,'a])| \ ($=\{1,2,a\}$)
+
+$\{f(x):x\in X,x>0\}\approx$\verb?brace([f(x) for x in X | x>0])?
+
+%*********************************************
+\hr\textbf{Special Functions}
+
+[fibonacci(k) for k in 0..] $\rightarrow$ [0,1,1,2,3,5,...]
+
+[legendre(i,11) for i in 0..5] $\rightarrow$ [0,1,- 1,1,1,1]
+
+[jacobi(i,15) for i in 0..5] $\rightarrow$ [0,1,1,0,1,0]
+
+[eulerPhi i for i in 1..] $\rightarrow$ [1,1,2,2,4,2,...]
+
+[moebiusMu i for i in 1..] $\rightarrow$ [1,- 1,- 1,0,- 1,1,...]
+
+E1(0.01) $\rightarrow$ 4.0379295765381134
+
+Gamma(0.01) $\rightarrow$ 99.432585119150588
+
+%*********************************************
+\hr\textbf{Stream}
+
+)set streams calculate 6
+
+\verb|ints := [i for i in 1..]| $\rightarrow$ \verb|[1,2,3,4,5,6,...]|
+
+ints.20 $\rightarrow$ 20
+
+\verb;[i for i in ints | odd? i]; $\rightarrow$ \verb|[1,3,5,7,9,11,...]|
+
+%*********************************************
+\hr\textbf{String}
+
+creation: \ s:= \verb|"Hello"| 
+
+concatenate \verb|"He" "llo"| $\rightarrow$ \verb|"Hello"| 
+
+\texttt{s(1)='H' \quad s.1='H' \quad s(2..3)='el' \quad s(4..)='lo'}
+
+split("hi there",char " ") $\rightarrow$ \verb|["hi","there"]|
+
+prefix?("He","Hello") $\rightarrow$ true
+
+substring?("ll","Hello",3) $\rightarrow$ true
+
+%*********************************************
+\hr\textbf{TwoDimensionalArray}
+
+creation: \verb|arr:ARRAY2 INT:=new(2,3,0)| $\rightarrow$ 
+$
+\left[
+\begin{array}{ccc}
+0 & 0 & 0\\
+0 & 0 & 0\\
+\end{array}
+\right]
+$
+
+nrows arr $\rightarrow$ 2
+
+ncols arr $\rightarrow$ 3
+
+setelt(arr,1,1,17) $\rightarrow$
+$
+\left[
+\begin{array}{ccc}
+17 & 0 & 0\\
+0 & 0 & 0\\
+\end{array}
+\right]
+$
+
+arr(1,1) $\rightarrow$ 17
+
+%*********************************************
+\hr\textbf{Univariate Polynomial}
+
+creation: \verb|p:UP(x,INT):=(3*x-1)^2*(2*x+8)|\\
+\hbox{\hskip 1.5cm}\verb|q:UP(x,INT):=(1-6*x+9*x^2)^2|
+
+leadingCoefficient p $\rightarrow$ 18
+
+degree p $\rightarrow$ 3
+
+reductum p $\rightarrow$ $60x^2-46x+8$
+
+gcd(p,q) $\rightarrow$ $9x^2-6x+1$
+
+lcm(p,q) $\rightarrow$ $162x^5+432x^4-756x^3+408x^2-94x+8$
+
+resultant(p,q) $\rightarrow$ 0
+
+p(2) $\rightarrow$ 300 (used as function)
+
+D(p) $\rightarrow$ $54x^2+120x-46$ (derivative)
+
+%*********************************************
+\hr\textbf{Vector}
+
+creation: \verb|v := vector([1,2,3,4,5])| $\rightarrow$ $[1,2,3,4,5]$
+
+length: \verb|#v| $\rightarrow$ 5
+
+access: v.2 $\rightarrow$ 2
+
+add: v+v $\rightarrow$ $[2,4,6,8,10]$
+
+multiply: 5*v $\rightarrow$ $[5,10,15,20,25]$
+
+assign: v.2 := 7 $\rightarrow$ $[1,7,3,4,5]$
+
+\end{multicols*}
+
+\end{document}
\ No newline at end of file



