diff --git a/changelog b/changelog index d486296..cdd0688 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,6 @@ +20080131 tpd src/input/Makefile add ei.input for Ei regression test +20080131 tpd src/input/ei.input regression test function Ei +20080131 tpd src/algebra/special.spad add special function Ei 20080127 tpd src/doc/Makefile add refcard 20080127 tpd src/doc/refcard added 20080125 tpd --patch-55 (January 2008) release diff --git a/src/algebra/special.spad.pamphlet b/src/algebra/special.spad.pamphlet index 6b173ae..aab10e4 100644 --- a/src/algebra/special.spad.pamphlet +++ b/src/algebra/special.spad.pamphlet @@ -49,6 +49,36 @@ DoubleFloatSpecialFunctions(): Exports == Impl where En: (PI,R) -> OPR ++ En(n,x) is the nth Exponential Integral Function + Ei: (OPR) -> OPR + ++ Ei is the Exponential Integral function + ++ This is computed using a 6 part piecewise approximation. + ++ DoubleFloat can only preserve about 16 digits but the + ++ Chebyshev approximation used can give 30 digits. + + Ei1: (OPR) -> OPR + ++ Ei1 is the first approximation of Ei where the result is + ++ x*%e^-x*Ei(x) from -infinity to -10 (preserves digits) + + Ei2: (OPR) -> OPR + ++ Ei2 is the first approximation of Ei where the result is + ++ x*%e^-x*Ei(x) from -10 to -4 (preserves digits) + + Ei3: (OPR) -> OPR + ++ Ei3 is the first approximation of Ei where the result is + ++ (Ei(x)-log |x| - gamma)/x from -4 to 4 (preserves digits) + + Ei4: (OPR) -> OPR + ++ Ei4 is the first approximation of Ei where the result is + ++ x*%e^-x*Ei(x) from 4 to 12 (preserves digits) + + Ei5: (OPR) -> OPR + ++ Ei5 is the first approximation of Ei where the result is + ++ x*%e^-x*Ei(x) from 12 to 32 (preserves digits) + + Ei6: (OPR) -> OPR + ++ Ei6 is the first approximation of Ei where the result is + ++ x*%e^-x*Ei(x) from 32 to infinity (preserves digits) + 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)}. @@ -171,6 +201,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where @ \section{The Exponential Integral} +\subsection{The E1 function} (Quoted from Segletes\cite{2}): A number of useful integrals exist for which no exact solutions have @@ -313,7 +344,7 @@ 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 +work, with the use of Chebyshev\cite{18,19} 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 @@ -504,6 +535,1514 @@ The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}. w:R:=1/(n-1)*(exp(-x)-x*v) w::OPR +@ +\section{The Ei Function} +This function is based on Kin L. Lee's work\cite{8}. See also \cite{21}. +\subsection{Abstract} +The exponential integral Ei(x) is evaluated via Chebyshev series +expansion of its associated functions to achieve high relative +accuracy throughout the entire real line. The Chebyshev coefficients +for these functions are given to 30 significant digits. Clenshaw's\cite{20} +method is modified to furnish an efficient procedure for the accurate +solution of linear systems having near-triangular coefficient +matrices. +\subsection{Introduction} +The evaulation of the exponential integral +\begin{equation} +Ei(x)=\int_{-\infty}^{X}{\frac{e^u}{u}}\ du=-E_1(-x), x \ne 0 +\end{equation} +is usually based on the value of its associated functions, for +example, $xe^{-x}Ei(x)$. High accuracy tabulations of integral (1) by +means of Taylor series techniques are given by Harris \cite{9} and +Miller and Hurst \cite{10}. The evaluation of $Ei(x)$ for +$-4 \le x \le \infty$ by means of Chebyshev series is provided by +Clenshaw \cite{11} to have the absolute accuracy of 20 decimal +places. The evaluation of the same integral (1) by rational +approximation of its associated functions is furnished by Cody and +Thacher \cite{12,13} for $-\infty < x < \infty$, and has the relative +accuracy of 17 significant figures. + +The approximation of Cody and Thacher from the point of view of +efficient function evaluation are preferable to those of +Clenshaw. However, the accuracy of the latter's procedure, unlike +those of the former, is not limited by the accuracy or the +availability of a master function, which is a means of explicitly +evaluating the function in question. + +In this paper $Ei(x)$ (or equivalently $-E_1(-x)$) for the entire real +line is evaluted via Chebyshev series expansion of its associated +functions that are accurate to 30 significant figures by a +modification of Clenshaw's procedure. To verify the accuracy of the +several Chebyshev series, values of the associated functions were +checked against those computed by Taylor series and those of Murnaghan +and Wrench \cite{14} (see Remarks on Convergence and Accuracy). + +Although for most purposes fewer than 30 figures of accuracy are +required, such high accuracy is desirable for the following +reasons. In order to further reduce the number of arithmetical +operations in the evaluation of a function, the Chebyshev series in +question can either be converted into a rational function or +rearranged into an ordinary polynomial. Since several figures may be +lost in either of these procedures, it is necessary to provide the +Chebyshev series with a sufficient number of figures to achieve the +desired accuracy. Furthermore, general function approximation +routines, such as those used for minimax rational function +approximations, require the explicit evaluation of the function to be +approximated. To take account of the errors commited by these +routines, the function values must have an accuracy higher than the +approximation to be determined. Consequently, high-precision results +are useful as a master function for finding approximations for (or +involving) $Ei(x)$ (e.g. \cite{12,13}) where prescribed accuracy is +less than 30 figures. + +\subsection{Discussion} + +It is proposed here to provide for the evaluation of $Ei(x)$ by +obtaining Chebyshev coefficients for the associated functions given by +table 1. + +\noindent +{\bf Table 1}: Associated Functions of $Ei(x)$ and their ranges of Chebyshev +Series Expansions + +\begin{tabular}{clc} +& Associated function & Range of expansion\\ +Ei1 & $xe^{-x}Ei(x)$ & $-\infty < x \le -10$\\ +Ei2 & $xe^{-x}Ei(x)$ & $-10 \le x \le -4$\\ +Ei3 & $\frac{Ei(x)-log\vert x\vert - \gamma}{x}$ & $-4 \le x \le d42$\\ +Ei4 & $xe^{-x}Ei(x)$ & $4 \le x \le 12$\\ +Ei5 & $xe^{-x}Ei(x)$ & $12 \le x \le 32$\\ +Ei6 & $xe^{-x}Ei(x)$ & $32 \le x < \infty$\\ +\end{tabular}\\ +\hbox{\hskip 2cm}($\gamma$ = 0.5772156649... is Euler's constant.) + +<>= + + Ei(y:OPR):OPR == + infinite? y => 1 + x:R:=retract(y) + x < -10.0::R => + ei:R:=retract(Ei1(y)) + (ei/(x*exp(-x)))::OPR + x < -4.0::R => + ei:R:=retract(Ei2(y)) + (ei/(x*exp(-x)))::OPR + x < 4.0::R => + ei3:R:=retract(Ei3(y)) + gamma:R:=0.577215664901532860606512090082::R + (ei3*x+log(abs(x))+gamma)::OPR + x < 12.0::R => + ei:R:=retract(Ei4(y)) + (ei/(x*exp(-x)))::OPR + x < 32.0::R => + ei:R:=retract(Ei5(y)) + (ei/(x*exp(-x)))::OPR + ei:R:=retract(Ei6(y)) + (ei/(x*exp(-x)))::OPR + +@ +Note that the functions $[Ei(x)-log\vert x\vert - \gamma]/x$ and +$xe^{-x}Ei(x)$ have the limiting values of unity at the origin and at +infinity, respectively, and that the range of the associated function +values is close to unity (see table 4). This makes for the evaluation +of the associated functions over the indicated ranges in table 1 (and +thus $Ei(x)$ over the entire real line) with high relative accuracy by +means of the Chebyshev series. The reason for this will become +apparent later. + +Some remarks about the choice of the intervals of expansion for the +serveral Chebyshev series are in order here. The partition of the real +line indicated by table 1 is chosen to allow for the approximation of +the associated functions with a maximum error of $0.5\times 10^{-30}$ +by polynomials of degress $< 50$. The real line has also been +partitioned with the objective of providing the interval about zero +with the lowest degree of polynomial approximation of the six +intervals. This should compensate for the computation of +$log\vert x\vert$ required in the evaluation of $Ei(x)$ over that +interval. The ranges $-\infty < x \le -4$ and $4 \le x < \infty$ are +partitioned into 2 and 3 intervals, respectively, to provide +approximations to $xe^{-x}Ei(x)$ by polynomials of about the same +degree. + +\subsection{Expansions in Chebyshev Series} + +Let $\phi(t)$ be a differentiable function defined on [-1,1]. To +facilitate discussion, denote its Chebyshev series and that of its +derivative by +\begin{equation} +\phi(t)=\sum_{k=0}^{\infty}\ ^{'}{A_k^{(0)}T_k(t)}\quad +\phi^{'}(t)=\sum_{k=0}^{\infty}\ ^{'}{A_k^{(1)}T_k(t)} +\end{equation} +where $T_k(t)$ are Chebyshev polynomials defined by +\begin{equation} +T_k(t)=cos(k\ \arccos\ t),\quad -1 \le t \le 1 +\end{equation} +(A prime over a summation sign indicates that the first term is to be +halved.) + +If $\phi(t)$ and $\phi^{'}(t)$ are continuous, the Chebyshev +coefficients $A_k^{(0)}$ and $A_k^{(1)}$ can be obtained analytically +(if possible) or by numerical quadrature. However, since each function +in table 1 satisfies a linear differential equation with polynomial +coefficients, the Chebyshev coefficients can be more readily evaluated +by the method of Clenshaw \cite{16}. + +There are several variations of Clenshaw's procedure (see, +e.g. \cite{17}), but for high-precision computation, where multiple +precision arithmetic is employed, we find his original procedure +easiest to implement. However, straightforward application of it may +result in a loss of accuracy if the trial solutions selected are not +sufficiently independent. How the difficulty is overcome will be +pointed out subsequently. + +\subsection{The function $xe^{-x}Ei(x)$ on the Finite Interval} + +We consider first the Chebyshev series expansion of +\begin{equation} +f(x)=xe^{-x}Ei(x),\quad (a \le x \le b) +\end{equation} +with $x\ne 0$. One can easily verify that after the change of +variables +\begin{equation} +x=[(b-a)T + a + b]/2,\quad(-1 \le t \le 1) +\end{equation} +the function +\begin{equation} +\phi(t)=f\left[\frac{(b-a)t + a + b}{2}\right]=f(x) +\end{equation} +satisfies the differential equation +\begin{equation} +2(pt+q)\phi^{'}(t)+p(pt+q-2)\phi(t)=p(pt+q) +\end{equation} +with\footnote{The value of $Ei(a)$ may be evaluated by means of the +Taylor series. In this report $Ei(a)$ is computed by first finding the +Chebyshev series approximation to $[Ei(x)-log\vert x\vert-\gamma]/x$ +to get $Ei(a)$. The quantities $e^a$ and $\log\vert a\vert$ for +integral values of $a$ may be found in existing tables} +\begin{equation} +\phi(-1)=ae^{-a}Ei(a) +\end{equation} +where $p=b-a$ and $q=b+a$. Replacing $\phi(t)$ and $\phi^{'}(t)$ in +equations 7 by their Chebyshev series, we obtain +\begin{equation} +\sum_{k=0}^{\infty}\ ^{'}{(-1)^kA_k^{(0)}}=\phi(-1) +\end{equation} +\begin{equation} +2\sum_{k=0}^{\infty}\ {'}{A_k^{(1)}(pt+q)T_k(t)}+ +p\sum_{k=0}^{\infty}\ {'}{A_k^{(0)}(pt+q-2)T_k(t)}= +p(pt+q) +\end{equation} +It can be demonstrated that if $B_k$ are the Chebyshev coefficients of +a function $\Psi(t)$, then $C_k$, the Chebyshev coefficients of +$t^r\Psi(t)$ for positive integers r, are given by \cite{16} +\begin{equation} +C_k=2^{-r}\sum_{i=0}^r\binom{r}{i}B_{\vert k-r+2i\vert} +\end{equation} +Consequently, the left member of equation 15 can be rearranged into a +single series involving $T_k(t)$. The comparison of the coefficients +of $T_k(t)$ that yields the infinite system of equations +\begin{equation} +\left. +\begin{array}{c} +\displaystyle\sum_{k=0}^{\infty}\ ^{'}{(-1)^kA_k^{(0)}} = \phi(-1)\\ +\\ +2pA_{\vert k-1\vert}^{(1)}+ +4qA_k^{(1)}+ +2pA_{k+1}^{(1)}+ +p^2A_{\vert k-1\vert}^{(0)}+ +2p(q-2)A_k^{(0)}+ +p^2A_{k+1}^{(0)}\\ +\\ +=\left\{ +\begin{array}{rcl} +4pq & , & k=0\\ +2p^2 & , & k=1\\ +0 & , & k=2,3,\ldots +\end{array} +\right. +\end{array} +\right\} +\end{equation} +The relation \cite{16} +\begin{equation} +2kA_k^{(0)}=A_{k-1}^{(1)}-A_{k+1}^{(1)} +\end{equation} +can be used to reduce equation 18 to a system of equations involving +only $A_k^{(0)}$. Thus, replacing $k$ of equations 18 by $k+2$ and +subtracting the resulting equation from equations 18, we have, by +means of equation 19, the system of equations +\begin{equation} +\left. +\begin{array}{c} +\displaystyle\sum_{k=0}^{\infty}\ ^{'}{(-1)^kA_k^{(0)}} = \phi(-1)\\ +\\ +2p(q-2)A_0+(8q+p^2)A_1+2p(6-q)A_2-p^2A_3=4pq\\ +\\ +p^2A_{k-1}+2p(2k+q-2)A_k+8q(k+1)A_{k+1}+2p(2k-q+6)A_{k+2}-p^2A_{k+3}\\ +\\ +=\left\{ +\begin{array}{rcl} +2p^2 & , & k=1\\ +0 & , & k=2,3,\ldots +\end{array} +\right. +\end{array} +\right\} +\end{equation} +The superscript of $A_k^{(0)}$ is dropped for simplicity. In order to +solve the infinite system 20, Clenshaw \cite{11} essentially +considered the required solution as the limiting solution of the +sequence of truncated systems consisting of the first $M+1$ equations +of the same system, that is, the solution of the system +\begin{equation} +\sum_{k=0}^M\ ^{'}{(-1)^kA_k}=\phi(-1) +\end{equation} +\begin{equation} +2p(q-2)A_0+(8q+p^2)A_1+2p(q-6)A_2-p^2A_3=4pq +\end{equation} +\begin{equation} +\left. +\begin{array}{c} +p^2A_{k-1}+2p(2k+q-2)A_k+8q(k+1)A_{k+1}+2p(2k-q+6)A_{k+2}-p^2A_{k+3}\\ +\\ +=\left\{ +\begin{array}{rcl} +2p^2 & , & k=1\\ +0 & , & k=2,3,\ldots,M-3 +\end{array} +\right.