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[Axiom-developer] 20080119.01.tpd.patch
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From: |
daly |
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Subject: |
[Axiom-developer] 20080119.01.tpd.patch |
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Date: |
Sat, 19 Jan 2008 08:27:36 -0600 |
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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