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To: Martin Rubey <martin.rubey@univie.ac.at>
Subject: Re: [Axiom-developer] how does the interpreter choose signatures?
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I understand a tiny little bit now.  There is a function orderMms in
i-funsel.boot that orders the pre-selected signatures by cost, which is
computed in mmCost.  In the case of integrate(p, s), the costs turn out to be
all the same, namely 86999.  In the case of foo$TEST, they are

|cost: |
(((|Test|) (|Expression| (|Integer|)) (|Expression| (|Integer|)))
 ((|Expression| (|Integer|)) (|Expression| (|Integer|))) (NIL))
53600
|cost: |
(((|Test|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|)))
 ((|Fraction| (|Integer|)) (|Fraction| (|Integer|))) (NIL))
52500
|cost: |
(((|Test|) (|PrimeField| 5) (|PrimeField| 5))
 ((|PrimeField| 5) (|PrimeField| 5)) (NIL))
51500

I do not understand yet how cost is assigned to the various signatures, but the
results look slightly fishy to me...

Martin

Martin Rubey <martin.rubey@univie.ac.at> writes:

> Dear all,
> 
> I'd like to understand the following behaviour.  Consider
> 
> -------------------------------------------------------------------------------
> )abb package TEST Test
> Test(): with
> 
>      foo: Fraction Integer -> Fraction Integer
>      foo: PrimeField 5 -> PrimeField 5
>      foo: Expression Integer -> Expression Integer
> 
>   == add
> 
>     foo(n: Fraction Integer) == n
>     foo(n: PrimeField 5) == n
>     foo(n: Expression Integer) == n
> -------------------------------------------------------------------------------
> 
> Then we have:
> 
> (4) -> foo 3
> 
>  Function Selection for foo
>       Arguments: PI 
>    -> no appropriate foo found in PositiveInteger 
>    -> no appropriate foo found in Integer 
>    -> no appropriate foo found in PositiveInteger 
>    -> no appropriate foo found in Integer 
> 
>  Modemaps from Associated Packages 
>    no modemaps
> 
>  Remaining General Modemaps 
>    [1] Expression Integer -> Expression Integer from Test
>    [2] PrimeField 5 -> PrimeField 5 from Test
>    [3] Fraction Integer -> Fraction Integer from Test
>  
>  [1]  signature:   PF 5 -> PF 5
>       implemented: slot (PrimeField 5)(PrimeField 5) from TEST
>  [2]  signature:   FRAC INT -> FRAC INT
>       implemented: slot (Fraction (Integer))(Fraction (Integer)) from TEST
>  [3]  signature:   EXPR INT -> EXPR INT
>       implemented: slot (Expression (Integer))(Expression (Integer)) from TEST
>  
> 
>    (4)  3
>                                                            Type: PrimeField 5
> -------------------------------------------------------------------------------
> 
> Note that the interpreter seems to prefer PF 5 over FRAC INT over EXPR INT.
> Now consider:
> 
> p := z^10; s:=z=0..1; 
> 
> (6) -> integrate(p, s)
> 
>  Function Selection for integrate
>       Arguments: (POLY INT,SEGBIND NNI) 
>    -> no appropriate integrate found in Polynomial Integer 
>    -> no appropriate integrate found in SegmentBinding NonNegativeInteger 
>    -> no appropriate integrate found in Polynomial Integer 
>    -> no appropriate integrate found in SegmentBinding NonNegativeInteger 
> 
>  Modemaps from Associated Packages 
>    [1] (Polynomial D2,Symbol) -> Polynomial D2 from Polynomial D2
>             if D2 has ALGEBRA FRAC INT and D2 has RING