\\ +\\ +p^2A_{M-3}+2p(2M+q-6)A_{M-2}+8q(M-1)A_{M-1}+2p(2M+4-q)A_M=0\\ +\\ +p^2A_{M-2}+2p(2M+q-4)A_{M-1}+8qMA_M=0 +\end{array} +\right\} +\end{equation} +where $A_k$ is assumed to vanish for $K \ge M+1$. To solve system +(21,22,23) consider first the subsystem 23 consisting of $M-2$ +equations in $M$ unknowns. Here use is made of the fact that the +subsystem 23 is satisfied by +\begin{equation} +A_k=c_1\alpha_k+c_2\beta_k+\gamma_k\quad(k=0,1,2,\ldots) +\end{equation} +for arbitrary constants $c_1$ and $c_2$, where $\gamma_k$ is a +particular solution of 23 and where $\alpha_k$ and $\beta_k$ are two +independent solutions of the homogeneous equations (23 with $2p^2$ +deleted) of the same subsystem. Hence, if $\alpha_k$, $\beta_k$, and +$\gamma_k$ are available, the solution of system (21,22,23) reduces to +the determinant of $c_1$ and $c_2$ from equations 21 and 22. + +To solve equations (21,22,23), we note that +\begin{equation} +\gamma_0=2,\quad \gamma_k=0,\quad {\textrm for\ }k=1(1)M +\end{equation} +is obviously a particular solution of equation 23. The two independent +solutions $\gamma_k$ and $\beta_k$ of the homogeneous equations of the +same subsystem can be generated in turn by backward recurrence if we +set +\begin{equation} +\left. +\begin{array}{l} +\hbox{\hskip 4cm}\alpha_{M-1}=0,\quad\alpha_M=1\\ +\textrm{and}\\ +\hbox{\hskip 4cm}\beta_{M-1}=1,\quad\beta_M=0\\ +\end{array} +\right\} +\end{equation} +or choose any $\alpha{M-1}$, $\alpha_M$, and $\beta_{M-1}$, $\beta_M$ +for which $\alpha_{M-1}\beta_M-\alpha_M\beta_{M-1}\ne 0$. The +arbitrary constants $c_1$ and $c_2$ are determined, and consequently +the solution of equations (21,22,23) if equation 24 is substituted +into equation 21 and 22 and the resulting equations +\begin{equation} +c_1R(\alpha)+c_2R(\beta)=\phi(-1)-1 +\end{equation} +\begin{equation} +c_1S(\alpha)+c_2S(\beta)=8p +\end{equation} +are solved as two equations in two unknowns. The terms $R(\alpha)$ and +$S(\alpha)$ are equal, respectively, to the left members of equations +21 and 22 corresponding to solution $\alpha_k$. (The identical +designation holds for $R(\beta)$ and $S(\beta)$.) + +The quantities $\alpha_k$ and $\beta_k$ are known as trial solutions +in reference \cite{12}. Clenshaw has pointed out that if $\alpha_k$ +and $\beta_k$ are not sufficiently independent, loss of significance +will occur in the formation of the linear combination 24, with +consequent loss of accuracy. Clenshaw suggested the Gauss-Seidel +iteration procedure to improve the accuracy of the solution. However, +this requires the application of an additional computing procedure and +may prove to be extremely slow. A simpler procedure which does not +alter the basic computing scheme given above is proposed here. The +loss of accuracy can effectively be regained if we first generate a +third trial solution $\delta_k$ (k=0,1,$\ldots$,M), where +$\delta_{M-1}$ and $\delta_M$ are equal to +$c_1\alpha_{M-1}+c_2\beta_{M-1}$ and +$c_1\alpha_M+c_2\beta_M$, respectively, and where $\delta_k$ +(k=M-2,M-3,$\ldots$,0) is determined using backward recurrence as +before by means of equation 23. Then either $\alpha_k$ or $\beta_k$ is +replaced by $\delta_k$ and a new set of $c_1$ and $c_2$ is determined +by equations 27 and 28. Such a procedure can be repeated until the +required accuracy is reached. However, only one application of it was +necessary in the computation of the coefficients of this report. + +As an example, consider the case for $4 \le x \le 12$ with $M=15$. The +right member of equation 27 and of equation 28 assume, respectively, +the values of $0.43820800$ and $64$. The trial solutions $\alpha_k$ +and $\beta_k$ generated with $\alpha_{14}=8$, $\alpha_{15}=9$ and +$\beta_{14}=7$, $\beta_{15}=8$ are certainly independent, since +$\alpha_{14}\beta_{15}-\alpha_{15}\beta_{14}=1\ne 0$. A check of table +2 shows that equations 27 and 28 have, respectively, the residuals of +$-0.137\times 10^-4$ and $-0.976\times 10^{-3}$. The same table also +shows that $c_1\alpha_k$ is opposite in sign but nearly equal in +magnitude to $c_2\beta_k$. Cancellations in the formation of the +linear combination 24 causes a loss of significance of 2 to 6 figures +in the computed $A_k$. In the second iteration, where a new set of +$\beta_k$ is generated replacing $\beta_{14}$ and $\beta_{15}$, +respectively, by $c_1\alpha_{14}+c_2\beta_{14}$ and +$c_1\alpha_{15}+c_2\beta_{15}$ of the first iteration, the new +$c_1\alpha_k$ and $c_2\beta_k$ differed from 2 to 5 orders of +magnitude. Consequently, no cancellation of significant figures in the +computation of $A_k$ occurred. Notice that equations 27 and 28 are now +satisfied exactly. Further note that the new $c_1$ and $c_2$ are near +zero and unity, respectively, for the reason that if equations 21, 22, +and 23 are satisfied by equation 24 exactly in the first iteration, +the new $c_1$ and $c_2$ should have the precise values zero and 1, +respectively. The results of the third iteration show that the $A_k$ +of the second iteration are already accurate to eight decimal places, +since the $A_k$ in the two iterations differ in less that +$0.5\times 10^{-8}$. Notice that for the third iteration, equations +27 and 28 are also satisfied exactly and that $c_1=1$ and $c_2=0$ +(relative to 8 places of accuracy). + +\noindent +{\bf Table 2}: Computation of Chebyshev Coefficients for $xe^{-x}Ei(x)$ + +\hrule +First iteration: $\alpha_{14}=8$, $\alpha_{15}=9$; $\beta_{14}=7$, +$\beta_{15}=8$ +\hrule +\begin{tabular}{|r|r|r|r|} +k & $c_1\alpha_k\hbox{\hskip 1cm}$ & $c_2\beta_k\hbox{\hskip 1cm}$ & +$A_k\hbox{\hskip 1cm}$\\ +&&&\\ + 0 & 0.71690285E 03 & -0.71644773E 03 & 0.24551200E 01\\ + 1 & -0.33302683E 03 & 0.33286440E 03 & -0.16243000E 00\\ + 2 & 0.13469341E 03 & -0.13464845E 03 & 0.44960000E-01\\ + 3 & -0.43211869E 02 & 0.43205127E 02 & -0.67420000E-02\\ + 4 & 0.99929173E 01 & -0.99942238E 01 & -0.13065000E-02\\ + 5 & -0.11670764E 01 & 0.11684574E 01 & 0.13810000E-02\\ + 6 & -0.25552137E 00 & 0.25493635E 00 & -0.58502000E-02\\ + 7 & 0.20617247E 00 & -0.20599754E 00 & 0.17493000E-03\\ + 8 & -0.75797238E-01 & 0.75756767E-01 & -0.40471000E-04\\ + 9 & 0.20550680E-01 & -0.20543463E-01 & 0.72170000E-05\\ +10 & -0.45192333E-02 & 0.45183721E-02 & -0.86120000E-06\\ +11 & 0.82656562E-03 & -0.82656589E-03 & -0.27000000E-09\\ +12 & -0.12333571E-03 & 0.12337366E-03 & 0.37950000E-07\\ +13 & 0.13300910E-04 & -0.13315328E-04 & -0.14418000E-07\\ +14 & -0.29699001E-06 & 0.30091136E-06 & 0.39213500E-08\\ +15 & -0.33941716E-06 & 0.33852528E-06 & -0.89188000E-09\\ +\end{tabular}\\ +\hbox{\hskip 3.0cm}$c_1=0.37613920E-07$\\ +\hbox{\hskip 3.0cm}$c_2=-0.42427144E-07$\\ +\hbox{\hskip 1.0cm}$c_1R(\alpha)+c_2R(\beta)-0.43820800E\ 00=-0.13700000E-04$\\ +\hbox{\hskip 1.0cm}$c_1S(\alpha)+c_2S(\beta)-0.64000000E\ 00=-0.97600000E-03$ + +\hrule +Second iteration: $\alpha_{14}=8$, $\alpha_{15}=9$; \\ +$\beta_{14}=0.39213500E-08$, $\beta_{15}=-0.89188000E-09$ +\hrule +\begin{tabular}{|r|r|r|r|} +k & $c_1\alpha_k\hbox{\hskip 1cm}$ & $c_2\beta_k\hbox{\hskip 1cm}$ & +$A_k\hbox{\hskip 1cm}$\\ +&&&\\ + 0 & 0.36701576E-05 & 0.45512986E 00 & 0.24551335E 01\\ + 1 & -0.17051695E-05 & -0.16243666E 00 & -0.16243837E 00\\ + 2 & 0.68976566E-06 & 0.44956834E-01 & 0.44957523E-01\\ + 3 & -0.22132756E-06 & -0.67413538E-02 & -0.67415751E-02\\ + 4 & 0.51197561E-07 & -0.13067496E-02 & -0.13066984E-02\\ + 5 & -0.59856744E-08 & 0.13810895E-02 & 0.13810835E-02\\ + 6 & -0.13059663E-08 & -0.58502164E-03 & -0.58502294E-03\\ + 7 & 0.10552667E-08 & 0.17492889E-03 & 0.17492994E-03\\ + 8 & -0.38808033E-09 & -0.40472426E-04 & -0.40472814E-04\\ + 9 & 0.10523831E-09 & 0.72169965E-05 & 0.72171017E-05\\ +10 & -0.23146333E-10 & -0.86125438E-06 & -0.86127752E-06\\ +11 & 0.42342615E-11 & -0.25542252E-09 & -0.25118825E-09\\ +12 & -0.63200810E-12 & 0.37946968E-07 & 0.37946336E-07\\ +13 & 0.68210630E-13 & -0.14417584E-07 & -0.14417516E-07\\ +14 & -0.15414832E-14 & 0.39212981E-08 & 0.39212965E-08\\ +15 & -0.17341686E-14 & -0.89186818E-09 & -0.89186991E-09\\ +\end{tabular}\\ +\hbox{\hskip 3.0cm}$c_1=-0.19268540E-15$\\ +\hbox{\hskip 3.0cm}$c_2=0.99998675E\ 00$\\ +\hbox{\hskip 1.0cm}$c_1R(\alpha)+c_2R(\beta)-0.43820800E\ 00=0.0$\\ +\hbox{\hskip 1.0cm}$c_1S(\alpha)+c_2S(\beta)-0.64000000E\ 00=0.0$ + +\noindent +{\bf Table 2}: Computation of Chebyshev Coefficients for +$xe^{-x}Ei(x)$ - Concluded\\ +\hbox{\hskip 0.5cm}$[4 \le x \le 12 with M=15; \gamma_0=2, \gamma_k=0 +\textrm{for\ }k=1(1)15]$ + +\hrule +Third iteration: $\alpha_{14}=8$, $\alpha_{15}=9$;\\ +\hbox{\hskip 0.5cm}$\beta_{14}=0.39212965E-08$, $\beta_{15}=-0.89186991E-09$ +\hrule +\begin{tabular}{|r|r|r|r|} +k & $c_1\alpha_k\hbox{\hskip 1cm}$ & $c_2\beta_k\hbox{\hskip 1cm}$ & +$A_k\hbox{\hskip 1cm}$\\ +&&&\\ + 0 & -0.23083059E-07 & 0.45513355E 00 & 0.24551335E 01\\ + 1 & 0.10724479E-07 & -0.16243838E 00 & -0.16243837E 00\\ + 2 & -0.43382065E-08 & 0.44957526E-01 & 0.44957522E-01\\ + 3 & 0.13920157E-08 & -0.67415759E-02 & -0.67415745E-02\\ + 4 & -0.32200152E-09 & -0.13066983E-02 & -0.13066986E-02\\ + 5 & 0.37646251E-10 & 0.13810835E-02 & 0.13810836E-02\\ + 6 & 0.82137336E-11 & -0.58502297E-03 & -0.58502296E-03\\ + 7 & -0.66369857E-11 & 0.17492995E-03 & 0.17492994E-03\\ + 8 & 0.24407892E-11 & -0.40472817E-04 & -0.40472814E-04\\ + 9 & -0.66188494E-12 & 0.72171023E-05 & 0.72171017E-05\\ +10 & 0.14557636E-12 & -0.86127766E-06 & -0.86127751E-06\\ +11 & -0.26630930E-13 & -0.25116620E-09 & -0.25119283E-09\\ +12 & 0.39749465E-14 & 0.37946334E-07 & 0.37946337E-07\\ +13 & -0.42900337E-15 & -0.14417516E-07 & -0.14417516E-07\\ +14 & 0.96949915E-17 & 0.39212966E-08 & 0.39212966E-08\\ +15 & 0.10906865E-16 & -0.89186992E-09 & -0.89186990E-09\\ +\end{tabular}\\ +\hbox{\hskip 3.0cm}$c_1=0.12118739E-17$\\ +\hbox{\hskip 3.0cm}$c_2=0.10000000E\ 01$\\ +\hbox{\hskip 1.0cm}$c_1R(\alpha)+c_2R(\beta)-0.43820800E\ 00=0.0$\\ +\hbox{\hskip 1.0cm}$c_1S(\alpha)+c_2S(\beta)-0.64000000E\ 00=0.0$\\ +\hrule + +It is worth noting that the coefficient matrix of system (21,22,23) +yields an upper triangular matrix of order $M-1$ after the deletion of +the first two rows and the last two columns. Consequently, the +procedure of this section is applicable to any linear system having +this property. As a matter of fact, the same procedure can be +generalized to solve linear systems having coefficient matrices of +order N, the deletion of whose first $r$ ($r < N$) rows and last $r$ +columns yields upper triangular matrices of order $N-r$. + +\subsection{The Function $(1/x)[Ei(x)-log\vert x\vert-\gamma]$} + +Let +\begin{equation} +f(x)=(1/x)[Ei(x)-log\vert x\vert-\gamma],\quad g(x)=e^x,\quad +\vert x\vert \le b +\end{equation} +These functions, with the change of variable $x=bt$, simultaneously +satisfy the differential equations +\begin{equation} +bt^2\phi^{'}(t)+bt\phi(t)-\psi(t)=-1 +\end{equation} +\begin{equation} +\psi^{'}(t)-b\psi(t)=0,\quad -1 \le t \le 1 +\end{equation} +Conversely,\footnote{The general solution of the differential +equations has the form +$$ +\phi(t)=(c_1/t)+[Ei(bt)-log\vert bt\vert-\gamma]/bt +$$ +$$ +\psi(t)=c_2e^{bt} +$$ +where the first and second terms of $\phi(t)$ are, respectively, the +complementary solution and a particular integral of equation 30. The +requirement that $\phi(t)$ is bounded makes the constant $c_1=0$. The +fact that $\psi(0)=1$ is implicit in equation 30.} any solution of +equations 30 and 31 is equal to the functions given by equations 29 +for the change of variable $x=bt$. Therefore, boundary conditions need +not be imposed for the solution of the differential equations. + +A procedure similar to that of the previous section gives the coupled +infinite recurrence relations +\begin{equation} +bA_1+bA_3-B_0+B_2=-2 +\end{equation} +\begin{equation} +\left. +\begin{array}{c} +kbA_{k-1}+2(k+1)bA_{k+1}+(k+2)bA_{k+3}-2B_k+2B_{k+2}=0\\ +\\ +bB_{k-1}-2kB_k-bB_{k+1}=0,\quad k=1,2,\ldots +\end{array} +\right\} +\end{equation} +where $A_k$ and $B_k$ are the Chebyshev coefficients of $\phi(t)$ and +$\psi(t)$, respectively. + +Consider first the subsystem 33. If $A_k=\alpha_k$ and $B_k=\beta_k$ +are a simultaneous solution of the system, which is homogeneous, then +\begin{equation} +\left. +\begin{array}{l} +\hbox{\hskip 4cm}A_k=c\alpha_k\\ +\textrm{and}\\ +\hbox{\hskip 4cm}B_k=c\beta_k\\ +\end{array} +\right\} +\end{equation} +are also a solution for an arbitrary constant $c$. Thus based on +considerations analogous to the solution of equations 21, 22, and 23, +one can initiate an approximate solution of equations 32 and 33 by +setting +\begin{equation} +\left. +\begin{array}{l} +\alpha_M=0,\quad\alpha_k=0\quad\textrm{for }k \ge M+1\\ +\\ +\beta_M=1,\quad\beta_k=0\quad\textrm{for }k \ge M+1 +\end{array} +\right\} +\end{equation} +and then determining $\alpha_k$ and $\beta_k$ ($k=M-1, M-2, \ldots, +0$) by backward recurrence by means of equation 33. The arbitrary +constant $c$ is determined by substituting 34 into 32. + +\subsection{The Function $xe^{-x}Ei(x)$ on the Infinite Interval} +Let +\begin{equation} +f(x)=xe^{-x}Ei(x),\quad -\infty < x \le b < 0,\quad or 0 < b \le x < \infty +\end{equation} +By making the change of variables, +\begin{equation} +x=2b/(t+1) +\end{equation} +we can easily demonstrate that +\begin{equation} +f(x)=f[2b/(t+1)]=\phi(t) +\end{equation} +satisfies the differential equation +\begin{equation} +(t+1)^2\phi^{'}(t)+(t+1-2b)\phi(t)=-2b +\end{equation} +with +\begin{equation} +\phi(1)=be^{-b}Ei(b) +\end{equation} +An infinite system of equations involving the Chebyshev coefficients +$A_k$ of $\phi(t)$ is deducible from equations 39 and 40 by the same +procedure as applied to equations 13 and 14 to obtain the infinite +system 20; it is given as follows. +\begin{equation} +\sum_{k=0}^\infty\ ^{'}A_k=\phi(1)=be^{-b}Ei(b) +\end{equation} +\begin{equation} +(1-2b)A_0+3A_1+(3+2b)A_2+A_3=-4b +\end{equation} +\begin{equation} +\begin{array}{l} +kA_{k-1}+2[(2k+1)-2b]A_k+6(k+1)A_{k+1}+2(2k+3+2b)A_{k+2}\\ +\hbox{\hskip 4.0cm}+(k+2)A_{k+3}=0,\quad k=1,2,\ldots +\end{array} +\end{equation} +As in the case of equations 21, 22 and 23, the solution of 41, 42 and +43 can be assumed to be +\begin{equation} +A_k=c_1\alpha_k+c_2\beta_k +\end{equation} +with $A_k$ vanishing for a $k \ge M$. Thus, we can set, say +\begin{equation} +\left. +\begin{array}{ccc} +\alpha_{M-1}=0 & , & \alpha_M=1\\ +\beta_{M-1}=1 & , & \beta_M=0 +\end{array} +\right\} +\end{equation} +and determine the trial solutions $\alpha_k$ and $\beta_k$ +(k=M-1,M-2,$\ldots$,0) by means of equation 43 by backward +recurrence. The required solution of equations 41,42,adn 43 is then +determined by substituting equation 44 in equations 41 and 42 and +solving the resulting equations for $c_1$ and $c_2$. + +Loss of accuracy in the computation of $A_k$ can also occur here, as +in the solution of equations 21, 22 and 23, if the trial solutions are +not sufficiently independent. The process used to improve the accuracy +of $A_k$ of the system 21, 22 and 23 can also be applied here. + +For efficiency in computation, it is worth noting that for $b < 0$ +($-\infty < x \le b < 0$) the boundary condition 40 is not required +for the solution of equation 39 and 40. This follows from the fact +that any solution\footnote{The general solution of the differential +equation 39. Since equation 39 has no bounded complementary solution +for $-\infty < x \le b < 0$, every solution of it is equal to the +particular integral $xe^{-x}Ei(x)$. On the other hand, a solution of +equation 39 for $0 < x \le b < \infty$ would, in general, involve the +complementary function. Hence, boundary condition 40 is required to +guarantee that the solution of equation 39 is equal to +$xe^{-x}Ei(x)$.