> 
>  Remaining General Modemaps 
>    [1] (D,D1) -> D from D
>             if D1 = SYMBOL and D has UTSCAT D2 and D2 has RING and D2 
>             has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
>             has ALGEBRA FRAC INT or D1 = SYMBOL and D has UTSCAT D2 and
>             D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
>             integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
>    [2] (D,D1) -> D from D
>             if D1 = SYMBOL and D has UPXSCAT D2 and D2 has RING and D2 
>             has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
>             has ALGEBRA FRAC INT or D1 = SYMBOL and D has UPXSCAT D2 
>             and D2 has RING and D2 has variables: D2 -> List D1 and D2 
>             has integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
>          
>    [3] (D,D1) -> D from D
>             if D1 = SYMBOL and D has ULSCAT D2 and D2 has RING and D2 
>             has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 
>             has ALGEBRA FRAC INT or D1 = SYMBOL and D has ULSCAT D2 and
>             D2 has RING and D2 has variables: D2 -> List D1 and D2 has 
>             integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT
>    [4] (D,D1) -> D from D
>             if D has MTSCAT(D2,D1) and D2 has RING and D1 has ORDSET 
>             and D2 has ALGEBRA FRAC INT
>    [5] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List 
>             Expression D4)
>             from IntegrationResultRFToFunction D4
>             if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo 
>             Integer,OrderedSet,LinearlyExplicitRingOver Integer)
>    [6] (GeneralUnivariatePowerSeries(D2,D3,D4),Variable D3) -> 
>             GeneralUnivariatePowerSeries(D2,D3,D4)
>             from GeneralUnivariatePowerSeries(D2,D3,D4)
>             if D3: SYMBOL and D2 has ALGEBRA FRAC INT and D2 has RING 
>             and D4: D2
>    [7] (D2,Symbol) -> Union(D2,List D2) from FunctionSpaceIntegration(
>             D4,D2)
>             if D4 has Join(EuclideanDomain,OrderedSet,
>             CharacteristicZero,RetractableTo Integer,
>             LinearlyExplicitRingOver Integer) and D2 has Join(
>             TranscendentalFunctionCategory,PrimitiveFunctionCategory,
>             AlgebraicallyClosedFunctionSpace D4)
>    [8] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
>             Fraction Polynomial D4) -> Union(f1: OrderedCompletion Expression
>             D4,f2: List OrderedCompletion Expression D4,fail: failed,
>             pole: potentialPole)
>             from RationalFunctionDefiniteIntegration D4
>             if D4 has Join(EuclideanDomain,OrderedSet,
>             CharacteristicZero,RetractableTo Integer,
>             LinearlyExplicitRingOver Integer)
>    [9] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
>             Expression D4) -> Union(f1: OrderedCompletion Expression D4,f2: 
>             List OrderedCompletion Expression D4,fail: failed,pole: 
>             potentialPole)
>             from RationalFunctionDefiniteIntegration D4
>             if D4 has Join(EuclideanDomain,OrderedSet,
>             CharacteristicZero,RetractableTo Integer,
>             LinearlyExplicitRingOver Integer)
>    [10] (D2,SegmentBinding OrderedCompletion D2) -> Union(f1: 
>             OrderedCompletion D2,f2: List OrderedCompletion D2,fail: failed,