} of the differential equation 39 is equal to +$xe^{-x}Ei(x)$ ($x=2b/(t+1)$). Hence the $A_k$ of $xe^{-x}Ei(x)$ for +$-\infty < x \le b < 0$ can be obtained without the use of equation +39 and can be assumed to have the form +\begin{equation} +A_k = c\alpha_k,\quad(k=0,1,\ldots,M) +\end{equation} +The M+1 values of $\alpha_k$ can be generated by setting $\alpha_M=1$ +and computing $\alpha_k$ (k=0,1,$\ldots$,M-1) by means of equation +43 by backward recurrence. The substitution of equation 46 into 42 +then enables one to determine $c$ from the resulting equation. + +\subsection{Remarks on Convergence and Accuracy} + +The Chebyshev coefficients of table 3 were computed on the IBM 7094 +with 50-digit normalized floating-point arithmetic. In order to assure +that the sequence of approximate solutions (see Discusion) converged +to the limiting solution of the differential equation in question, a +trial M was incremented by 4 until the approximate Chebyshev +coefficients showed no change greter than or equal to +$0.5\times 10^{-35}$. Hence the maximum error is bounded by +\begin{equation} +0.5(M+1)\times 10^{-35}+\sum_{M+1}^\infty{\vert A_k\vert} +\end{equation} +where the first term is the maximum error of the M+1 approximate +Chebyshev coefficients, and the sum is the maximum error of the +truncated Chebyshev series of M+1 terms. If the Chebyshev series is +rapidly convergent, the maximum error of the approximate Chebyshev +series should be of the order of $10^{-30}$. The coefficients of table +3 have been rounded to 30 digits, and higher terms for $k > N$ giving +the maximum residual +\begin{equation} +\sum_{k=N+1}^M{\vert A_k\vert} < 0.5\times 10^{-30} +\end{equation} +have been dropped. This should allow for evaluation of the relevant +function that is accurate to 30 decimal places. Since the range of +values of each function is bounded between 2/5 and 5, the evaluated +function should be good to 30 significant digits. Taylor series +evaluation also checks with that of the function values of table 4 +(computed with 30-digit floating-point arithmetic using the +coefficients of table 3) for at least 28-1/2 significant +digits. Evaluation of Ei(x) using the coefficients of table 3 also +checked with Murnaghan and Wrench \cite{14} for 28-1/2 significant +figures. + +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients} +(a) +$$ +xe^{-x}Ei(x)=\sum_{k-0}^{40}\ {'}{A_kT_k(t)},\ \ t=(-20/x)-1,\ \ +(-\infty < x \le -10) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.1912173225 8605534539 1519326510E 01\\ + 1 & -0.4208355052 8684843755 0974986680E-01\\ + 2 & 0.1722819627 2843267833 7118157835E-02\\ + 3 & -0.9915782173 4445636455 9842322973E-04\\ + 4 & 0.7176093168 0227750526 5590665592E-05\\ + 5 & -0.6152733145 0951269682 7956791331E-06\\ + 6 & 0.6024857106 5627583129 3999701610E-07\\ + 7 & -0.6573848845 2883048229 5894189637E-08\\ + 8 & 0.7853167541 8323998199 4810079871E-09\\ + 9 & -0.1013730288 0038789855 4202774257E-09\\ +10 & 0.1399770413 2267686027 7823488623E-10\\ +11 & -0.2051008376 7838189961 8962318711E-11\\ +12 & 0.3168388726 0024778181 4907985818E-12\\ +13 & -0.5132760082 8391806541 5984751899E-13\\ +14 & 0.8680933040 7665493418 7433687383E-14\\ +15 & -0.1527015040 9030849719 8572355351E-14\\ +16 & 0.2784686251 6493573965 0105251453E-15\\ +17 & -0.5249890437 4217669680 8472933696E-16\\ +18 & 0.1020717991 2485612924 7455787226E-16\\ +\end{tabular} +\begin{tabular}{|r|r|} +19 & -0.2042264679 8997184130 8462421876E-17\\ +20 & 0.4197064172 7264847440 8827228562E-18\\ +21 & -0.8844508176 1728105081 6483737536E-19\\ +22 & 0.1908272629 5947174199 5060168262E-19\\ +23 & -0.4209746222 9351995033 6450865676E-20\\ +24 & 0.9483904058 1983732764 1500214512E-21\\ +25 & -0.2179467860 1366743199 4032574014E-21\\ +26 & 0.5103936869 0714509499 3452562741E-22\\ +27 & -0.1216883113 3344150908 9746779693E-22\\ +28 & 0.2951289166 4478751929 4773757144E-23\\ +29 & -0.7275353763 7728468971 4438950920E-24\\ +30 & 0.1821639048 6230739612 1667115976E-24\\ +31 & -0.4629629963 1633171661 2753482064E-25\\ +32 & 0.1193539790 9715779152 3052371292E-25\\ +33 & -0.3119493285 2201424493 1062147473E-26\\ +34 & 0.8261419734 5334664228 4170028518E-27\\ +35 & -0.2215803373 6609829830 2591177697E-27\\ +36 & 0.6016031671 6542638904 5303124429E-28\\ +37 & -0.1652725098 3821265964 9744302314E-28\\ +38 & 0.4592230358 7730270279 5636377166E-29\\ +39 & -0.1290062767 2132638473 7453212670E-29\\ +40 & 0.3662718481 0320025908 1177078922E-30\\ +\end{tabular} + +<>= + + Ei1(y:OPR):OPR == + infinite? y => 1 + x:R:=retract(y) + t:R:=acos((-20.0::R/x)-1.0::R)::R + t01:= 0.191217322586055345391519326510E1::R*cos(0.0::R)/2.0::R + t02:=t01-0.420835505286848437550974986680E-01::R*cos(t::R)::R + t03:=t02+0.172281962728432678337118157835E-02::R*cos( 2.0::R*t) + t04:=t03-0.991578217344456364559842322973E-04::R*cos( 3.0::R*t) + t05:=t04+0.717609316802277505265590665592E-05::R*cos( 4.0::R*t) + t06:=t05-0.615273314509512696827956791331E-06::R*cos( 5.0::R*t) + t07:=t06+0.602485710656275831293999701610E-07::R*cos( 6.0::R*t) + t08:=t07-0.657384884528830482295894189637E-08::R*cos( 7.0::R*t) + t09:=t08+0.785316754183239981994810079871E-09::R*cos( 8.0::R*t) + t10:=t09-0.101373028800387898554202774257E-09::R*cos( 9.0::R*t) + t11:=t10+0.139977041322676860277823488623E-10::R*cos(10.0::R*t) + t12:=t11-0.205100837678381899618962318711E-11::R*cos(11.0::R*t) + t13:=t12+0.316838872600247781814907985818E-12::R*cos(12.0::R*t) + t14:=t13-0.513276008283918065415984751899E-13::R*cos(13.0::R*t) + t15:=t14+0.868093304076654934187433687383E-14::R*cos(14.0::R*t) + t16:=t15-0.152701504090308497198572355351E-14::R*cos(15.0::R*t) + t17:=t16+0.278468625164935739650105251453E-15::R*cos(16.0::R*t) + t18:=t17-0.524989043742176696808472933696E-16::R*cos(17.0::R*t) + t19:=t18+0.102071799124856129247455787226E-16::R*cos(18.0::R*t) + t20:=t19-0.204226467989971841308462421876E-17::R*cos(19.0::R*t) + t21:=t20+0.419706417272648474408827228562E-18::R*cos(20.0::R*t) + t22:=t21-0.884450817617281050816483737536E-19::R*cos(21.0::R*t) + t23:=t22+0.190827262959471741995060168262E-19::R*cos(22.0::R*t) + t24:=t23-0.420974622293519950336450865676E-20::R*cos(23.0::R*t) + t25:=t24+0.948390405819837327641500214512E-21::R*cos(24.0::R*t) + t26:=t25-0.217946786013667431994032574014E-21::R*cos(25.0::R*t) + t27:=t26+0.510393686907145094993452562741E-22::R*cos(26.0::R*t) + t28:=t27-0.121688311333441509089746779693E-22::R*cos(27.0::R*t) + t29:=t28+0.295128916644787519294773757144E-23::R*cos(28.0::R*t) + t30:=t29-0.727535376377284689714438950920E-24::R*cos(29.0::R*t) + t31:=t30+0.182163904862307396121667115976E-24::R*cos(30.0::R*t) + t32:=t31-0.462962996316331716612753482064E-25::R*cos(31.0::R*t) + t33:=t32+0.119353979097157791523052371292E-25::R*cos(32.0::R*t) + t34:=t33-0.311949328522014244931062147473E-26::R*cos(33.0::R*t) + t35:=t34+0.826141973453346642284170028518E-27::R*cos(34.0::R*t) + t36:=t35-0.221580337366098298302591177697E-27::R*cos(35.0::R*t) + t37:=t36+0.601603167165426389045303124429E-28::R*cos(36.0::R*t) + t38:=t37-0.165272509838212659649744302314E-28::R*cos(37.0::R*t) + t39:=t38+0.459223035877302702795636377166E-29::R*cos(38.0::R*t) + t40:=t39-0.129006276721326384737453212670E-29::R*cos(39.0::R*t) + t41:=t40+0.366271848103200259081177078922E-30::R*cos(40.0::R*t) + t41::OPR + +@ + +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients - Continued} +(b) +$$ +xe^{-x}Ei(x)=\sum_{k-0}^{40}\ {'}{A_kT_k(t)},\ \ t=(x+7)/3,\ \ +(-10 \le x \le -4) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.1757556496 0612937384 8762834691E 011\\ + 1 & -0.4358541517 7361661170 5001867964E-01\\ + 2 & -0.7979507139 5584254013 3217027492E-02\\ + 3 & -0.1484372327 3037121385 0970210001E-02\\ + 4 & -0.2800301984 3775145748 6203954948E-03\\ + 5 & -0.5348648512 8657932303 9177361553E-04\\ + 6 & -0.1032867243 5735548661 0233266460E-04\\ + 7 & -0.2014083313 0055368773 2226198639E-05\\ + 8 & -0.3961758434 2738664582 2338443500E-06\\ + 9 & -0.7853872767 0966316306 7607656069E-07\\ +10 & -0.1567925981 0074698262 4616270279E-07\\ +11 & -0.3150055939 3763998825 0007372851E-08\\ +12 & -0.6365096822 5242037304 0380263972E-09\\ +13 & -0.1292888113 2805631835 6593121259E-09\\ +14 & -0.2638690999 6592557613 2149942808E-10\\ +15 & -0.5408958287 0450687349 1922207896E-11\\ +16 & -0.1113222784 6010898999 7676692708E-11\\ +17 & -0.2299624726 0744624618 4338864145E-12\\ +18 & -0.4766682389 4951902622 3913482091E-13\\ +19 & -0.9911756747 3352709450 6246643371E-14\\ +20 & -0.2067103580 4957072400 0900805021E-14\\ +\end{tabular} +\begin{tabular}{|r|r|} +21 & -0.4322776783 3833850564 5764394579E-15\\ +22 & -0.9063014799 6650172551 4905603356E-16\\ +23 & -0.1904669979 5816613974 4015963342E-16\\ +24 & -0.4011792326 3502786634 6744227520E-17\\ +25 & -0.8467772130 0168322313 4166334685E-18\\ +26 & -0.1790842733 6586966555 5826492204E-18\\ +27 & -0.3794490638 1714782440 1106175166E-19\\ +28 & -0.8053999236 7982798526 0999654058E-20\\ +29 & -0.1712339011 2362012974 3228671244E-20\\ +30 & -0.3646274058 7749686208 6576562816E-21\\ +31 & -0.7775969638 8939479435 3098157647E-22\\ +32 & -0.1660628498 4484020566 2531950966E-22\\ +33 & -0.3551178625 7882509300 5927145352E-23\\ +34 & -0.7603722685 9413580929 5734653294E-24\\ +35 & -0.1630074137 2584900288 9638374755E-24\\ +36 & -0.3498575202 7286322350 7538497255E-25\\ +37 & -0.7517179627 8900988246 0645145143E-26\\ +38 & -0.1616877440 0527227629 8777317918E-26\\ +39 & -0.3481270085 7247569174 8202271565E-27\\ +40 & -0.7502707775 5024654701 0642233720E-28\\ +41 & -0.1618454364 4959102680 7612330206E-28\\ +42 & -0.3494366771 7051616674 9482836452E-29\\ +43 & -0.7551036906 1261678585 6037026797E-30\\ +\end{tabular} + +<>= + + Ei2(y:OPR):OPR == + x:R:=retract(y) + t:R:=acos((x+7.0::R)/3.0::R)::R + t01:= 0.175755649606129373848762834691E1::R*cos(0.0::R)/2.0::R + t02:=t01-0.435854151773616611705001867964E-01::R*cos(t) + t03:=t02-0.797950713955842540133217027492E-02::R*cos( 2.0::R*t) + t04:=t03-0.148437232730371213850970210001E-02::R*cos( 3.0::R*t) + t05:=t04-0.280030198437751457486203954948E-03::R*cos( 4.0::R*t) + t06:=t05-0.534864851286579323039177361553E-04::R*cos( 5.0::R*t) + t07:=t06-0.103286724357355486610233266460E-04::R*cos( 6.0::R*t) + t08:=t07-0.201408331300553687732226198639E-05::R*cos( 7.0::R*t) + t09:=t08-0.396175843427386645822338443500E-06::R*cos( 8.0::R*t) + t10:=t09-0.785387276709663163067607656069E-07::R*cos( 9.0::R*t) + t11:=t10-0.156792598100746982624616270279E-07::R*cos(10.0::R*t) + t12:=t11-0.315005593937639988250007372851E-08::R*cos(11.0::R*t) + t13:=t12-0.636509682252420373040380263972E-09::R*cos(12.0::R*t) + t14:=t13-0.129288811328056318356593121259E-09::R*cos(13.0::R*t) + t15:=t14-0.263869099965925576132149942808E-10::R*cos(14.0::R*t) + t16:=t15-0.540895828704506873491922207896E-11::R*cos(15.0::R*t) + t17:=t16-0.111322278460108989997676692708E-11::R*cos(16.0::R*t) + t18:=t17-0.229962472607446246184338864145E-12::R*cos(17.0::R*t) + t19:=t18-0.476668238949519026223913482091E-13::R*cos(18.0::R*t) + t20:=t19-0.991175674733527094506246643371E-14::R*cos(19.0::R*t) + t21:=t20-0.206710358049570724000900805021E-14::R*cos(20.0::R*t) + t22:=t21-0.432277678338338505645764394579E-15::R*cos(21.0::R*t) + t23:=t22-0.906301479966501725514905603356E-16::R*cos(22.0::R*t) + t24:=t23-0.190466997958166139744015963342E-16::R*cos(23.0::R*t) + t25:=t24-0.401179232635027866346744227520E-17::R*cos(24.0::R*t) + t26:=t25-0.846777213001683223134166334685E-18::R*cos(25.0::R*t) + t27:=t26-0.179084273365869665555826492204E-18::R*cos(26.0::R*t) + t28:=t27-0.379449063817147824401106175166E-19::R*cos(27.0::R*t) + t29:=t28-0.805399923679827985260999654058E-20::R*cos(28.0::R*t) + t30:=t29-0.171233901123620129743228671244E-20::R*cos(29.0::R*t) + t31:=t30-0.364627405877496862086576562816E-21::R*cos(30.0::R*t) + t32:=t31-0.777596963889394794353098157647E-22::R*cos(31.0::R*t) + t33:=t32-0.166062849844840205662531950966E-22::R*cos(32.0::R*t) + t34:=t33-0.355117862578825093005927145352E-23::R*cos(33.0::R*t) + t35:=t34-0.760372268594135809295734653294E-24::R*cos(34.0::R*t) + t36:=t35-0.163007413725849002889638374755E-24::R*cos(35.0::R*t) + t37:=t36-0.349857520272863223507538497255E-25::R*cos(36.0::R*t) + t38:=t37-0.751717962789009882460645145143E-26::R*cos(37.0::R*t) + t39:=t38-0.161687744005272276298777317918E-26::R*cos(38.0::R*t) + t40:=t39-0.348127008572475691748202271565E-27::R*cos(39.0::R*t) + t41:=t40-0.750270777550246547010642233720E-28::R*cos(40.0::R*t) + t42:=t41-0.161845436449591026807612330206E-28::R*cos(41.0::R*t) + t43:=t42-0.349436677170516166749482836452E-29::R*cos(42.0::R*t) + t44:=t43-0.755103690612616785856037026797E-30::R*cos(43.0::R*t) + t44::OPR + +@ +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients - Continued} +(c) +$$ +[Ei-log\vert x\vert-\gamma]/x= +\sum_{k-0}^{33}\ {'}{A_kT_k(t)},\ \ t=x/4,\ \ +(-4 \le x \le 4) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.3293700103 7673912939 3905231421E 01\\ + 1 & 0.1679835052 3713029156 5505796064E 01\\ + 2 & 0.7220436105 6787543524 0299679644E 00\\ + 3 & 0.2600312360 5480956171 3740181192E 00\\ + 4 & 0.8010494308 1737502239 4742889237E-01\\ + 5 & 0.2151403663 9763337548 0552483005E-01\\ + 6 & 0.5116207789 9303312062 1968910894E-02\\ + 7 & 0.1090932861 0073913560 5066199014E-02\\ + 8 & 0.2107415320 2393891631 8348675226E-03\\ + 9 & 0.3719904516 6518885709 5940815956E-04\\ +10 & 0.6043491637 1238787570 4767032866E-05\\ +11 & 0.9092954273 9626095264 9596541772E-06\\ +12 & 0.1273805160 6592647886 5567184969E-06\\ +13 & 0.1669185748 4109890739 0896143814E-07\\ +14 & 0.2054417026 4010479254 7612484551E-08\\ +15 & 0.2383584444 4668176591 4052321417E-09\\ +\end{tabular} +\begin{tabular}{|r|r|} +16 & 0.2615386378 8854429666 9068664148E-10\\ +17 & 0.2721858622 8541670644 6550268995E-11\\ +18 & 0.2693750031 9835792992 5326427442E-12\\ +19 & 0.2541220946 7072635546 7884089307E-13\\ +20 & 0.2290130406 8650370941 8510620516E-14\\ +21 & 0.1975465739 0746229940 1057650412E-15\\ +22 & 0.1634024551 9289317406 8635419984E-16\\ +23 & 0.1298235437 0796376099 1961293204E-17\\ +24 & 0.9922587925 0737105964 4632581302E-19\\ +25 & 0.7306252806 7221032944 7230880087E-20\\ +26 & 0.5189676834 6043451272 0780080019E-21\\ +27 & 0.3560409454 0997068112 8043162227E-22\\ +28 & 0.2361979432 5793864237 0187203948E-23\\ +29 & 0.1516837767 7214529754 9624516819E-24\\ +30 & 0.9439089722 2448744292 5310405245E-26\\ +31 & 0.5697227559 5036921198 9581737831E-27\\ +32 & 0.3338333627 7954330315 6597939562E-28\\ +33 & 0.1900626012 8161914852 6680482237E-29\\ +\end{tabular} + +\noindent +($\gamma$=0.5772156649\ 0153286060\ 6512090082\ E\ 00) + +<>= + + Ei3(y:OPR):OPR == + x:R:=retract(y) + x = 0.0::R => 1 + t:R:=acos(x/4.0::R)::R + t01:= 0.329370010376739129393905231421E1::R*cos(0.0::R)/2.0::R + t02:=t01+0.167983505237130291565505796064E1::R*cos(t) + t03:=t02+0.722043610567875435240299679644E0::R*cos( 2.0::R*t) + t04:=t03+0.260031236054809561713740181192E0::R*cos( 3.0::R*t) + t05:=t04+0.801049430817375022394742889237E-01::R*cos( 4.0::R*t) + t06:=t05+0.215140366397633375480552483005E-01::R*cos( 5.0::R*t) + t07:=t06+0.511620778993033120621968910894E-02::R*cos( 6.0::R*t) + t08:=t07+0.109093286100739135605066199014E-02::R*cos( 7.0::R*t) + t09:=t08+0.210741532023938916318348675226E-03::R*cos( 8.0::R*t) + t10:=t09+0.371990451665188857095940815956E-04::R*cos( 9.0::R*t) + t11:=t10+0.604349163712387875704767032866E-05::R*cos(10.0::R*t) + t12:=t11+0.909295427396260952649596541772E-06::R*cos(11.0::R*t) + t13:=t12+0.127380516065926478865567184969E-06::R*cos(12.0::R*t) + t14:=t13+0.166918574841098907390896143814E-07::R*cos(13.0::R*t) + t15:=t14+0.205441702640104792547612484551E-08::R*cos(14.0::R*t) + t16:=t15+0.238358444446681765914052321417E-09::R*cos(15.0::R*t) + t17:=t16+0.261538637888544296669068664148E-10::R*cos(16.0::R*t) + t18:=t17+0.272185862285416706446550268995E-11::R*cos(17.0::R*t) + t19:=t18+0.269375003198357929925326427442E-12::R*cos(18.0::R*t) + t20:=t19+0.254122094670726355467884089307E-13::R*cos(19.0::R*t) + t21:=t20+0.229013040686503709418510620516E-14::R*cos(20.0::R*t) + t22:=t21+0.197546573907462299401057650412E-15::R*cos(21.0::R*t) + t23:=t22+0.163402455192893174068635419984E-16::R*cos(22.0::R*t) + t24:=t23+0.129823543707963760991961293204E-17::R*cos(23.0::R*t) + t25:=t24+0.992258792507371059644632581302E-19::R*cos(24.0::R*t) + t26:=t25+0.730625280672210329447230880087E-20::R*cos(25.0::R*t) + t27:=t26+0.518967683460434512720780080019E-21::R*cos(26.0::R*t) + t28:=t27+0.356040945409970681128043162227E-22::R*cos(27.0::R*t) + t29:=t28+0.236197943257938642370187203948E-23::R*cos(28.0::R*t) + t30:=t29+0.151683776772145297549624516819E-24::R*cos(29.0::R*t) + t31:=t30+0.943908972224487442925310405245E-26::R*cos(30.0::R*t) + t32:=t31+0.569722755950369211989581737831E-27::R*cos(31.0::R*t) + t33:=t32+0.333833362779543303156597939562E-28::R*cos(32.0::R*t) + t34:=t33+0.190062601281619148526680482237E-29::R*cos(33.0::R*t) + t34::OPR + +@ +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients - Continued} +(d) +$$ +xe^{-x}Ei(x)=\sum_{k-0}^{49}\ {'}{A_kT_k(t)},\ \ t=(x-8)/4,\ \ +(4 \le x \le 12) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.2455133538 7812952867 3420457043E 01\\ + 1 & -0.1624383791 3037652439 6002276856E 00\\ + 2 & 0.4495753080 9357264148 0785417193E-01\\ + 3 & -0.6741578679 9892299884 8718835050E-02\\ + 4 & -0.1306697142 8032942805 1599341387E-02\\ + 5 & 0.1381083146 0007257602 0202089820E-02\\ + 6 & -0.5850228790 1596579868 7368242394E-03\\ + 7 & 0.1749299341 0789197003 8740976432E-03\\ + 8 & -0.4047281499 0529303552 2869333800E-04\\ + 9 & 0.7217102412 1709975003 5752600049E-05\\ +10 & -0.8612776970 1986775241 4815450193E-06\\ +11 & -0.2514475296 5322559777 9084739054E-09 \\ +12 & 0.3794747138 2014951081 4074505574E-07\\ +13 & -0.1442117969 5211980616 0265640172E-07\\ +14 & 0.3935049295 9761013108 7190848042E-08\\ +15 & -0.9284689401 0633175304 7289210353E-09\\ +16 & 0.2031789568 0065461336 6090995698E-09\\ +17 & -0.4292498504 9923683142 7918026902E-10\\ +18 & 0.8992647177 7812393526 8001544182E-11\\ +19 & -0.1900869118 4121097524 2396635722E-11\\ +20 & 0.4092198912 2237383452 6121178338E-12\\ +21 & -0.8999253437 2931901982 5435824585E-13\\ +22 & 0.2019654670 8242638335 4948543451E-13\\ +23 & -0.4612930261 3830820719 4950531726E-14\\ +\end{tabular} +\begin{tabular}{|r|r|} +24 & 0.1069023072 9386369566 8857256409E-14\\ +25 & -0.2507030070 5700729569 2572254042E-15\\ +26 & 0.5937322503 7915516070 6073763509E-16\\ +27 & -0.1417734582 4376625234 4732005648E-16\\ +28 & 0.3409203754 3608089342 6806402093E-17\\ +29 & -0.8248290269 5054937928 8702529656E-18\\ +30 & 0.2006369712 6214423139 8824095937E-18\\ +31 & -0.4903851667 9674222440 3498152027E-19\\ +32 & 0.1203734482 3483321716 6664609324E-19\\ +33 & -0.2966282447 1413682538 1453572575E-20\\ +34 & 0.7335512384 2880759924 2142328436E-21\\ +35 & -0.1819924142 9085112734 4263485604E-21\\ +36 & 0.4528629374 2957606021 7359526404E-22\\ +37 & -0.1129980043 7506096133 8906717853E-22\\ +38 & 0.2826681251 2901165692 3764408445E-23\\ +39 & -0.7087717977 1690496166 6732640699E-24\\ +40 & 0.1781104524 0187095153 4401530034E-24\\ +41 & -0.4485004076 6189635731 2006142358E-25\\ +42 & 0.1131540292 5754766224 5053090840E-25\\ +43 & -0.2859957899 7793216379 0414326136E-26\\ +44 & 0.7240775806 9226736175 8172726753E-27\\ +45 & -0.1836132234 1257789805 0666710105E-27\\ +46 & 0.4663128735 2273048658 2600122073E-28\\ +47 & -0.1185959588 9190288794 6724005478E-28\\ +48 & 0.3020290590 5567131073 1137614875E-29\\ +49 & -0.7701650548 1663660609 8827057102E-30\\ +\end{tabular} + +<>= + + Ei4(y:OPR):OPR == + x:R:=retract(y) + t:R:=acos((x-8.0::R)/4.0::R)::R + t01:= 0.245513353878129528673420457043E1::R*cos(0.0::R)/2.0::R + t02:=t01-0.162438379130376524396002276856E0::R*cos(t) + t03:=t02+0.449575308093572641480785417193E-01::R*cos( 2.0::R*t) + t04:=t03-0.674157867998922998848718835050E-02::R*cos( 3.0::R*t) + t05:=t04-0.130669714280329428051599341387E-02::R*cos( 4.0::R*t) + t06:=t05+0.138108314600072576020202089820E-02::R*cos( 5.0::R*t) + t07:=t06-0.585022879015965798687368242394E-03::R*cos( 6.0::R*t) + t08:=t07+0.174929934107891970038740976432E-03::R*cos( 7.0::R*t) + t09:=t08-0.404728149905293035522869333800E-04::R*cos( 8.0::R*t) + t10:=t09+0.721710241217099750035752600049E-05::R*cos( 9.0::R*t) + t11:=t10-0.861277697019867752414815450193E-06::R*cos(10.0::R*t) + t12:=t11-0.251447529653225597779084739054E-09::R*cos(11.0::R*t) + t13:=t12+0.379474713820149510814074505574E-07::R*cos(12.0::R*t) + t14:=t13-0.144211796952119806160265640172E-07::R*cos(13.0::R*t) + t15:=t14+0.393504929597610131087190848042E-08::R*cos(14.0::R*t) + t16:=t15-0.928468940106331753047289210353E-09::R*cos(15.0::R*t) + t17:=t16+0.203178956800654613366090995698E-09::R*cos(16.0::R*t) + t18:=t17-0.429249850499236831427918026902E-10::R*cos(17.0::R*t) + t19:=t18+0.899264717778123935268001544182E-11::R*cos(18.0::R*t) + t20:=t19-0.190086911841210975242396635722E-11::R*cos(19.0::R*t) + t21:=t20+0.409219891222373834526121178338E-12::R*cos(20.0::R*t) + t22:=t21-0.899925343729319019825435824585E-13::R*cos(21.0::R*t) + t23:=t22+0.201965467082426383354948543451E-13::R*cos(22.0::R*t) + t24:=t23-0.461293026138308207194950531726E-14::R*cos(23.0::R*t) + t25:=t24+0.106902307293863695668857256409E-14::R*cos(24.0::R*t) + t26:=t25-0.250703007057007295692572254042E-15::R*cos(25.0::R*t) + t27:=t26+0.593732250379155160706073763509E-16::R*cos(26.0::R*t) + t28:=t27-0.141773458243766252344732005648E-16::R*cos(27.0::R*t) + t29:=t28+0.340920375436080893426806402093E-17::R*cos(28.0::R*t) + t30:=t29-0.824829026950549379288702529656E-18::R*cos(29.0::R*t) + t31:=t30+0.200636971262144231398824095937E-18::R*cos(30.0::R*t) + t32:=t31-0.490385166796742224403498152027E-19::R*cos(31.0::R*t) + t33:=t32+0.120373448234833217166664609324E-19::R*cos(32.0::R*t) + t34:=t33-0.296628244714136825381453572575E-20::R*cos(33.0::R*t) + t35:=t34+0.733551238428807599242142328436E-21::R*cos(34.0::R*t) + t36:=t35-0.181992414290851127344263485604E-21::R*cos(35.0::R*t) + t37:=t36+0.452862937429576060217359526404E-22::R*cos(36.0::R*t) + t38:=t37-0.112998004375060961338906717853E-22::R*cos(37.0::R*t) + t39:=t38+0.282668125129011656923764408445E-23::R*cos(38.0::R*t) + t40:=t39-0.708771797716904961666732640699E-24::R*cos(39.0::R*t) + t41:=t40+0.178110452401870951534401530034E-24::R*cos(40.0::R*t) + t42:=t41-0.448500407661896357312006142358E-25::R*cos(41.0::R*t) + t43:=t42+0.113154029257547662245053090840E-25::R*cos(42.0::R*t) + t44:=t43-0.285995789977932163790414326136E-26::R*cos(43.0::R*t) + t45:=t44+0.724077580692267361758172726753E-27::R*cos(44.0::R*t) + t46:=t45-0.183613223412577898050666710105E-27::R*cos(45.0::R*t) + t47:=t46+0.466312873522730486582600122073E-28::R*cos(46.0::R*t) + t48:=t47-0.118595958891902887946724005478E-28::R*cos(47.0::R*t) + t49:=t48+0.302029059055671310731137614875E-29::R*cos(48.0::R*t) + t50:=t49-0.770165054816636606098827057102E-30::R*cos(49.0::R*t) + t50::OPR + +@ + +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients - Continued} +(e) +$$ xe^{-x}Ei(x)=\sum_{k-0}^{47}\ {'}{A_kT_k(t)},\ \ t=(x-22)/10,\ \ +(12 \le x \le 32) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.2117028640 4369866832 9789991614E 01\\ + 1 & -0.3204237273 7548579499 0618303177E-01\\ + 2 & 0.8891732077 3531683589 0182400335E-02\\ + 3 & -0.2507952805 1892993708 8352442063E-02\\ + 4 & 0.7202789465 9598754887 5760902487E-03\\ + 5 & -0.2103490058 5011305342 3531441256E-03\\ + 6 & 0.6205732318 2769321658 8857730842E-04\\ + 7 & -0.1826566749 8167026544 9155689733E-04\\ + 8 & 0.5270651575 2893637580 7788296811E-05\\ + 9 & -0.1459666547 6199457532 3066719367E-05\\ +10 & 0.3781719973 5896367198 0484193981E-06\\ +11 & -0.8842581282 8407192007 7971589012E-07\\ +12 & 0.1741749198 5383936137 7350309156E-07\\ +13 & -0.2313517747 0436906350 6474480152E-08\\ +14 & -0.1228609819 1808623883 2104835230E-09\\ +15 & 0.2349966236 3228637047 8311381926E-09\\ +16 & -0.1100719401 0272628769 0738963049E-09\\ +17 & 0.3848275157 8612071114 9705563369E-10\\ +18 & -0.1148440967 4900158965 8439301603E-10\\ +19 & 0.3056876293 0885208263 0893626200E-11\\ +20 & -0.7388278729 2847356645 4163131431E-12\\ +21 & 0.1630933094 1659411056 4148013749E-12\\ +22 & -0.3276989373 3127124965 7111774748E-13\\ +\end{tabular} +\begin{tabular}{|r|r|} +23 & 0.5898114347 0713196171 1164283918E-14\\ +24 & -0.9099707635 9564920464 3554720718E-15\\ +25 & 0.1040752382 6695538658 5405697541E-15\\ +26 & -0.1809815426 0592279322 7163355935E-17\\ +27 & -0.3777098842 5639477336 9593494417E-17\\ +28 & 0.1580332901 0284795713 6759888420E-17\\ +29 & -0.4684291758 8088273064 8433752957E-18\\ +30 & 0.1199516852 5919809370 7533478542E-18\\ +31 & -0.2823594749 8418651767 9349931117E-19\\ +32 & 0.6293738065 6446352262 7520190349E-20\\ +33 & -0.1352410249 5047975630 5343973177E-20\\ +34 & 0.2837106053 8552914159 0980426210E-21\\ +35 & -0.5867007420 2463832353 1936371015E-22\\ +36 & 0.1205247636 0954731111 2449686917E-22\\ +37 & -0.2474446616 9988486972 8416011246E-23\\ +38 & 0.5099962585 8378500814 2986465688E-24\\ +39 & -0.1058382578 7754224088 7093294733E-24\\ +40 & 0.2215276245 0704827856 6429387155E-25\\ +41 & -0.4679278754 7569625867 1852546231E-26\\ +42 & 0.9972872990 6020770482 4269828079E-27\\ +43 & -0.2143267945 2167880459 1907805844E-27\\ +44 & 0.4640656908 8381811433 8414829515E-28\\ +45 & -0.1011447349 2115139094 8461800780E-28\\ +46 & 0.2217211522 7100771109 3046878345E-29\\ +47 & -0.4884890469 2437855322 4914645512E-30\\ +\end{tabular} + +<>= + + Ei5(y:OPR):OPR == + x:R:=retract(y) + t:R:=acos((x-22.0::R)/10.0::R)::R + t01:= 0.211702864043698668329789991614E1::R*cos(0.0::R)::R/2.0::R + t02:=t01-0.320423727375485794990618303177E-01::R*cos(t) + t03:=t02+0.889173207735316835890182400335E-02::R*cos( 2.0::R*t) + t04:=t03-0.250795280518929937088352442063E-02::R*cos( 3.0::R*t) + t05:=t04+0.720278946595987548875760902487E-03::R*cos( 4.0::R*t) + t06:=t05-0.210349005850113053423531441256E-03::R*cos( 5.0::R*t) + t07:=t06+0.620573231827693216588857730842E-04::R*cos( 6.0::R*t) + t08:=t07-0.182656674981670265449155689733E-04::R*cos( 7.0::R*t) + t09:=t08+0.527065157528936375807788296811E-05::R*cos( 8.0::R*t) + t10:=t09-0.145966654761994575323066719367E-05::R*cos( 9.0::R*t) + t11:=t10+0.378171997358963671980484193981E-06::R*cos(10.0::R*t) + t12:=t11-0.884258128284071920077971589012E-07::R*cos(11.0::R*t) + t13:=t12+0.174174919853839361377350309156E-07::R*cos(12.0::R*t) + t14:=t13-0.231351774704369063506474480152E-08::R*cos(13.0::R*t) + t15:=t14-0.122860981918086238832104835230E-09::R*cos(14.0::R*t) + t16:=t15+0.234996623632286370478311381926E-09::R*cos(15.0::R*t) + t17:=t16-0.110071940102726287690738963049E-09::R*cos(16.0::R*t) + t18:=t17+0.384827515786120711149705563369E-10::R*cos(17.0::R*t) + t19:=t18-0.114844096749001589658439301603E-10::R*cos(18.0::R*t) + t20:=t19+0.305687629308852082630893626200E-11::R*cos(19.0::R*t) + t21:=t20-0.738827872928473566454163131431E-12::R*cos(20.0::R*t) + t22:=t21+0.163093309416594110564148013749E-12::R*cos(21.0::R*t) + t23:=t22-0.327698937331271249657111774748E-13::R*cos(22.0::R*t) + t24:=t23+0.589811434707131961711164283918E-14::R*cos(23.0::R*t) + t25:=t24-0.909970763595649204643554720718E-15::R*cos(24.0::R*t) + t26:=t25+0.104075238266955386585405697541E-15::R*cos(25.0::R*t) + t27:=t26-0.180981542605922793227163355935E-17::R*cos(26.0::R*t) + t28:=t27-0.377709884256394773369593494417E-17::R*cos(27.0::R*t) + t29:=t28+0.158033290102847957136759888420E-17::R*cos(28.0::R*t) + t30:=t29-0.468429175880882730648433752957E-18::R*cos(29.0::R*t) + t31:=t30+0.119951685259198093707533478542E-18::R*cos(30.0::R*t) + t32:=t31-0.282359474984186517679349931117E-19::R*cos(31.0::R*t) + t33:=t32+0.629373806564463522627520190349E-20::R*cos(32.0::R*t) + t34:=t33-0.135241024950479756305343973177E-20::R*cos(33.0::R*t) + t35:=t34+0.283710605385529141590980426210E-21::R*cos(34.0::R*t) + t36:=t35-0.586700742024638323531936371015E-22::R*cos(35.0::R*t) + t37:=t36+0.120524763609547311112449686917E-22::R*cos(36.0::R*t) + t38:=t37-0.247444661699884869728416011246E-23::R*cos(37.0::R*t) + t39:=t38+0.509996258583785008142986465688E-24::R*cos(38.0::R*t) + t40:=t39-0.105838257877542240887093294733E-24::R*cos(39.0::R*t) + t41:=t40+0.221527624507048278566429387155E-25::R*cos(40.0::R*t) + t42:=t41-0.467927875475696258671852546231E-26::R*cos(41.0::R*t) + t43:=t42+0.997287299060207704824269828079E-27::R*cos(42.0::R*t) + t44:=t42-0.214326794521678804591907805844E-27::R*cos(43.0::R*t) + t45:=t42+0.464065690883818114338414829515E-28::R*cos(44.0::R*t) + t46:=t42-0.101144734921151390948461800780E-28::R*cos(45.0::R*t) + t47:=t42+0.221721152271007711093046878345E-29::R*cos(46.0::R*t) + t48:=t42-0.488489046924378553224914645512E-30::R*cos(47.0::R*t) + t48::OPR + +@ +{\vbox{\vskip 1cm}} + +{\bf Table 3: Chebyshev Coefficients - Continued} +(f) +$$ xe^{-x}Ei(x)=\sum_{k-0}^{46}\ {'}{A_kT_k(t)},\ \ t=(64/x)-1,\ \ +(32 \le x < \infty) +$$ +\begin{tabular}{|r|r|} +k & $A_k$\hbox{\hskip 3cm}\\ +&\\ + 0 & 0.2032843945 7961669908 7873844202E 01\\ + 1 & 0.1669920452 0313628514 7618434339E-01\\ + 2 & 0.2845284724 3613468074 2489985325E-03\\ + 3 & 0.7563944358 5162064894 8786693854E-05\\ + 4 & 0.2798971289 4508591575 0484318090E-06\\ + 5 & 0.1357901828 5345310695 2556392593E-07\\ + 6 & 0.8343596202 0404692558 5610289412E-09\\ + 7 & 0.6370971727 6402484382 7524337306E-10\\ + 8 & 0.6007247608 8118612357 6083084850E-11\\ + 9 & 0.7022876174 6797735907 5059216588E-12\\ +10 & 0.1018302673 7036876930 9667322152E-12\\ +11 & 0.1761812903 4308800404 0656741554E-13\\ +12 & 0.3250828614 2353606942 4072007647E-14\\ +13 & 0.5071770025 5058186788 1479300685E-15\\ +14 & 0.1665177387 0432942985 3520036957E-16\\ +15 & -0.3166753890 7975144007 2410018963E-16\\ +16 & -0.1588403763 6641415154 8423134074E-16\\ +17 & -0.4175513256 1380188308 9626455063E-17\\ +18 & -0.2892347749 7071418820 2868862358E-18\\ +19 & 0.2800625903 3966080728 9978777339E-18\\ +20 & 0.1322938639 5392708914 0532005364E-18\\ +21 & 0.1804447444 1773019958 5334811191E-19\\ +22 & -0.7905384086 5226165620 2021080364E-20\\ +23 & -0.4435711366 3695734471 8167314045E-20\\ +\end{tabular} +\begin{tabular}{|r|r|} +24 & -0.4264103994 9781026176 0579779746E-21\\ +25 & 0.3920101766 9371439072 5625388636E-21\\ +26 & 0.1527378051 3439636447 2804486402E-21\\ +27 & -0.1024849527 0494906078 6953149788E-22\\ +28 & -0.2134907874 7710893794 8904287231E-22\\ +29 & -0.3239139475 1602368761 4279789345E-23\\ +30 & 0.2142183762 2964597029 6249355934E-23\\ +31 & 0.8234609419 6189955316 9207838151E-24\\ +32 & -0.1524652829 6206721081 1495038147E-24\\ +33 & -0.1378208282 4882440129 0438126477E-24\\ +34 & 0.2131311201 4287370679 1513005998E-26\\ +35 & 0.2012649651 8713266585 9213006507E-25\\ +36 & 0.1995535662 0563740232 0607178286E-26\\ +37 & -0.2798995812 2017971142 6020884464E-26\\ +38 & -0.5534511830 5070025094 9784942560E-27\\ +39 & 0.3884995422 6845525312 9749000696E-27\\ +40 & 0.1121304407 2330701254 0043264712E-27\\ +41 & -0.5566568286 7445948805 7823816866E-28\\ +42 & -0.2045482612 4651357628 8865878722E-28\\ +43 & 0.8453814064 4893808943 7361193598E-29\\ +44 & 0.3565755151 2015152659 0791715785E-29\\ +45 & -0.1383652423 4779775181 