>             pole: potentialPole)
>             from ElementaryFunctionDefiniteIntegration(D4,D2)
>             if D2 has Join(TranscendentalFunctionCategory,
>             PrimitiveFunctionCategory,AlgebraicallyClosedFunctionSpace 
>             D4) and D4 has Join(EuclideanDomain,OrderedSet,
>             CharacteristicZero,RetractableTo Integer,
>             LinearlyExplicitRingOver Integer)
>  
>  [1]  signature:   (EXPR INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Expression (Integer))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTEF(INT,EXPR INT)
>  [2]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTRF INT
>  [3]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Fraction (Polynomial (Integer))))) from DEFINTRF INT
>  
> 
>  Function Selection for map by coercion facility (map) 
>       Arguments: ((NNI -> ORDCOMP EXPR INT),SEGBIND NNI) 
>       Target type: SEGBIND ORDCOMP EXPR INT 
>    -> no appropriate map found in SegmentBinding NonNegativeInteger 
>    -> no appropriate map found in SegmentBinding OrderedCompletion Expression Integer 
>    -> no appropriate map found in OrderedCompletion Expression Integer 
>    -> no appropriate map found in NonNegativeInteger 
>    -> no appropriate map found in OrderedCompletion Expression Integer 
> 
>  Modemaps from Associated Packages 
>    [1] ((D4 -> D5),SegmentBinding D4) -> SegmentBinding D5
>             from SegmentBindingFunctions2(D4,D5)
>             if D4 has TYPE and D5 has TYPE
>  
>  [1]  signature:   ((NNI -> ORDCOMP EXPR INT),SEGBIND NNI) -> SEGBIND ORDCOMP EXPR INT
>       implemented: slot (SegmentBinding (OrderedCompletion (Expression (Integer))))(Mapping (OrderedCompletion (Expression (Integer))) (NonNegativeInteger))(SegmentBinding (NonNegativeInteger)) from SEGBIND2(NNI,ORDCOMP EXPR INT)
>  
> 
>          1
>    (6)  --
>         11
>                     Type: Union(f1: OrderedCompletion Expression Integer,...)
> 
> There are many things I do not understand:
> 
> 1) why doesn't the interpreter choose the "General Modemap"
> 
>    [8] (Fraction Polynomial D4,SegmentBinding OrderedCompletion 
>             Fraction Polynomial D4)
> 
>   ?
> 
> 2) why is its preference different now:
> 
> 
>  [1]  signature:   (EXPR INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Expression (Integer))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTEF(INT,EXPR INT)
>  [2]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP EXPR INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Expression (Integer)))) from DEFINTRF INT
>  [3]  signature:   (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT) -> Union(f1: ORDCOMP EXPR INT,f2: LIST ORDCOMP EXPR INT,fail: failed,pole: potentialPole)
>       implemented: slot (Union (: f1 (OrderedCompletion (Expression (Integer)))) (: f2 (List (OrderedCompletion (Expression (Integer))))) (: fail failed) (: pole potentialPole))(Fraction (Polynomial (Integer)))(SegmentBinding (OrderedCompletion (Fraction (Polynomial (Integer))))) from DEFINTRF INT
>  
> I.e., in package DEFINTRF, it seems to prefer (FRAC POLY INT,SEGBIND ORDCOMP
> EXPR INT) over (FRAC POLY INT,SEGBIND ORDCOMP FRAC POLY INT).
> 
> 3) why does it prefer DEFINTEF(INT,EXPR INT) over DEFINTRF INT ?
> 
> 
> ANY hints?
> 
> Martin
> 
> 
> 
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer




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From: "Bill Page" <bill.page@newsynthesis.org>
To: "William Sit" <wyscc@sci.ccny.cuny.edu>
Subject: Re: [Axiom-developer] Re: Axiom: fixed and fixed proposed bit vectors
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William,

Unfortunately your proposal below is quite dissonant with how the
current ZWiki/Zoped-based Axiom Wiki actually works. Unlike some of
the other options discussed by Martin, implementing this cannot be
done simply by adding new entries to a table of allowed values for
various status fields already defined the issuetracker object. Instead
I think it would require some deep changes to the underlying python
code which are not a likely to be generally applicable to other users
of the ZWiki package who use this same issuetracker code. But of
course if someone is willing to make and maintain such changes to the
software that we use for Axiom Wiki, I am quite willing to discuss it
further.

Regards,
Bill Page.

On 1/27/08, William Sit <wyscc@sci.ccny.cuny.edu> wrote:
> I consider "fixed xxx" and "fixed proposed" as two
> statuses, not 16 (but that is up to your interpretation
> what counts as a status). One reason for proposing this
> "cryptic" version is to enable automatic sorting. One can
> write a program to search "fixed" and "fixed proposed" and
> then parse xxx to filter out those that are fixed or not
> fixed, or proposed or not proposed for any of the flavors.
> This program will allow users to do searches and be
> presented only with a particular xxx view.
>
> As long as the bit vector is documented, it is no more
> cryptic than other code. A bit vector is the most
> efficient of data representation. Since each axiom has its
> own bit, it is easy to even spot the status visually.
>
> If you use comments after "fixed" and "fixed proposed",
> someone will have to read the comments manually and sort
> the bug reports manually. It would be better to know ahead
> of reading the comments what one is about to read.
>
> In the best scenario, the code xxx would only go through 4
> values (starting with 000 and increasing by changing a 0
> to a 1, at most three times).
>
>
> William
>
> On Sat, 26 Jan 2008 08:54:55 -0500
>   "Bill Page" <bill.page@newsynthesis.org> wrote:
> >I must say that I do not like the sound of "fixed
> >somewhere" - it
> >sounds too vague - but I understand the intent. Maybe it
> >would be
> >better to write:
> >
> >   "fixed as noted ..."
> >
> >And then expect to find a comment saying in which
> >versions/forks the
> >fix occurs. This is almost like you are doing now except
> >for the
> >distinction between "as noted" and "proposed" (patch
> >attached to the
> >bug report).
> >
> >I don't think William intended 7 different statuses. The
> >parameter can
> >be included as a comment. But I do not see any point in
> >being so
> >cryptic.
> >
> >Regards,
> >Bill Page.
> >
> >On 26 Jan 2008 09:47:43 +0100, Martin Rubey wrote:
> >> Martin Rubey <martin.rubey@univie.ac.at> writes:
> >>
> >> > Yes, I like this proposal equally well.  Comments?
> >>
> >> Oops, sorry, I don't understand it:  it would imply
> >>having 7
> >> additional statuses too.  I don't think it's possible to
> >>have
> >> a "variable" status?
> >>
> >> The point is being able to filter out all issues that
> >>are
> >> 'fixed somewhere.'
> >>
> >> Martin
> >> >
> >> > "William Sit" <wyscc@sci.ccny.cuny.edu> writes:
> >> >
> >> > > Dear Martin:
> >> > >
> >> > > May I suggest using a bit vector to notate the
> >>fixes? I propose
> >> > > that the bits of the bit vector be assigned
> >>according to chronological
> >> > > order of axiom branches, so that bit 0 is for Tim's
> >>Axiom, bit 1 for
> >> > > FriCAS and bit 2 for OpenAxiom, etc. So we need only
> >>two status:
> >> > > fixed xxx and fixed proposed xxx for now. The bit
> >>vector is extensible
> >> > > if more flavors come along.
> >> > >
> >> > > Example: "fixed 101" means patches exist and
> >>implemented for
> >> > > OpenAxiom and for Tim's Axiom, each patch according
> >>to the
> >> > > requirements of the flavor.
> >> > >
> >> > > "fixed proposed 010" means a patch exist (proposed)
> >>but not
> >> > > implemented (for whatever reasons) on FriCAS.
> >> > >
> >> > > "fixed 000" means no patch implemented
> >> > >
> >> > > "fixed proposed 000" means no patch proposed
> >>(implies of
> >> > > course "fixed 000").
> >> > >
> >> > > You may forward this to the groups or disregard as
> >>you
> >> > > please.
> >> > >
> >> > > William
> >> > >
> >> > >
> >> > > On 25 Jan 2008 22:11:55 +0100
> >> > >   Martin Rubey <martin.rubey@univie.ac.at> wrote:
> >> > > >I have just added a new bug status "fixed
> >>somewhere".
> >> > > >
> >> > > > Please use "fix proposed" and "fixed somewhere"
> >>according
> >> > > > to the following rules:
> >> > > >
> >> > > >> 1) fix proposed
> >> > > >>    would be used if a patch is attached, but it
> >>hasn't been
> >> > > >> fixed in one of the systems
> >> > > >> 2) fixed somewhere would be used if the issue is
> >>fixed
> >> > > >> somewhere, regardless whether a patch is included
> >>or
> >> > > >> not.
> >> > > >> ...
> >
> >
> >_______________________________________________
> >Axiom-developer mailing list
> >Axiom-developer@nongnu.org
> >http://lists.nongnu.org/mailman/listinfo/axiom-developer
>
> William Sit,
> Professor of Mathematics, City College of New York
>
> Office: R6/202C Tel: 212-650-5179, Fax: 212-862-0004
> Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
>



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I just installed Axiom from the SVN repository. When I start it, the 
"standard" Hyperdoc comes up. How do I get it to start up the Firefox 
version?