0195772006E-29\\ +46 & -0.6062142653 2093450576 7865286306E-30\\ +\end{tabular} + +<>= + + Ei6(y:OPR):OPR == + infinite? y => 1 + x:R:=retract(y) + m:R:=64.0::R/x-1.0::R + t:R:=acos(m::R)::R + t01:= 0.203284394579616699087873844202E1::R*cos(0.0::R)::R/2.0::R + t02:=t01+0.166992045203136285147618434339E-01::R*cos(t) + t03:=t02+0.284528472436134680742489985325E-03::R*cos( 2.0::R*t) + t04:=t03+0.756394435851620648948786693854E-05::R*cos( 3.0::R*t) + t05:=t04+0.279897128945085915750484318090E-06::R*cos( 4.0::R*t) + t06:=t05+0.135790182853453106952556392593E-07::R*cos( 5.0::R*t) + t07:=t06+0.834359620204046925585610289412E-09::R*cos( 6.0::R*t) + t08:=t07+0.637097172764024843827524337306E-10::R*cos( 7.0::R*t) + t09:=t08+0.600724760881186123576083084850E-11::R*cos( 8.0::R*t) + t10:=t09+0.702287617467977359075059216588E-12::R*cos( 9.0::R*t) + t11:=t10+0.101830267370368769309667322152E-12::R*cos(10.0::R*t) + t12:=t11+0.176181290343088004040656741554E-13::R*cos(11.0::R*t) + t13:=t12+0.325082861423536069424072007647E-14::R*cos(12.0::R*t) + t14:=t13+0.507177002550581867881479300685E-15::R*cos(13.0::R*t) + t15:=t14+0.166517738704329429853520036957E-16::R*cos(14.0::R*t) + t16:=t15-0.316675389079751440072410018963E-16::R*cos(15.0::R*t) + t17:=t16-0.158840376366414151548423134074E-16::R*cos(16.0::R*t) + t18:=t17-0.417551325613801883089626455063E-17::R*cos(17.0::R*t) + t19:=t18-0.289234774970714188202868862358E-18::R*cos(18.0::R*t) + t20:=t19+0.280062590339660807289978777339E-18::R*cos(19.0::R*t) + t21:=t20+0.132293863953927089140532005364E-18::R*cos(20.0::R*t) + t22:=t21+0.180444744417730199585334811191E-19::R*cos(21.0::R*t) + t23:=t22-0.790538408652261656202021080364E-20::R*cos(22.0::R*t) + t24:=t23-0.443571136636957344718167314045E-20::R*cos(23.0::R*t) + t25:=t24-0.426410399497810261760579779746E-21::R*cos(24.0::R*t) + t26:=t25+0.392010176693714390725625388636E-21::R*cos(25.0::R*t) + t27:=t26+0.152737805134396364472804486402E-21::R*cos(26.0::R*t) + t28:=t27-0.102484952704949060786953149788E-22::R*cos(27.0::R*t) + t29:=t28-0.213490787477108937948904287231E-22::R*cos(28.0::R*t) + t30:=t29-0.323913947516023687614279789345E-23::R*cos(29.0::R*t) + t31:=t30+0.214218376229645970296249355934E-23::R*cos(30.0::R*t) + t32:=t31+0.823460941961899553169207838151E-24::R*cos(31.0::R*t) + t33:=t32-0.152465282962067210811495038147E-24::R*cos(32.0::R*t) + t34:=t33-0.137820828248824401290438126477E-24::R*cos(33.0::R*t) + t35:=t34+0.213131120142873706791513005998E-26::R*cos(34.0::R*t) + t36:=t35+0.201264965187132665859213006507E-25::R*cos(35.0::R*t) + t37:=t36+0.199553566205637402320607178286E-26::R*cos(36.0::R*t) + t38:=t37-0.279899581220179711426020884464E-26::R*cos(37.0::R*t) + t39:=t38-0.553451183050700250949784942560E-27::R*cos(38.0::R*t) + t40:=t39+0.388499542268455253129749000696E-27::R*cos(39.0::R*t) + t41:=t40+0.112130440723307012540043264712E-27::R*cos(40.0::R*t) + t42:=t41-0.556656828674459488057823816866E-28::R*cos(41.0::R*t) + t43:=t42-0.204548261246513576288865878722E-28::R*cos(42.0::R*t) + t44:=t43+0.845381406448938089437361193598E-29::R*cos(43.0::R*t) + t45:=t44+0.356575515120151526590791715785E-29::R*cos(44.0::R*t) + t46:=t45-0.138365242347797751810195772006E-29::R*cos(45.0::R*t) + t47:=t46-0.606214265320934505767865286306E-30::R*cos(46.0::R*t) + t47::OPR + +@ + +{\vbox{\vskip 1cm}} + +{\bf Table 4: Function Values of the Associated Functions} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=-(20/x)-1$ & $xe^{-x}Ei(x)$\hbox{\hskip 3cm}\\ +&&\\ +$-\infty$ & -1.000 & 0.1000000000 0000000000 0000000000 E 01\\ +-160 & -0.875 & 0.9938266956 7406127387 8797850088 E 00\\ +-80 & -0.750 & 0.9878013330 9428877356 4522608410 E 00\\ +-53 1/3 & -0.625 & 0.9819162901 4319443961 7735426105 E 00\\ +-40 & -0.500 & 0.9761646031 8514305080 8000604060 E 00\\ +-32 & -0.375 & 0.9705398840 7466392046 2584664361 E 00\\ +-26 2/3 & -0.250 & 0.9650362511 2337703576 3536593528 E 00\\ +-22 6/7 & -0.125 & 0.9596482710 7936727616 5478970820 E 00\\ +-20 & -0.000 & 0.9543709099 1921683397 5195829433 E 00\\ +-17 7/9 & 0.125 & 0.9491994907 7974574460 6445346803 E 00\\ +-16 & 0.250 & 0.9441296577 3690297898 4149471583 E 00\\ +-14 6/11 & 0.375 & 0.9391573444 1928424124 0422409988 E 00\\ +-13 1/3 & 0.500 & 0.9342787466 5341046480 9375801650 E 00\\ +-12 4/13 & 0.625 & 0.9294902984 9721403772 5319679042 E 00\\ +-11 3/7 & 0.750 & 0.9247886511 4084169605 5993585492 E 00\\ +-10 2/3 & 0.875 & 0.9201706542 4944567620 2148012149 E 00\\ +-10 & 1.000 & 0.9156333393 9788081876 0698157666 E 00 +\end{tabular} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=-(x+7)/3$ & $xe^{-x}Ei(x)$\hbox{\hskip 3cm}\\ +&&\\ +-10.000 & -1.000 & 0.9156333393 9788081876 0698157661 E 01\\ + -9.625 & -0.875 & 0.9128444614 6799341885 6575662217 E 00\\ + -9.250 & -0.750 & 0.9098627515 2542413937 8954274597 E 00\\ + -8.875 & -0.625 & 0.9066672706 5475388033 4995756418 E 00\\ + -8.500 & -0.500 & 0.9032339019 7320784414 4682926135 E 00\\ + -8.125 & -0.375 & 0.8995347176 8847383630 1415777697 E 00\\ + -7.750 & -0.250 & 0.8955371870 8753915717 9475513219 E 00\\ + -7.375 & -0.125 & 0.8912031763 2125431626 7087476258 E 00\\ + -7.000 & -0.000 & 0.8864876725 3642935289 3993846569 E 00\\ + -6.625 & 0.125 & 0.8813371384 6821020039 4305706270 E 00\\ + -6.250 & 0.250 & 0.8756873647 8846593227 6462155532 E 00\\ + -5.875 & 0.375 & 0.8694606294 5411341030 2047153364 E 00\\ + -5.500 & 0.500 & 0.8625618846 9070142209 0918986586 E 00\\ + -5.125 & 0.625 & 0.8548735538 9019954239 2425567234 E 00\\ + -4.750 & 0.750 & 0.8462482991 0358736117 1665798810 E 00\\ + -4.375 & 0.875 & 0.8364987545 5629874174 2152267582 E 00\\ + -4.000 & 1.000 & 0.8253825996 0422333240 8183035504 E 00 +\end{tabular} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=x/4$ & +$[Ei(x)-log\vert x\vert - \gamma]/x$\hbox{\hskip 2cm}\\ +&&\\ + -4.0 & -1.000 & 0.4918223446 0781809647 9962798267 E 00\\ + -3.5 & -0.875 & 0.5248425066 4412835691 8258753311 E 00\\ + -3.0 & -0.750 & 0.5629587782 2127986313 8086024270 E 00\\ + -2.5 & -0.625 & 0.6073685258 5838306451 4266925640 E 00\\ + -2.0 & -0.500 & 0.6596316780 8476964479 5492023380 E 00\\ + -1.5 & -0.375 & 0.7218002369 4421992965 7623030310 E 00\\ + -1.0 & -0.250 & 0.7965995992 9705313428 3675865540 E 00\\ + -0.5 & -0.125 & 0.8876841582 3549672587 2151815870 E 00\\ + 0.0 & -0.000 & 0.1000000000 0000000000 0000000000 E 01\\ + 0.5 & 0.125 & 0.1140302841 0431720574 6248768807 E 01\\ + 1.0 & 0.250 & 0.1317902151 4544038948 6000884424 E 01\\ + 1.5 & 0.375 & 0.1545736450 7467337302 4859074039 E 01\\ + 2.0 & 0.500 & 0.1841935755 2702059966 7788045934 E 01\\ + 2.5 & 0.625 & 0.2232103799 1211651144 5340506423 E 01\\ + 3.0 & 0.750 & 0.2752668205 6852580020 0219289740 E 01\\ + 3.5 & 0.875 & 0.3455821531 9301241243 7300898811 E 01\\ + 4.0 & 1.000 & 0.4416841111 0086991358 0118598668 E 01 +\end{tabular} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=(x-8)/4$ &$xe^{-x}Ei(x)$\hbox{\hskip 3cm}\\ +&&\\ + 4.0 & -1.000 & 0.1438208031 4544827847 0968670330 E 01\\ + 4.5 & -0.875 & 0.1396419029 6297460710 0674523183 E 01\\ + 5.0 & -0.750 & 0.1353831277 4552859779 0189174047 E 01\\ + 5.5 & -0.625 & 0.1314143565 7421192454 1219816991 E 01\\ + 6.0 & -0.500 & 0.1278883860 4895616189 2314099578 E 01\\ + 6.5 & -0.375 & 0.1248391155 0017014864 0741941387 E 01\\ + 7.0 & -0.250 & 0.1222408052 3605310590 3656846622 E 01\\ + 7.5 & -0.125 & 0.1200421499 5996307864 3879158950 E 01\\ + 8.0 & -0.000 & 0.1181847986 9872079731 7739362644 E 01\\ + 8.5 & 0.125 & 0.1166126525 8117484943 9918142965 E 01\\ + 9.0 & 0.250 & 0.1152759208 7089248132 2396814952 E 01\\ + 9.5 & 0.375 & 0.1141323475 9526242015 5338560641 E 01\\ +10.0 & 0.500 & 0.1131470204 7341077803 4051681355 E 01\\ +10.5 & 0.625 & 0.1122915570 0177606064 2888630755 E 01\\ +11.0 & 0.750 & 0.1115430938 9980384416 4779434229 E 01\\ +11.5 & 0.875 & 0.1108832926 3050773058 6855234934 E 01\\ +12.0 & 1.000 & 0.1102974544 9067590726 7241234953 E 01\\ +\end{tabular} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=(x-22)/10$ &$xe^{-x}Ei(x)$\hbox{\hskip 3cm}\\ +&&\\ +12.00 & -1.000 & 0.1102974544 9067590726 7241234952 E 01\\ +13.25 & -0.875 & 0.1090844898 2154756926 6468614954 E 01\\ +14.50 & -0.750 & 0.1081351395 7351912850 6346643795 E 01\\ +15.75 & -0.625 & 0.1073701384 1997572371 2157900374 E 01\\ +17.00 & -0.500 & 0.1067393691 9585378312 9572196197 E 01\\ +18.25 & -0.375 & 0.1062096608 6221502426 8372647556 E 01\\ +19.50 & -0.250 & 0.1057581342 1587250319 5393949410 E 01\\ +20.75 & -0.125 & 0.1053684451 2894094408 2102194964 E 01\\ +22.00 & -0.000 & 0.1050285719 6851897941 1780664532 E 01\\ +23.25 & 0.125 & 0.1047294551 7053248581 1492365591 E 01\\ +24.50 & 0.250 & 0.1044641267 9046436368 9761075289 E 01\\ +25.75 & 0.375 & 0.1042271337 2023202388 5710928048 E 01\\ +27.00 & 0.500 & 0.1040141438 3230104381 3713899754 E 01\\ +28.25 & 0.625 & 0.1038216700 3601458768 0056548394 E 01\\ +29.50 & 0.750 & 0.1036468726 2924118457 5154685419 E 01\\ +30.75 & 0.875 & 0.1034874149 8964796947 2990938990 E 01\\ +32.00 & 1.000 & 0.1033413564 2162410494 3493552567 E 01\\ +\end{tabular} + +{\vbox{\vskip 1cm}} + +\begin{tabular}{|r|c|r|} +x\hbox{\hskip 0.5cm} & $t=(64/x)-1$ &$xe^{-x}Ei(x)$\hbox{\hskip 3cm}\\ +&&\\ +$\infty$ & -1.000 & 0.100000000 0000000000 00000000001 E 01\\ +512 & -0.875 & 0.100196079 9450711925 31337468473 E 01\\ +256 & -0.750 & 0.100393713 0905698627 88009078297 E 01\\ +170 2/3 & -0.625 & 0.100592927 5692929112 94663030932 E 01\\ +128 & -0.500 & 0.100793752 4408140182 81776821694 E 01\\ +102 2/5 & -0.375 & 0.100996217 7406449755 74367545570 E 01\\ +85 1/3 & -0.250 & 0.101200354 5332988482 01864466702 E 01\\ +73 1/7 & -0.125 & 0.101406194 9696971331 45942329335 E 01\\ +64 & -0.000 & 0.101613772 3494325321 70357100831 E 01\\ +56 8/9 & 0.125 & 0.101823121 1884832696 82337017143 E 01\\ +51 1/5 & 0.250 & 0.102034277 2930783774 87217829808 E 01\\ +46 6/11 & 0.375 & 0.102247277 8405420595 91275364791 E 01\\ +42 2/3 & 0.500 & 0.102462161 4681078391 01187804247 E 01\\ +39 5/13 & 0.625 & 0.102678968 3709028524 50984510823 E 01\\ +36 4/7 & 0.750 & 0.102897740 4105808008 63378435059 E 01\\ +34 2/15 & 0.875 & 0.103118521 2364659263 55875784663 E 01\\ +32 & 1.000 & 0.103341356 4216241049 43493552567 E 01\\ +\end{tabular} + +<>= + polygamma(k,z) == CPSI(k, z)$Lisp polygamma(k,x) == RPSI(k, x)$Lisp @@ -811,5 +2350,43 @@ 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 +\bibitem{8} Lee, K.L.,``High-precision Chebyshev series approximation +to the exponential integral'', NASA-TN-D-5953, A-3571, No Copyright +Doc. ID=19700026648, Accession ID=70N35964, Aug 1970 +\bibitem{9} Harris, Frank E.: Tables of the Exponential Integral +Ei(x). Math. Tables and Other Aids to Computation, vol. 11, 1957, +pp.9-16 +\bibitem{10} Miller, James; and Hurst, R.P.: Simplified Calculation of +the Exponential Integral. Math. Tables and Other Aids to Computation, +vol. 12, 1958, pp 187-193. +\bibitem{11} Clenshaw, C.W.: Chebyshev Series for Mathematical +Functions. Mathematical Tables, vol. 5, National Physical Laboratory, +Her Majesty's Stationery Office, London, 1962, p. 29. +\bibitem{12} Cody, W.J.; and Thacher, H.C., Jr.: Rational +Approximations for the Exponential Integral $E_1(x)$. Math. Comp., +vol.22, July 1968, pp. 641-649. +\bibitem{13} Cody, W.J.; and Thacher, H.C., Jr.: Rational +Approximations for the Exponential Integral $Ei(x)$. Math. Comp., +vol.22, April 1969, pp. 289-303. +\bibitem{14} Murnaghan, F.D.; and Wrench, J.W., Jr.: The Converging +Factor for the Exponential Integral. Rep. 1535, David Taylor Model +Basin Applied Mathematics Lab., Jan. 1963. +\bibitem{16} Clenshaw, C.W.: The Numerical Solution of Linear +Differential Equation in Chebyshev Series. Proc. Cambridge Phil. Soc., +vol. 53, 1957, pp 134-149 +\bibitem{17} Fox, L.; and Parker, I.B.: Chebyshev Polynomials in +Numerical Analysis. Oxford Univ. Press, London, 1968. +\bibitem{18} Jeffrey, Alan ``Handbook of Mathematical Formulas and +Integrals'' Elsevier Academic Press 2004 3rd Edition ISBN +0-12-382256-4 pp167-171 +\bibitem{19} Press, William, et.al., ``Numerical Recipes in C'' +Press Syndicate Univ. of Cambridge, 1995 ISBN 0-521-43108-5 +pp190-194 +\bibitem{20} Press, William, et.al., ``Numerical Recipes in C'' +Press Syndicate Univ. of Cambridge, 1995 ISBN 0-521-43108-5 +p196 +\bibitem{21} Press, William, et.al., ``Numerical Recipes in C'' +Press Syndicate Univ. of Cambridge, 1995 ISBN 0-521-43108-5 +p222-225 \end{thebibliography} \end{document} diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 6cc1513..a54a9e7 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -302,8 +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 e1.regress easter.regress \ - efi.regress \ + dmp.regress dpol.regress e1.regress ei.regress \ + easter.regress efi.regress \ eigen.regress elemfun.regress elemnum.regress elfuts.regress \ elt.regress en.regress \ eq.regress eqtbl.regress equation2.regress \ @@ -542,6 +542,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/drawcfun.input ${OUT}/drawcurv.input \ ${OUT}/draw.input ${OUT}/drawcx.input ${OUT}/drawex.input \ ${OUT}/drawpoly.input ${OUT}/drawx.input ${OUT}/e1.input \ + ${OUT}/ei.input \ ${OUT}/easter.input ${OUT}/efi.input ${OUT}/egg.input \ ${OUT}/eigen.input \ ${OUT}/elemfun.input ${OUT}/elemnum.input ${OUT}/elfuts.input \ @@ -776,7 +777,7 @@ DOCFILES= \ ${DOC}/e04jaf.input.dvi ${DOC}/e04mbf.input.dvi \ ${DOC}/e04naf.input.dvi ${DOC}/e04ucf.input.dvi \ ${DOC}/e04ycf.input.dvi ${DOC}/e1.input.dvi \ - ${DOC}/easter.input.dvi \ + ${DOC}/ei.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 ${DOC}/elemnum.input.dvi \ diff --git a/src/input/ei.input.pamphlet b/src/input/ei.input.pamphlet new file mode 100644 index 0000000..9c84341 --- /dev/null +++ b/src/input/ei.input.pamphlet @@ -0,0 +1,2311 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input ei.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +The Ei implementation in Axiom uses Chebyshev\cite{1} polynomials +to approximate the function. The coefficients are not used +here but kept here for reference purposes. + +The values generated are compared against the values in +Abramowitz and Stegun\cite{2}. +<<*>>= +)spool ei.output +)set message test on +)set message auto off +)clear all +digits 35 + +--S 1 of 20 +gamma:=0.577215664901532860606512090082 +--R +--R +--R (2) 0.5772156649 0153286060 6512090082 +--R Type: Float +--E 1 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(-\infty < x \le -10)$ in the polynomial +$$\sum_{k=0}^{40}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=(-20/x)-1$ + +<<*>>= + +--S 2 of 20 +aChebyshev:=_ +[0.191217322586055345391519326510E1,_ +-0.420835505286848437550974986680E-01,_ + 0.172281962728432678337118157835E-02,_ +-0.991578217344456364559842322973E-04,_ + 0.717609316802277505265590665592E-05,_ +-0.615273314509512696827956791331E-06,_ + 0.602485710656275831293999701610E-07,_ +-0.657384884528830482295894189637E-08,_ + 0.785316754183239981994810079871E-09,_ +-0.101373028800387898554202774257E-09,_ + 0.139977041322676860277823488623E-10,_ +-0.205100837678381899618962318711E-11,_ + 0.316838872600247781814907985818E-12,_ +-0.513276008283918065415984751899E-13,_ + 0.868093304076654934187433687383E-14,_ +-0.152701504090308497198572355351E-14,_ + 0.278468625164935739650105251453E-15,_ +-0.524989043742176696808472933696E-16,_ + 0.102071799124856129247455787226E-16,_ +-0.204226467989971841308462421876E-17,_ + 0.419706417272648474408827228562E-18,_ +-0.884450817617281050816483737536E-19,_ + 0.190827262959471741995060168262E-19,_ +-0.420974622293519950336450865676E-20,_ + 0.948390405819837327641500214512E-21,_ +-0.217946786013667431994032574014E-21,_ + 0.510393686907145094993452562741E-22,_ +-0.121688311333441509089746779693E-22,_ + 0.295128916644787519294773757144E-23,_ +-0.727535376377284689714438950920E-24,_ + 0.182163904862307396121667115976E-24,_ +-0.462962996316331716612753482064E-25,_ + 0.119353979097157791523052371292E-25,_ +-0.311949328522014244931062147473E-26,_ + 0.826141973453346642284170028518E-27,_ +-0.221580337366098298302591177697E-27,_ + 0.601603167165426389045303124429E-28,_ +-0.165272509838212659649744302314E-28,_ + 0.459223035877302702795636377166E-29,_ +-0.129006276721326384737453212670E-29,_ + 0.366271848103200259081177078922E-30] +--R +--R +--R (3) +--R [1.9121732258 6055345391 51932651, - 0.0420835505 2868484375 5097498668, +--R 0.0017228196 2728432678 3371181578 35, +--R - 0.0000991578 2173444563 6455984232 2973, +--R 0.0000071760 9316802277 5052655906 65592, +--R - 0.6152733145 0951269682 7956791331 E -6, +--R 0.6024857106 5627583129 399970161 E -7, +--R - 0.6573848845 2883048229 5894189637 E -8, +--R 0.7853167541 8323998199 4810079871 E -9, +--R - 0.1013730288 0038789855 4202774257 E -9, +--R 0.1399770413 2267686027 7823488623 E -10, +--R - 0.2051008376 7838189961 8962318711 E -11, +--R 0.3168388726 0024778181 4907985818 E -12, +--R - 0.5132760082 8391806541 5984751899 E -13, +--R 0.8680933040 7665493418 7433687383 E -14, +--R - 0.1527015040 9030849719 8572355351 E -14, +--R 0.2784686251 6493573965 0105251453 E -15, +--R - 0.5249890437 4217669680 8472933696 E -16, +--R 0.1020717991 2485612924 7455787226 E -16, +--R - 0.2042264679 8997184130 8462421876 E -17, +--R 0.4197064172 7264847440 8827228562 E -18, +--R - 0.8844508176 1728105081 6483737536 E -19, +--R 0.1908272629 5947174199 5060168262 E -19, +--R - 0.4209746222 9351995033 6450865676 E -20, +--R 0.9483904058 1983732764 1500214512 E -21, +--R - 0.2179467860 1366743199 4032574014 E -21, +--R 0.5103936869 0714509499 3452562741 E -22, +--R - 0.1216883113 3344150908 9746779693 E -22, +--R 0.2951289166 4478751929 4773757144 E -23, +--R - 0.7275353763 7728468971 443895092 E -24, +--R 0.1821639048 6230739612 1667115976 E -24, +--R - 0.4629629963 1633171661 2753482064 E -25, +--R 0.1193539790 9715779152 3052371292 E -25, +--R - 0.3119493285 2201424493 1062147473 E -26, +--R 0.8261419734 5334664228 4170028518 E -27, +--R - 0.2215803373 6609829830 2591177697 E -27, +--R 0.6016031671 6542638904 5303124429 E -28, +--R - 0.1652725098 3821265964 9744302314 E -28, +--R 0.4592230358 7730270279 5636377166 E -29, +--R - 0.1290062767 2132638473 745321267 E -29, +--R 0.3662718481 0320025908 1177078922 E -30] +--R Type: List Float +--E 2 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 3 of 20 +[[-160.,0.993826695674061273878797850088,_ + Ei1(-160.0),Ei1(-160.0)-0.993826695674061273878797850088],_ +[-80.0,0.987801333094288773564522608410,_ + Ei1(-80.0),Ei1(-80.0)-0.987801333094288773564522608410],_ +[-53.0-1.0/3.0,0.981916290143194439617735426105,_ + Ei1(-53.0-1.0/3.0),Ei1(-53.0-1.0/3.0)-0.981916290143194439617735426105],_ +[-40.0,0.976164603185143050808000604060,_ + Ei1(-40.0),Ei1(-40.0)-0.976164603185143050808000604060],_ +[-32.0,0.970539884074663920462584664361,_ + Ei1(-32.0),Ei1(-32.0)-0.970539884074663920462584664361],_ +[-26.0-2.0/3.0,0.965036251123377035763536593528,_ + Ei1(-26.0-2.0/3.0),Ei1(-26.0-2.0/3.0)-0.965036251123377035763536593528],_ +[-22.0-6.0/7.0,0.959648271079367276165478970820,_ + Ei1(-22.0-6.0/7.0),Ei1(-22.0-6.0/7.0)-0.959648271079367276165478970820],_ +[-20.0,0.954370909919216833975195829433,_ + Ei1(-20.0),Ei1(-20.0)-0.954370909919216833975195829433],_ +[-17.0-7.0/9.0,0.949199490779745744606445346803,_ + Ei1(-17.0-7.0/9.0),Ei1(-17.0-7.0/9.0)-0.949199490779745744606445346803],_ +[-16.0,0.944129657736902978984149471583,_ + Ei1(-16.0),Ei1(-16.0)-0.944129657736902978984149471583],_ +[-14.0-6.0/11.0,0.939157344419284241240422409988,_ + Ei1(-14.0-6.0/11.0),Ei1(-14.0-6.0/11.0)-0.939157344419284241240422409988],_ +[-13.0-1.0/3.0,0.934278746653410464809375801650,_ + Ei1(-13.0-1.0/3.0),Ei1(-13.0-1.0/3.0)-0.934278746653410464809375801650],_ +[-12.0-4.0/13.0,0.929490298497214037725319679042,_ + Ei1(-12.0-4.0/13.0),Ei1(-12.0-4.0/13.0)-0.929490298497214037725319679042],_ +[-11.0-3.0/7.0,0.924788651140841696055993585492,_ + Ei1(-11.0-3.0/7.0),Ei1(-11.0-3.0/7.0)-0.924788651140841696055993585492],_ +[-10.0-2.0/3.0,0.920170654249445676202148012149,_ + Ei1(-10.0-2.0/3.0),Ei1(-10.0-2.0/3.0)-0.920170654249445676202148012149],_ +[-10.0,0.915633339397880818760698157666,_ + Ei1(-10.0),Ei1(-10.0)-0.915633339397880818760698157666]] +--R +--R +--R (4) +--R [[- 160.,0.99382669567406123,0.99382669567406123,0.], +--R [- 80.,0.98780133309428875,0.98780133309428886,1.1102230246251565E-16], +--R [- 53.333333333333336,0.98191629014319448,0.98191629014319448,0.], +--R [- 40.,0.97616460318514309,0.97616460318514309,0.], +--R [- 32.,0.97053988407466396,0.97053988407466363,- 3.3306690738754696E-16], +--R [- 26.666666666666668,0.96503625112337699,0.96503625112337699,0.], +--R +--R [- 22.857142857142858, 0.95964827107936723, 0.95964827107936734, +--R 1.1102230246251565E-16] +--R , +--R [- 20.,0.9543709099192168,0.95437090991921691,1.1102230246251565E-16], +--R [- 17.777777777777779,0.94919949077974575,0.94919949077974575,0.], +--R [- 16.,0.94412965773690294,0.94412965773690294,0.], +--R +--R [- 14.545454545454545, 0.93915734441928422, 0.93915734441928411, +--R - 1.1102230246251565E-16] +--R , +--R +--R [- 13.333333333333334, 0.93427874665341049, 0.9342787466534106, +--R 1.1102230246251565E-16] +--R , +--R +--R [- 12.307692307692308, 0.92949029849721398, 0.92949029849721387, +--R - 1.1102230246251565E-16] +--R , +--R [- 11.428571428571429,0.92478865114084174,0.92478865114084174,0.], +--R +--R [- 10.666666666666666, 0.92017065424944566, 0.92017065424944577, +--R 1.1102230246251565E-16] +--R , +--R [- 10.,0.91563333939788083,0.91563333939788094,1.1102230246251565E-16]] +--R Type: List List OnePointCompletion DoubleFloat +--E 3 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(-10 \le x \le -4)$ in the polynomial +$$\sum_{k=0}^{43}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=(x+7)/3$ + +<<*>>= +--S 4 of 20 +bChebyshev:=[_ + 0.175755649606129373848762834691E1,_ +-0.435854151773616611705001867964E-01,_ +-0.797950713955842540133217027492E-02,_ +-0.148437232730371213850970210001E-02,_ +-0.280030198437751457486203954948E-03,_ +-0.534864851286579323039177361553E-04,_ +-0.103286724357355486610233266460E-04,_ +-0.201408331300553687732226198639E-05,_ +-0.396175843427386645822338443500E-06,_ +-0.785387276709663163067607656069E-07,_ +-0.156792598100746982624616270279E-07,_ +-0.315005593937639988250007372851E-08,_ +-0.636509682252420373040380263972E-09,_ +-0.129288811328056318356593121259E-09,_ +-0.263869099965925576132149942808E-10,_ +-0.540895828704506873491922207896E-11,_ +-0.111322278460108989997676692708E-11,_ +-0.229962472607446246184338864145E-12,_ +-0.476668238949519026223913482091E-13,_ +-0.991175674733527094506246643371E-14,_ +-0.206710358049570724000900805021E-14,_ +-0.432277678338338505645764394579E-15,_ +-0.906301479966501725514905603356E-16,_ +-0.190466997958166139744015963342E-16,_ +-0.401179232635027866346744227520E-17,_ +-0.846777213001683223134166334685E-18,_ +-0.179084273365869665555826492204E-18,_ +-0.379449063817147824401106175166E-19,_ +-0.805399923679827985260999654058E-20,_ +-0.171233901123620129743228671244E-20,_ +-0.364627405877496862086576562816E-21,_ +-0.777596963889394794353098157647E-22,_ +-0.166062849844840205662531950966E-22,_ +-0.355117862578825093005927145352E-23,_ +-0.760372268594135809295734653294E-24,_ +-0.163007413725849002889638374755E-24,_ +-0.349857520272863223507538497255E-25,_ +-0.751717962789009882460645145143E-26,_ +-0.161687744005272276298777317918E-26,_ +-0.348127008572475691748202271565E-27,_ +-0.750270777550246547010642233720E-28,_ +-0.161845436449591026807612330206E-28,_ +-0.349436677170516166749482836452E-29,_ +-0.755103690612616785856037026797E-30] +--R +--R +--R (5) +--R [1.7575564960 6129373848 762834691, - 0.0435854151 7736166117 0500186796 4, +--R - 0.0079795071 3955842540 1332170274 92, +--R - 0.0014843723 2730371213 8509702100 01, +--R - 0.0002800301 9843775145 7486203954 948, +--R - 0.0000534864 8512865793 2303917736 1553, +--R - 0.0000103286 7243573554 8661023326 646, +--R - 0.0000020140 8331300553 6877322261 98639, +--R - 0.3961758434 2738664582 23384435 E -6, +--R - 0.7853872767 0966316306 7607656069 E -7, +--R - 0.1567925981 0074698262 4616270279 E -7, +--R - 0.3150055939 3763998825 0007372851 E -8, +--R - 0.6365096822 5242037304 0380263972 E -9, +--R - 0.1292888113 2805631835 6593121259 E -9, +--R - 0.2638690999 6592557613 2149942808 E -10, +--R - 0.5408958287 0450687349 1922207896 E -11, +--R - 0.1113222784 6010898999 7676692708 E -11, +--R - 0.2299624726 0744624618 4338864145 E -12, +--R - 0.4766682389 4951902622 3913482091 E -13, +--R - 0.9911756747 3352709450 6246643371 E -14, +--R - 0.2067103580 4957072400 0900805021 E -14, +--R - 0.4322776783 3833850564 5764394579 E -15, +--R - 0.9063014799 6650172551 4905603356 E -16, +--R - 0.1904669979 5816613974 4015963342 E -16, +--R - 0.4011792326 3502786634 674422752 E -17, +--R - 0.8467772130 0168322313 4166334685 E -18, +--R - 0.1790842733 6586966555 5826492204 E -18, +--R - 0.3794490638 1714782440 1106175166 E -19, +--R - 0.8053999236 7982798526 0999654058 E -20, +--R - 0.1712339011 2362012974 3228671244 E -20, +--R - 0.3646274058 7749686208 6576562816 E -21, +--R - 0.7775969638 8939479435 3098157647 E -22, +--R - 0.1660628498 4484020566 2531950966 E -22, +--R - 0.3551178625 7882509300 5927145352 E -23, +--R - 0.7603722685 9413580929 5734653294 E -24, +--R - 0.1630074137 2584900288 9638374755 E -24, +--R - 0.3498575202 7286322350 7538497255 E -25, +--R - 0.7517179627 8900988246 0645145143 E -26, +--R - 0.1616877440 0527227629 8777317918 E -26, +--R - 0.3481270085 7247569174 8202271565 E -27, +--R - 0.7502707775 5024654701 064223372 E -28, +--R - 0.1618454364 4959102680 7612330206 E -28, +--R - 0.3494366771 7051616674 9482836452 E -29, +--R - 0.7551036906 1261678585 6037026797 E -30] +--R Type: List Float +--E 4 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 5 of 20 +[[-10.000,0.915633339397880818760698157661,_ + Ei2(-10.000),Ei2(-10.000)-0.915633339397880818760698157661],_ +[ -9.625,0.912844461467993418856575662217,_ + Ei2( -9.625),Ei2( -9.625)-0.912844461467993418856575662217],_ +[ -9.250,0.909862751525424139378954274597,_ + Ei2( -9.250),Ei2( -9.250)-0.909862751525424139378954274597],_ +[ -8.875,0.906667270654753880334995756418,_ + Ei2( -8.875),Ei2( -8.875)-0.906667270654753880334995756418],_ +[ -8.500,0.903233901973207844144682926135,_ + Ei2( -8.500),Ei2( -8.500)-0.903233901973207844144682926135],_ +[ -8.125,0.899534717688473836301415777697,_ + Ei2( -8.125),Ei2( -8.125)-0.899534717688473836301415777697],_ +[ -7.750,0.895537187087539157179475513219,_ + Ei2( -7.750),Ei2( -7.750)-0.895537187087539157179475513219],_ +[ -7.375,0.891203176321254316267087476258,_ + Ei2( -7.375),Ei2( -7.375)-0.891203176321254316267087476258],_ +[ -7.000,0.886487672536429352893993846569,_ + Ei2( -7.000),Ei2( -7.000)-0.886487672536429352893993846569],_ +[ -6.625,0.881337138468210200394305706270,_ + Ei2( -6.625),Ei2( -6.625)-0.881337138468210200394305706270],_ +[ -6.250,0.875687364788465932276462155532,_ + Ei2( -6.250),Ei2( -6.250)-0.875687364788465932276462155532],_ +[ -5.875,0.869460629454113410302047153364,_ + Ei2( -5.875),Ei2( -5.875)-0.869460629454113410302047153364],_ +[ -5.500,0.862561884690701422090918986586,_ + Ei2( -5.500),Ei2( -5.500)-0.862561884690701422090918986586],_ +[ -5.125,0.854873553890199542392425567234,_ + Ei2( -5.125),Ei2( -5.125)-0.854873553890199542392425567234],_ +[ -4.750,0.846248299103587361171665798810,_ + Ei2( -4.750),Ei2( -4.750)-0.846248299103587361171665798810],_ +[ -4.375,0.836498754556298741742152267582,_ + Ei2( -4.375),Ei2( -4.375)-0.836498754556298741742152267582],_ +[ -4.000,0.825382599604223332408183035504,_ + Ei2( -4.000),Ei2( -4.000)-0.825382599604223332408183035504]] +--R +--R +--R (6) +--R [[- 10.,0.91563333939788083,0.91563333939788083,0.], +--R [- 9.625,0.91284446146799347,0.91284446146799336,- 1.1102230246251565E-16], +--R [- 9.25,0.90986275152542417,0.90986275152542395,- 2.2204460492503131E-16], +--R [- 8.875,0.90666727065475383,0.90666727065475394,1.1102230246251565E-16], +--R [- 8.5,0.90323390197320785,0.90323390197320796,1.1102230246251565E-16], +--R [- 8.125,0.89953471768847382,0.89953471768847415,3.3306690738754696E-16], +--R [- 7.75,0.89553718708753915,0.89553718708753927,1.1102230246251565E-16], +--R [- 7.375,0.89120317632125434,0.89120317632125423,- 1.1102230246251565E-16], +--R [- 7.,0.88648767253642935,0.88648767253642924,- 1.1102230246251565E-16], +--R [- 6.625,0.88133713846821016,0.88133713846821005,- 1.1102230246251565E-16], +--R [- 6.25,0.87568736478846598,0.87568736478846598,0.], +--R [- 5.875,0.8694606294541134,0.86946062945411307,- 3.3306690738754696E-16], +--R [- 5.5,0.86256188469070139,0.86256188469070139,0.], +--R [- 5.125,0.85487355389019959,0.85487355389019937,- 2.2204460492503131E-16], +--R [- 4.75,0.84624829910358734,0.84624829910358745,1.1102230246251565E-16], +--R [- 4.375,0.83649875455629874,0.83649875455629874,0.], +--R [- 4.,0.82538259960422333,0.82538259960422322,- 1.1102230246251565E-16]] +--R Type: List List OnePointCompletion DoubleFloat +--E 5 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(-4 \le x \le 4)$ in the polynomial +$$\sum_{k=0}^{33}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=x/4$ + +<<*>>= +--S 6 of 20 +cChebyshev:=[_ +0.329370010376739129393905231421E1,_ +0.167983505237130291565505796064E1,_ +0.722043610567875435240299679644E0,_ +0.260031236054809561713740181192E0,_ +0.801049430817375022394742889237E-01,_ +0.215140366397633375480552483005E-01,_ +0.511620778993033120621968910894E-02,_ +0.109093286100739135605066199014E-02,_ +0.210741532023938916318348675226E-03,_ +0.371990451665188857095940815956E-04,_ +0.604349163712387875704767032866E-05,_ +0.909295427396260952649596541772E-06,_ +0.127380516065926478865567184969E-06,_ +0.166918574841098907390896143814E-07,_ +0.205441702640104792547612484551E-08,_ +0.238358444446681765914052321417E-09,_ +0.261538637888544296669068664148E-10,_ +0.272185862285416706446550268995E-11,_ +0.269375003198357929925326427442E-12,_ +0.254122094670726355467884089307E-13,_ +0.229013040686503709418510620516E-14,_ +0.197546573907462299401057650412E-15,_ +0.163402455192893174068635419984E-16,_ +0.129823543707963760991961293204E-17,_ +0.992258792507371059644632581302E-19,_ +0.730625280672210329447230880087E-20,_ +0.518967683460434512720780080019E-21,_ +0.356040945409970681128043162227E-22,_ +0.236197943257938642370187203948E-23,_ +0.151683776772145297549624516819E-24,_ +0.943908972224487442925310405245E-26,_ +0.569722755950369211989581737831E-27,_ +0.333833362779543303156597939562E-28,_ +0.190062601281619148526680482237E-29] +--R +--R +--R (7) +--R [3.2937001037 6739129393 905231421, 1.6798350523 7130291565 505796064, +--R 0.7220436105 6787543524 0299679644, 0.2600312360 5480956171 3740181192, +--R 0.0801049430 8173750223 9474288923 7, 0.0215140366 3976333754 8055248300 5, +--R 0.0051162077 8993033120 6219689108 94, +--R 0.0010909328 6100739135 6050661990 14, +--R 0.0002107415 3202393891 6318348675 226, +--R 0.0000371990 4516651888 5709594081 5956, +--R 0.0000060434 9163712387 8757047670 32866, +--R 0.9092954273 9626095264 9596541772 E -6, +--R 0.1273805160 6592647886 5567184969 E -6, +--R 0.1669185748 4109890739 0896143814 E -7, +--R 0.2054417026 4010479254 7612484551 E -8, +--R 0.2383584444 4668176591 4052321417 E -9, +--R 0.2615386378 8854429666 9068664148 E -10, +--R 0.2721858622 8541670644 6550268995 E -11, +--R 0.2693750031 9835792992 5326427442 E -12, +--R 0.2541220946 7072635546 7884089307 E -13, +--R 0.2290130406 8650370941 8510620516 E -14, +--R 0.1975465739 0746229940 1057650412 E -15, +--R 0.1634024551 9289317406 8635419984 E -16, +--R 0.1298235437 0796376099 1961293204 E -17, +--R 0.9922587925 0737105964 4632581302 E -19, +--R 0.7306252806 7221032944 7230880087 E -20, +--R 0.5189676834 6043451272 0780080019 E -21, +--R 0.3560409454 0997068112 8043162227 E -22, +--R 0.2361979432 5793864237 0187203948 E -23, +--R 0.1516837767 7214529754 9624516819 E -24, +--R 0.9439089722 2448744292 5310405245 E -26, +--R 0.5697227559 5036921198 9581737831 E -27, +--R 0.3338333627 7954330315 6597939562 E -28, +--R 0.1900626012 8161914852 6680482237 E -29] +--R Type: List Float +--E 6 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 7 of 20 +[[-4.0,0.491822344607818096479962798267,_ + Ei3(-4.0),Ei3(-4.0)-0.491822344607818096479962798267],_ +[-3.5,0.524842506644128356918258753311,_ + Ei3(-3.5),Ei3(-3.5)-0.524842506644128356918258753311],_ +[-3.0,0.562958778221279863138086024270,_ + Ei3(-3.0),Ei3(-3.0)-0.562958778221279863138086024270],_ +[-2.5,0.607368525858383064514266925640,_ + Ei3(-2.5),Ei3(-2.5)-0.607368525858383064514266925640],_ +[-2.0,0.659631678084769644795492023380,_ + Ei3(-2.0),Ei3(-2.0)-0.659631678084769644795492023380],_ +[-1.5,0.721800236944219929657623030310,_ + Ei3(-1.5),Ei3(-1.5)-0.721800236944219929657623030310],_ +[-1.0,0.796599599297053134283675865540,_ + Ei3(-1.0),Ei3(-1.0)-0.796599599297053134283675865540],_ +[-0.5,0.887684158235496725872151815870,_ + Ei3(-0.5),Ei3(-0.5)-0.887684158235496725872151815870],_ +[0.0,1.00000000000000000000000000000,_ + Ei3(0.0),Ei3(0.0)-1.00000000000000000000000000000],_ +[0.5,1.14030284104317205746248768807,_ + Ei3(0.5),Ei3(0.5)-1.14030284104317205746248768807],_ +[1.0,1.31790215145440389486000884424,_ + Ei3(1.0),Ei3(1.0)-1.31790215145440389486000884424],_ +[1.5,1.54573645074673373024859074039,_ + Ei3(1.5),Ei3(1.5)-1.54573645074673373024859074039],_ +[2.0,1.84193575527020599667788045934,_ + Ei3(2.0),Ei3(2.0)-1.84193575527020599667788045934],_ +[2.5,2.23210379912116511445340506423,_ + Ei3(2.5),Ei3(2.5)-2.23210379912116511445340506423],_ +[3.0,2.75266820568525800200219289740,_ + Ei3(3.0),Ei3(3.0)-2.75266820568525800200219289740],_ +[3.5,3.45582153193012412437300898811,_ + Ei3(3.5),Ei3(3.5)-3.45582153193012412437300898811],_ +[4.0,4.41684111100869913580118598668,_ + Ei3(4.0),Ei3(4.0)-4.41684111100869913580118598668]] +--R +--R +--R (8) +--R [[- 4.,0.4918223446078181,0.49182234460781826,1.6653345369377348E-16], +--R [- 3.5,0.52484250664412835,0.52484250664412835,0.], +--R [- 3.,0.56295877822127982,0.56295877822128015,3.3306690738754696E-16], +--R [- 2.5,0.60736852585838308,0.60736852585838341,3.3306690738754696E-16], +--R [- 2.,0.65963167808476963,0.65963167808476975,1.1102230246251565E-16], +--R [- 1.5,0.72180023694421991,0.72180023694422013,2.2204460492503131E-16], +--R [- 1.,0.79659959929705315,0.79659959929705293,- 2.2204460492503131E-16], +--R [- 0.5,0.88768415823549673,0.88768415823549696,2.2204460492503131E-16], +--R [0.,1.,1.,0.], +--R [0.5,1.1403028410431721,1.1403028410431715,- 6.6613381477509392E-16], +--R [1.,1.3179021514544038,1.3179021514544034,- 4.4408920985006262E-16], +--R [1.5,1.5457364507467337,1.5457364507467335,- 2.2204460492503131E-16], +--R [2.,1.841935755270206,1.8419357552702071,1.1102230246251565E-15], +--R [2.5,2.2321037991211652,2.2321037991211647,- 4.4408920985006262E-16], +--R [3.,2.7526682056852581,2.7526682056852589,8.8817841970012523E-16], +--R [3.5,3.4558215319301242,3.4558215319301238,- 4.4408920985006262E-16], +--R [4.,4.4168411110086989,4.4168411110087007,1.7763568394002505E-15]] +--R Type: List List OnePointCompletion DoubleFloat +--E 7 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(4 \le x \le 12)$ in the polynomial +$$\sum_{k=0}^{49}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=(x-8)/4$ + +<<*>>= +--S 8 of 20 +dChebyshev:=[_ + 0.245513353878129528673420457043E1,_ +-0.162438379130376524396002276856E0,_ + 0.449575308093572641480785417193E-01,_ +-0.674157867998922998848718835050E-02,_ +-0.130669714280329428051599341387E-02,_ + 0.138108314600072576020202089820E-02,_ +-0.585022879015965798687368242394E-03,_ + 0.174929934107891970038740976432E-03,_ +-0.404728149905293035522869333800E-04,_ + 0.721710241217099750035752600049E-05,_ +-0.861277697019867752414815450193E-06,_ +-0.251447529653225597779084739054E-09,_ -- E-06? or wrong place? + 0.379474713820149510814074505574E-07,_ +-0.144211796952119806160265640172E-07,_ + 0.393504929597610131087190848042E-08,_ +-0.928468940106331753047289210353E-09,_ + 0.203178956800654613366090995698E-09,_ +-0.429249850499236831427918026902E-10,_ + 0.899264717778123935268001544182E-11,_ +-0.190086911841210975242396635722E-11,_ + 0.409219891222373834526121178338E-12,_ +-0.899925343729319019825435824585E-13,_ + 0.201965467082426383354948543451E-13,_ +-0.461293026138308207194950531726E-14,_ + 0.106902307293863695668857256409E-14,_ +-0.250703007057007295692572254042E-15,_ + 0.593732250379155160706073763509E-16,_ +-0.141773458243766252344732005648E-16,_ + 0.340920375436080893426806402093E-17,_ +-0.824829026950549379288702529656E-18,_ + 0.200636971262144231398824095937E-18,_ +-0.490385166796742224403498152027E-19,_ + 0.120373448234833217166664609324E-19,_ +-0.296628244714136825381453572575E-20,_ + 0.733551238428807599242142328436E-21,_ +-0.181992414290851127344263485604E-21,_ + 0.452862937429576060217359526404E-22,_ +-0.112998004375060961338906717853E-22,_ + 0.282668125129011656923764408445E-23,_ +-0.708771797716904961666732640699E-24,_ + 0.178110452401870951534401530034E-24,_ +-0.448500407661896357312006142358E-25,_ + 0.113154029257547662245053090840E-25,_ +-0.285995789977932163790414326136E-26,_ + 0.724077580692267361758172726753E-27,_ +-0.183613223412577898050666710105E-27,_ + 0.466312873522730486582600122073E-28,_ +-0.118595958891902887946724005478E-28,_ + 0.302029059055671310731137614875E-29,_ +-0.770165054816636606098827057102E-30] +--R +--R +--R (9) +--R [2.4551335387 8129528673 420457043, - 0.1624383791 3037652439 6002276856, +--R 0.0449575308 0935726414 8078541719 3, +--R - 0.0067415786 7998922998 8487188350 5, +--R - 0.0013066971 4280329428 0515993413 87, +--R 0.0013810831 4600072576 0202020898 2, +--R - 0.0005850228 7901596579 8687368242 394, +--R 0.0001749299 3410789197 0038740976 432, +--R - 0.0000404728 1499052930 3552286933 38, +--R 0.0000072171 0241217099 7500357526 00049, +--R - 0.8612776970 1986775241 4815450193 E -6, +--R - 0.2514475296 5322559777 9084739054 E -9, +--R 0.3794747138 2014951081 4074505574 E -7, +--R - 0.1442117969 5211980616 0265640172 E -7, +--R 0.3935049295 9761013108 7190848042 E -8, +--R - 0.9284689401 0633175304 7289210353 E -9, +--R 0.2031789568 0065461336 6090995698 E -9, +--R - 0.4292498504 9923683142 7918026902 E -10, +--R 0.8992647177 7812393526 8001544182 E -11, +--R - 0.1900869118 4121097524 2396635722 E -11, +--R 0.4092198912 2237383452 6121178338 E -12, +--R - 0.8999253437 2931901982 5435824585 E -13, +--R 0.2019654670 8242638335 4948543451 E -13, +--R - 0.4612930261 3830820719 4950531726 E -14, +--R 0.1069023072 9386369566 8857256409 E -14, +--R - 0.2507030070 5700729569 2572254042 E -15, +--R 0.5937322503 7915516070 6073763509 E -16, +--R - 0.1417734582 4376625234 4732005648 E -16, +--R 0.3409203754 3608089342 6806402093 E -17, +--R - 0.8248290269 5054937928 8702529656 E -18, +--R 0.2006369712 6214423139 8824095937 E -18, +--R - 0.4903851667 9674222440 3498152027 E -19, +--R 0.1203734482 3483321716 6664609324 E -19, +--R - 0.2966282447 1413682538 1453572575 E -20, +--R 0.7335512384 2880759924 2142328436 E -21, +--R - 0.1819924142 9085112734 4263485604 E -21, +--R 0.4528629374 2957606021 7359526404 E -22, +--R - 0.1129980043 7506096133 8906717853 E -22, +--R 0.2826681251 2901165692 3764408445 E -23, +--R - 0.7087717977 1690496166 6732640699 E -24, +--R 0.1781104524 0187095153 4401530034 E -24, +--R - 0.4485004076 6189635731 2006142358 E -25, +--R 0.1131540292 5754766224 505309084 E -25, +--R - 0.2859957899 7793216379 0414326136 E -26, +--R 0.7240775806 9226736175 8172726753 E -27, +--R - 0.1836132234 1257789805 0666710105 E -27, +--R 0.4663128735 2273048658 2600122073 E -28, +--R - 0.1185959588 9190288794 6724005478 E -28, +--R 0.3020290590 5567131073 1137614875 E -29, +--R - 0.7701650548 1663660609 8827057102 E -30] +--R Type: List Float +--E 8 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 9 of 20 +[[4.0,1.43820803145448278470968670330,_ + Ei4(4.0),Ei4(4.0)-1.43820803145448278470968670330],_ +[4.5,1.39641902962974607100674523183,_ + Ei4(4.5), Ei4(4.5)-1.39641902962974607100674523183],_ +[5.0,1.35383127745528597790189174047,_ + Ei4(5.0),Ei4(5.0)-1.35383127745528597790189174047],_ +[5.5,1.31414356574211924541219816991,_ + Ei4(5.5),Ei4(5.5)-1.31414356574211924541219816991],_ +[6.0,1.27888386048956161892314099578,_ + Ei4(6.0),Ei4(6.0)-1.27888386048956161892314099578],_ +[6.5,1.24839115500170148640741941387,_ + Ei4(6.5),Ei4(6.5)-1.24839115500170148640741941387],_ +[7.0,1.22240805236053105903656846622,_ + Ei4(7.0),Ei4(7.0)-1.22240805236053105903656846622],_ +[7.5,1.20042149959963078643879158950,_ + Ei4(7.5),Ei4(7.5)-1.20042149959963078643879158950],_ +[8.0,1.18184798698720797317739362644,_ + Ei4(8.0),Ei4(8.0)-1.18184798698720797317739362644],_ +[8.5,1.16612652581174849439918142965,_ + Ei4(8.5),Ei4(8.5)-1.16612652581174849439918142965],_ +[9.0,1.15275920870892481322396814952,_ + Ei4(9.0),Ei4(9.0)-1.15275920870892481322396814952],_ +[9.5,1.14132347595262420155338560641,_ + Ei4(9.5),Ei4(9.5)-1.14132347595262420155338560641],_ +[10.0,1.13147020473410778034051681355,_ + Ei4(10.0),Ei4(10.0)-1.13147020473410778034051681355],_ +[10.5,1.12291557001776060642888630755,_ + Ei4(10.5),Ei4(10.5)-1.12291557001776060642888630755],_ +[11.0,1.11543093899803844164779434229,_ + Ei4(11.0),Ei4(11.0)-1.11543093899803844164779434229],_ +[11.5,1.10883292630507730586855234934,_ + Ei4(11.5),Ei4(11.5)-1.10883292630507730586855234934],_ +[12.0,1.10297454490675907267241234953,_ + Ei4(12.0),Ei4(12.0)-1.10297454490675907267241234953]] +--R +--R +--R (10) +--R [[4.,1.4382080314544827,1.4382080314544827,0.], +--R [4.5,1.3964190296297461,1.3964190296297465,4.4408920985006262E-16], +--R [5.,1.3538312774552861,1.3538312774552856,- 4.4408920985006262E-16], +--R [5.5,1.3141435657421192,1.314143565742119,- 2.2204460492503131E-16], +--R [6.,1.2788838604895616,1.2788838604895618,2.2204460492503131E-16], +--R [6.5,1.2483911550017015,1.2483911550017011,- 4.4408920985006262E-16], +--R [7.,1.222408052360531,1.222408052360531,0.], +--R [7.5,1.2004214995996307,1.2004214995996305,- 2.2204460492503131E-16], +--R [8.,1.1818479869872081,1.1818479869872081,0.], +--R [8.5,1.1661265258117486,1.1661265258117477,- 8.8817841970012523E-16], +--R [9.,1.1527592087089249,1.1527592087089251,2.2204460492503131E-16], +--R [9.5,1.1413234759526243,1.1413234759526236,- 6.6613381477509392E-16], +--R [10.,1.1314702047341079,1.1314702047341079,0.], +--R [10.5,1.1229155700177607,1.1229155700177604,- 2.2204460492503131E-16], +--R [11.,1.1154309389980384,1.115430938998039,6.6613381477509392E-16], +--R [11.5,1.1088329263050773,1.1088329263050771,- 2.2204460492503131E-16], +--R [12.,1.1029745449067592,1.1029745449067592,0.]] +--R Type: List List OnePointCompletion DoubleFloat +--E 9 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(12 \le x \le 32)$ in the polynomial +$$\sum_{k=0}^{47}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=(x-22)/10$ + +<<*>>= +--S 10 of 20 +eChebyshev:=[_ + 0.211702864043698668329789991614E1,_ +-0.320423727375485794990618303177E-01,_ + 0.889173207735316835890182400335E-02,_ +-0.250795280518929937088352442063E-02,_ + 0.720278946595987548875760902487E-03,_ +-0.210349005850113053423531441256E-03,_ + 0.620573231827693216588857730842E-04,_ +-0.182656674981670265449155689733E-04,_ + 0.527065157528936375807788296811E-05,_ --? 