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>I just installed Axiom from the SVN repository. When I start it, the 
>"standard" Hyperdoc comes up. How do I get it to start up the Firefox 
>version?

The new rootpage lives in $AXIOM/doc/hypertex/rootpage.xhtml

So if we assume that $AXIOM is something like /usr/local/axiom/mnt/fedora5
you do two steps:

1) start axiom and type
    )browse
 
   you should see a message about listening on port 8085

2) start firefox and go to your local machine with:

http://127.0.0.1:8085/usr/local/axiom/mnt/fedora5/doc/hypertex/rootpage.xhtml

   (that is, http://127.0.0.1:8085/$AXIOM/doc/hypertex/rootpage.xhtml
    but the browser does not understand shell variables)

   and you should see the new rootpage. Note that the page name ends in
   ".xhtml", not ".html"

Future releases will know the location and the URL will just include
the pagename, not the path.

To exit browse, just interrupt Axiom.  Note that the socket is not
properly closed so you cannot yet restart the )browse in the same
session once you exit. This will be fixed shortly.

Tim



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>I just installed Axiom from the SVN repository. When I start it, the 
>"standard" Hyperdoc comes up. How do I get it to start up the Firefox 
>version?

Also note that:

o Static versions of these pages exist online at
  http://axiom.axiom-developer.org/axiom-website/hyperdoc/rootpage.xhtml

o not all of the hyperdoc pages have been converted yet
o there are new pages which do not exist in the old hyperdoc
o the Topics branch contains the most number of converted pages.
o the Jenks book branch is a Arthur's mathml version of the book.
o the Commands branch is in transition to act like a full Axiom console
  so the code there is minimal. Ultimately I expect it to be similar to
  the Sage notebook-like interface. 
o the Fonts link shows the characters Axiom uses. There is a file
    zips/axiomfonts.tgz
  that contains some of the fonts and instructions for finding 
  and installing others.

o You can write new pages. If you look at the bookvol11 with xdvi
    xdvi $AXIOM/doc/bookvol11.dvi
  the first chapter begins to explain how to do this.

Tim


Tim



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Tim,

When I build Axiom on OpenSuse I have to modify the
"Makefile.linux" section of Makefile.pamphlet to find the
X files.  /usr/X11R6/lib and /usr/X11R6/include are not
used, they need to be changed to /usr/lib and /usr/include.

Arthur



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>When I build Axiom on OpenSuse I have to modify the
>"Makefile.linux" section of Makefile.pamphlet to find the
>X files.  /usr/X11R6/lib and /usr/X11R6/include are not
>used, they need to be changed to /usr/lib and /usr/include.

Yes, that's also true for other distros. I have a branch for
that in the Makefile for the other distros. I'll add it for
OpenSuse.

Tim



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Peter,

Thank you very much for sending the revised version of src_aldor3.tgz!
I think eliminating the Java dependency is a very important thing if
we are to eventually include this as an optional part of the
Axiom/FriCAS/OpenAxiom build.  I am trying to use your revised build
system with OpenAxiom but I have some questions:

Could you explain the following lines in the makefile:

  echo ")co RNS.spad"  >> $@.lsp

I was not able to find 'RNS.spad' in the Axiom sources or in your
tarball. Perhaps you forgot to include it?

I notice also that there is still some mention of java in the makefile, e.g.

all_dirs += $(ALDOR_INT)/java_axiom

$(ALDOR_INT)/java_axiom/spadset.mk: $(ALDOR_INT)/java_axiom/.dir
$(ALDOR_INT)/java_axiom/spadset.mk: $(domain_dep_files) $(aldor_dep_files)
$(ALDOR_INT)/java_axiom/spadset.mk: $(ALDOR_INT)/java/Sort.class
$(ALDOR_INT)/java_axiom/spadset.mk: $(ALDOR_INT)/deps/axextend.dep
$(ALDOR_INT)/java_axiom/spadset.mk: $(ALDOR_INT)/libaxiom.lst
$(ALDOR_SRC)/extra_deps.lst
        (cd $(ALDOR_INT); java -cp `pwd`/java Sort makefile     \
                                        deps/axextend.dep       \
                                        libaxiom.lst            \
                                        $(ALDOR_SRC)/extra_deps.lst)

java_axiom: $(ALDOR_INT)/java_axiom/spadset.mk
.PHONY: java_axiom

...

#
# :: Java
#
all_dirs += $(ALDOR_INT)/java
$(ALDOR_INT)/java/Sort.class: $(ALDOR_INT)/java/.dir
$(ALDOR_INT)/java/Sort.class: $(ALDOR_SRC)/Sort.java
        javac -source 1.5 -d $(ALDOR_INT)/java Sort.java


---------

Am I misunderstanding or are these only a left-over stanza's from the
previous version which is not used any more?

Regards,
Bill Page.

On 1/17/08, Peter Broadbery wrote:
> Finally got round to looking at this.
> The code attached can be used to build the axiom/aldor interopability
> library (libaxiom.al).
>
> New features:
> Less abuse of make features (though I've only used a recent version of
> make, and have used some newer features)
> Dependency on java removed.
> Cleaner build process - all the dependency analysis is done in the
> aldor code.  It should be more data driven, but it isn't.
>
> There's a small aldor patch that I'll send along this evening.
>
> Peter
>
> _______________________________________________
> Axiom-developer mailing list
> Axiom-developer@nongnu.org
> http://lists.nongnu.org/mailman/listinfo/axiom-developer
>
>
>



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Peter,

Thanks for the explanation.

Where can I find the source for 'RNS.spad'?

Regards,
Bill Page.

On 1/30/08, Peter Broadbery <p.broadbery@googlemail.com> wrote:
> On Jan 30, 2008 1:48 AM, Bill Page <bill.page@newsynthesis.org> wrote:
> > Peter,
> >
> > Thank you very much for sending the revised version of src_aldor3.tgz!
> > I think eliminating the Java dependency is a very important thing if
> > we are to eventually include this as an optional part of the
> > Axiom/FriCAS/OpenAxiom build.  I am trying to use your revised build
> > system with OpenAxiom but I have some questions:
> >
> > Could you explain the following lines in the makefile:
> >
> >   echo ")co RNS.spad"  >> $@.lsp
> >
>
> This is needed to get axiom to autoload some functions required by
> later steps - I guess the clean thing would be to remove it and force
> the relevant bits manually
>
> > I was not able to find 'RNS.spad' in the Axiom sources or in your
> > tarball. Perhaps you forgot to include it?
> >
> > I notice also that there is still some mention of java in the makefile, e.g.
> >
> > ...
>
>
> Not used - the purpose was side by side comparison of java and aldor output.
> (the java_axiom directory isn't referenced anywhere)
>
> Peter
>



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On 1/30/08, Peter Broadbery wrote:
> On Jan 30, 2008 2:18 PM, Bill Page wrote:
> > Peter,
> >
> > Thanks for the explanation.
> >
> > Where can I find the source for 'RNS.spad'?
> >
> >
>
> I think you don't need it - the command is sufficient to load what's required.
>

Oh, ok. I think I see. You mean just the attempt to compile a spad
file is sufficient - even though there is an error message about the
missing file?

OpenAxiom (and FriCAS) have a somewhat different directory structure
than Axiom so I am having to look at more details than I otherwise
would.

Thanks again.

Regards,
Bill Page.