7560 or 7580? +-0.145966654761994575323066719367E-05,_ + 0.378171997358963671980484193981E-06,_ +-0.884258128284071920077971589012E-07,_ + 0.174174919853839361377350309156E-07,_ +-0.231351774704369063506474480152E-08,_ +-0.122860981918086238832104835230E-09,_ + 0.234996623632286370478311381926E-09,_ +-0.110071940102726287690738963049E-09,_ + 0.384827515786120711149705563369E-10,_ +-0.114844096749001589658439301603E-10,_ + 0.305687629308852082630893626200E-11,_ +-0.738827872928473566454163131431E-12,_ + 0.163093309416594110564148013749E-12,_ +-0.327698937331271249657111774748E-13,_ + 0.589811434707131961711164283918E-14,_ +-0.909970763595649204643554720718E-15,_ + 0.104075238266955386585405697541E-15,_ +-0.180981542605922793227163355935E-17,_ +-0.377709884256394773369593494417E-17,_ + 0.158033290102847957136759888420E-17,_ +-0.468429175880882730648433752957E-18,_ + 0.119951685259198093707533478542E-18,_ +-0.282359474984186517679349931117E-19,_ + 0.629373806564463522627520190349E-20,_ +-0.135241024950479756305343973177E-20,_ + 0.283710605385529141590980426210E-21,_ +-0.586700742024638323531936371015E-22,_ + 0.120524763609547311112449686917E-22,_ +-0.247444661699884869728416011246E-23,_ + 0.509996258583785008142986465688E-24,_ +-0.105838257877542240887093294733E-24,_ + 0.221527624507048278566429387155E-25,_ +-0.467927875475696258671852546231E-26,_ + 0.997287299060207704824269828079E-27,_ +-0.214326794521678804591907805844E-27,_ + 0.464065690883818114338414829515E-28,_ +-0.101144734921151390948461800780E-28,_ + 0.221721152271007711093046878345E-29,_ +-0.488489046924378553224914645512E-30] +--R +--R +--R (11) +--R [2.1170286404 3698668329 789991614, - 0.0320423727 3754857949 9061830317 7, +--R 0.0088917320 7735316835 8901824003 35, +--R - 0.0025079528 0518929937 0883524420 63, +--R 0.0007202789 4659598754 8875760902 487, +--R - 0.0002103490 0585011305 3423531441 256, +--R 0.0000620573 2318276932 1658885773 0842, +--R - 0.0000182656 6749816702 6544915568 9733, +--R 0.0000052706 5157528936 3758077882 96811, +--R - 0.0000014596 6654761994 5753230667 19367, +--R 0.3781719973 5896367198 0484193981 E -6, +--R - 0.8842581282 8407192007 7971589012 E -7, +--R 0.1741749198 5383936137 7350309156 E -7, +--R - 0.2313517747 0436906350 6474480152 E -8, +--R - 0.1228609819 1808623883 210483523 E -9, +--R 0.2349966236 3228637047 8311381926 E -9, +--R - 0.1100719401 0272628769 0738963049 E -9, +--R 0.3848275157 8612071114 9705563369 E -10, +--R - 0.1148440967 4900158965 8439301603 E -10, +--R 0.3056876293 0885208263 08936262 E -11, +--R - 0.7388278729 2847356645 4163131431 E -12, +--R 0.1630933094 1659411056 4148013749 E -12, +--R - 0.3276989373 3127124965 7111774748 E -13, +--R 0.5898114347 0713196171 1164283918 E -14, +--R - 0.9099707635 9564920464 3554720718 E -15, +--R 0.1040752382 6695538658 5405697541 E -15, +--R - 0.1809815426 0592279322 7163355935 E -17, +--R - 0.3777098842 5639477336 9593494417 E -17, +--R 0.1580332901 0284795713 675988842 E -17, +--R - 0.4684291758 8088273064 8433752957 E -18, +--R 0.1199516852 5919809370 7533478542 E -18, +--R - 0.2823594749 8418651767 9349931117 E -19, +--R 0.6293738065 6446352262 7520190349 E -20, +--R - 0.1352410249 5047975630 5343973177 E -20, +--R 0.2837106053 8552914159 098042621 E -21, +--R - 0.5867007420 2463832353 1936371015 E -22, +--R 0.1205247636 0954731111 2449686917 E -22, +--R - 0.2474446616 9988486972 8416011246 E -23, +--R 0.5099962585 8378500814 2986465688 E -24, +--R - 0.1058382578 7754224088 7093294733 E -24, +--R 0.2215276245 0704827856 6429387155 E -25, +--R - 0.4679278754 7569625867 1852546231 E -26, +--R 0.9972872990 6020770482 4269828079 E -27, +--R - 0.2143267945 2167880459 1907805844 E -27, +--R 0.4640656908 8381811433 8414829515 E -28, +--R - 0.1011447349 2115139094 846180078 E -28, +--R 0.2217211522 7100771109 3046878345 E -29, +--R - 0.4884890469 2437855322 4914645512 E -30] +--R Type: List Float +--E 10 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 11 of 20 +[[12.00,1.10297454490675907267241234952,_ + Ei5(12.00),Ei5(12.00)-1.10297454490675907267241234952],_ +[13.25,1.09084489821547569266468614954,_ + Ei5(13.25),Ei5(13.25)-1.09084489821547569266468614954],_ +[14.50,1.08135139573519128506346643795,_ + Ei5(14.50),Ei5(14.50)-1.08135139573519128506346643795],_ +[15.75,1.07370138419975723712157900374,_ + Ei5(15.75),Ei5(15.75)-1.07370138419975723712157900374],_ +[17.00,1.06739369195853783129572196197,_ + Ei5(17.00),Ei5(17.00)-1.06739369195853783129572196197],_ +[18.25,1.06209660862215024268372647556,_ + Ei5(18.25),Ei5(18.25)-1.06209660862215024268372647556],_ +[19.50,1.05758134215872503195393949410,_ + Ei5(19.50),Ei5(19.50)-1.05758134215872503195393949410],_ +[20.75,1.05368445128940944082102194964,_ + Ei5(20.75),Ei5(20.75)-1.05368445128940944082102194964],_ +[22.00,1.05028571968518979411780664532,_ + Ei5(22.00),Ei5(22.00)-1.05028571968518979411780664532],_ +[23.25,1.04729455170532485811492365591,_ + Ei5(23.25),Ei5(23.25)-1.04729455170532485811492365591],_ +[24.50,1.04464126790464363689761075289,_ + Ei5(24.50),Ei5(24.50)-1.04464126790464363689761075289],_ +[25.75,1.04227133720232023885710928048,_ + Ei5(25.75),Ei5(25.75)-1.04227133720232023885710928048],_ +[27.00,1.04014143832301043813713899754,_ + Ei5(27.00),Ei5(27.00)-1.04014143832301043813713899754],_ +[28.25,1.03821670036014587680056548394,_ + Ei5(28.25),Ei5(28.25)-1.03821670036014587680056548394],_ +[29.50,1.03646872629241184575154685419,_ + Ei5(29.50),Ei5(29.50)-1.03646872629241184575154685419],_ +[30.75,1.03487414989647969472990938990,_ + Ei5(30.75),Ei5(30.75)-1.03487414989647969472990938990],_ +[32.00,1.03341356421624104943493552567,_ + Ei5(32.00),Ei5(32.00)-1.03341356421624104943493552567]] +--R +--R +--R (12) +--R [[12.,1.1029745449067592,1.1029745449067585,- 6.6613381477509392E-16], +--R [13.25,1.0908448982154757,1.090844898215475,- 6.6613381477509392E-16], +--R [14.5,1.0813513957351912,1.0813513957351915,2.2204460492503131E-16], +--R [15.75,1.0737013841997571,1.0737013841997574,2.2204460492503131E-16], +--R [17.,1.0673936919585378,1.0673936919585385,6.6613381477509392E-16], +--R [18.25,1.0620966086221502,1.0620966086221502,0.], +--R [19.5,1.057581342158725,1.0575813421587252,2.2204460492503131E-16], +--R [20.75,1.0536844512894095,1.0536844512894095,0.], +--R [22.,1.0502857196851898,1.0502857196851898,0.], +--R [23.25,1.0472945517053249,1.0472945517053245,- 4.4408920985006262E-16], +--R [24.5,1.0446412679046437,1.0446412679046437,0.], +--R [25.75,1.0422713372023202,1.04227133720232,- 2.2204460492503131E-16], +--R [27.,1.0401414383230105,1.0401414383230101,- 4.4408920985006262E-16], +--R [28.25,1.0382167003601459,1.0382167003601459,0.], +--R [29.5,1.0364687262924119,1.0364687262924113,- 6.6613381477509392E-16], +--R [30.75,1.0348741498964797,1.0348741498964795,- 2.2204460492503131E-16], +--R [32.,1.033413564216241,1.0334135642162412,2.2204460492503131E-16]] +--R Type: List List OnePointCompletion DoubleFloat +--E 11 + +@ +These are the Chebyshev coefficients used by Axiom in the range +$(32 \le x < \infty)$ in the polynomial +$$\sum_{k=0}^{46}\ ^{'}{A_kT_k(t)}$$ +with the scaling factor $t=(64/X)-1$ + +<<*>>= + +--S 12 of 20 +fChebyshev:=[_ + 0.203284394579616699087873844202E1,_ + 0.166992045203136285147618434339E-01,_ + 0.284528472436134680742489985325E-03,_ + 0.756394435851620648948786693854E-05,_ + 0.279897128945085915750484318090E-06,_ + 0.135790182853453106952556392593E-07,_ + 0.834359620204046925585610289412E-09,_ + 0.637097172764024843827524337306E-10,_ + 0.600724760881186123576083084850E-11,_ + 0.702287617467977359075059216588E-12,_ + 0.101830267370368769309667322152E-12,_ + 0.176181290343088004040656741554E-13,_ + 0.325082861423536069424072007647E-14,_ + 0.507177002550581867881479300685E-15,_ + 0.166517738704329429853520036957E-16,_ +-0.316675389079751440072410018963E-16,_ +-0.158840376366414151548423134074E-16,_ +-0.417551325613801883089626455063E-17,_ +-0.289234774970714188202868862358E-18,_ + 0.280062590339660807289978777339E-18,_ + 0.132293863953927089140532005364E-18,_ + 0.180444744417730199585334811191E-19,_ +-0.790538408652261656202021080364E-20,_ +-0.443571136636957344718167314045E-20,_ +-0.426410399497810261760579779746E-21,_ + 0.392010176693714390725625388636E-21,_ + 0.152737805134396364472804486402E-21,_ +-0.102484952704949060786953149788E-22,_ +-0.213490787477108937948904287231E-22,_ +-0.323913947516023687614279789345E-23,_ + 0.214218376229645970296249355934E-23,_ + 0.823460941961899553169207838151E-24,_ +-0.152465282962067210811495038147E-24,_ +-0.137820828248824401290438126477E-24,_ + 0.213131120142873706791513005998E-26,_ + 0.201264965187132665859213006507E-25,_ + 0.199553566205637402320607178286E-26,_ +-0.279899581220179711426020884464E-26,_ +-0.553451183050700250949784942560E-27,_ + 0.388499542268455253129749000696E-27,_ + 0.112130440723307012540043264712E-27,_ +-0.556656828674459488057823816866E-28,_ +-0.204548261246513576288865878722E-28,_ + 0.845381406448938089437361193598E-29,_ + 0.356575515120151526590791715785E-29,_ +-0.138365242347797751810195772006E-29,_ +-0.606214265320934505767865286306E-30] +--R +--R +--R (13) +--R [2.0328439457 9616699087 873844202, 0.0166992045 2031362851 4761843433 9, +--R 0.0002845284 7243613468 0742489985 325, +--R 0.0000075639 4435851620 6489487866 93854, +--R 0.2798971289 4508591575 048431809 E -6, +--R 0.1357901828 5345310695 2556392593 E -7, +--R 0.8343596202 0404692558 5610289412 E -9, +--R 0.6370971727 6402484382 7524337306 E -10, +--R 0.6007247608 8118612357 608308485 E -11, +--R 0.7022876174 6797735907 5059216588 E -12, +--R 0.1018302673 7036876930 9667322152 E -12, +--R 0.1761812903 4308800404 0656741554 E -13, +--R 0.3250828614 2353606942 4072007647 E -14, +--R 0.5071770025 5058186788 1479300685 E -15, +--R 0.1665177387 0432942985 3520036957 E -16, +--R - 0.3166753890 7975144007 2410018963 E -16, +--R - 0.1588403763 6641415154 8423134074 E -16, +--R - 0.4175513256 1380188308 9626455063 E -17, +--R - 0.2892347749 7071418820 2868862358 E -18, +--R 0.2800625903 3966080728 9978777339 E -18, +--R 0.1322938639 5392708914 0532005364 E -18, +--R 0.1804447444 1773019958 5334811191 E -19, +--R - 0.7905384086 5226165620 2021080364 E -20, +--R - 0.4435711366 3695734471 8167314045 E -20, +--R - 0.4264103994 9781026176 0579779746 E -21, +--R 0.3920101766 9371439072 5625388636 E -21, +--R 0.1527378051 3439636447 2804486402 E -21, +--R - 0.1024849527 0494906078 6953149788 E -22, +--R - 0.2134907874 7710893794 8904287231 E -22, +--R - 0.3239139475 1602368761 4279789345 E -23, +--R 0.2142183762 2964597029 6249355934 E -23, +--R 0.8234609419 6189955316 9207838151 E -24, +--R - 0.1524652829 6206721081 1495038147 E -24, +--R - 0.1378208282 4882440129 0438126477 E -24, +--R 0.2131311201 4287370679 1513005998 E -26, +--R 0.2012649651 8713266585 9213006507 E -25, +--R 0.1995535662 0563740232 0607178286 E -26, +--R - 0.2798995812 2017971142 6020884464 E -26, +--R - 0.5534511830 5070025094 978494256 E -27, +--R 0.3884995422 6845525312 9749000696 E -27, +--R 0.1121304407 2330701254 0043264712 E -27, +--R - 0.5566568286 7445948805 7823816866 E -28, +--R - 0.2045482612 4651357628 8865878722 E -28, +--R 0.8453814064 4893808943 7361193598 E -29, +--R 0.3565755151 2015152659 0791715785 E -29, +--R - 0.1383652423 4779775181 0195772006 E -29, +--R - 0.6062142653 2093450576 7865286306 E -30] +--R Type: List Float +--E 12 + +@ +In the following table there are 4 columns. The first column +is the argument of Ei(x) shown in Table 4 in \cite{1}. The second +column is the exact value shown in the table. Column 3 is the +value returned by Axiom and column 4 is the difference. +See special.spad.dvi for details. + +<<*>>= +--S 13 of 20 +[[32,1.03341356421624104943493552567,_ + Ei6(32.0),Ei6(32.0)-1.03341356421624104943493552567],_ +[34+2/15,1.03118521236465926355875784663,_ + Ei6(34.0+2/15),Ei6(34.0+2/15)-1.03118521236465926355875784663],_ +[36+4/7,1.02897740410580800863378435059,_ + Ei6(36.0+4/7),Ei6(36.0+4/7)-1.02897740410580800863378435059],_ +[39+5/13,1.02678968370902852450984510823,_ + Ei6(39.0+5/13),Ei6(39.0+5/13)-1.02678968370902852450984510823],_ +[42+2/3,1.02462161468107839101187804247,_ + Ei6(42.0+2/3),Ei6(42.0+2/3)-1.02462161468107839101187804247],_ +[46+6/11,1.02247277840542059591275364791,_ + Ei6(46.0+6/11),Ei6(46.0+6/11)-1.02247277840542059591275364791],_ +[51+1/5,1.02034277293078377487217829808,_ + Ei6(51.0+1/5),Ei6(51.0+1/5)-1.02034277293078377487217829808],_ +[56+8/9,1.01823121188483269682337017143,_ + Ei6(56.0+8/9),Ei6(56.0+8/9)-1.01823121188483269682337017143],_ +[64,1.01613772349432532170357100831,_ + Ei6(64.0),Ei6(64.0)-1.01613772349432532170357100831],_ +[73+1/7,1.01406194969697133145942329335,_ + Ei6(73.0+1/7),Ei6(73.0+1/7)-1.01406194969697133145942329335],_ +[85+1/3,1.01200354533298848201864466702,_ + Ei6(85.0+1/3),Ei6(85.0+1/3)-1.01200354533298848201864466702],_ +[102+2/5,1.00996217740644975574367545570,_ + Ei6(102.0+2/5),Ei6(102.0+2/5)-1.00996217740644975574367545570],_ +[128,1.00793752440814018281776821694,_ + Ei6(128.0),Ei6(128.0)-1.00793752440814018281776821694],_ +[170+2/3,1.00592927569292911294663030932,_ + Ei6(170.0+2/3),Ei6(170.0+2/3)-1.00592927569292911294663030932],_ +[256,1.00393713090569862788009078297,_ + Ei6(256.0),Ei6(256.0)-1.00393713090569862788009078297],_ +[512,1.00196079945071192531337468473,_ + Ei6(512.0),Ei6(512.0)-1.00196079945071192531337468473],_ +[infinity(),1.00000000000000000000000000001,_ + Ei6(infinity()),Ei6(infinity())-1.00000000000000000000000000001]] +--R +--R +--R (14) +--R [[32.,1.033413564216241,1.0334135642162412,2.2204460492503131E-16], +--R +--R 512 +--R [---, 1.0311852123 6465926355 875784663, 1.0311852123646588, +--R 15 +--R - 4.4408920985006262E-16] +--R , +--R 256 +--R [---,1.0289774041 0580800863 378435059,1.028977404105808,0.], +--R 7 +--R 512 +--R [---,1.0267896837 0902852450 984510823,1.0267896837090285,0.], +--R 13 +--R +--R 128 +--R [---, 1.0246216146 8107839101 187804247, 1.0246216146810787, +--R 3 +--R 2.2204460492503131E-16] +--R , +--R 512 +--R [---,1.0224727784 0542059591 275364791,1.0224727784054206,0.], +--R 11 +--R 256 +--R [---,1.0203427729 3078377487 217829808,1.0203427729307837,0.], +--R 5 +--R +--R 512 +--R [---, 1.0182312118 8483269682 337017143, 1.0182312118848329, +--R 9 +--R 2.2204460492503131E-16] +--R , +--R [64.,1.0161377234943254,1.0161377234943252,- 2.2204460492503131E-16], +--R +--R 512 +--R [---, 1.0140619496 9697133145 942329335, 1.0140619496969712, +--R 7 +--R - 2.2204460492503131E-16] +--R , +--R 256 +--R [---,1.0120035453 3298848201 864466702,1.0120035453329885,0.], +--R 3 +--R +--R 512 +--R [---, 1.0099621774 0644975574 36754557, 1.0099621774064493, +--R 5 +--R - 4.4408920985006262E-16] +--R , +--R [128.,1.0079375244081401,1.0079375244081401,0.], +--R +--R 512 +--R [---, 1.0059292756 9292911294 663030932, 1.0059292756929286, +--R 3 +--R - 4.4408920985006262E-16] +--R , +--R [256.,1.0039371309056986,1.0039371309056981,- 4.4408920985006262E-16], +--R [512.,1.001960799450712,1.0019607994507116,- 4.4408920985006262E-16], +--R [infinity,1.,1.,0.]] +--R Type: List List Any +--E 13 + +@ +In the following table we show values returned by +$(Ei(x)-\log x-\gamma)/x$ where gamma is as shown above. +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. p 238 + +<<*>>= +--S 14 of 20 +h(x:DFLOAT):DFLOAT== + x=0.0::DFLOAT => 1.0 + y:DFLOAT:=retract(Ei(x)) + (y-log(x)-gamma)/x +--R +--R Function declaration h : DoubleFloat -> DoubleFloat has been added +--R to workspace. +--R Type: Void +--E 14 + +--S 15 of 20 +[[0.00,1.000000000,h(0.00),h(0.00)-1.000000000],_ + [0.01,1.002505566,h(0.01),h(0.01)-1.002505566],_ + [0.02,1.005022306,h(0.02),h(0.02)-1.005022306],_ + [0.03,1.007550283,h(0.03),h(0.03)-1.007550283],_ + [0.04,1.010089560,h(0.04),h(0.04)-1.010089560],_ + [0.05,1.012640202,h(0.05),h(0.05)-1.012640202],_ + [0.06,1.015202272,h(0.06),h(0.06)-1.015202272],_ + [0.07,1.017775836,h(0.07),h(0.07)-1.017775836],_ + [0.08,1.020360958,h(0.08),h(0.08)-1.020360958],_ + [0.09,1.022957705,h(0.09),h(0.09)-1.022957705],_ + [0.10,1.025566141,h(0.10),h(0.10)-1.025566141],_ + [0.11,1.028186335,h(0.11),h(0.11)-1.028186335],_ + [0.12,1.030818352,h(0.12),h(0.12)-1.030818352],_ + [0.13,1.033462259,h(0.13),h(0.13)-1.033462259],_ + [0.14,1.036118125,h(0.14),h(0.14)-1.036118125],_ + [0.15,1.038786018,h(0.15),h(0.15)-1.038786018],_ + [0.16,1.041466006,h(0.16),h(0.16)-1.041466006],_ + [0.17,1.044158158,h(0.17),h(0.17)-1.044158158],_ + [0.18,1.046862544,h(0.18),h(0.18)-1.046862544],_ + [0.19,1.049579234,h(0.19),h(0.19)-1.049579234],_ + [0.20,1.052308298,h(0.20),h(0.20)-1.052308298],_ + [0.21,1.055049807,h(0.21),h(0.21)-1.055049807],_ + [0.22,1.057803833,h(0.22),h(0.22)-1.057803833],_ + [0.23,1.060570446,h(0.23),h(0.23)-1.060570446],_ + [0.24,1.063349719,h(0.24),h(0.24)-1.063349719],_ + [0.25,1.066141726,h(0.25),h(0.25)-1.066141726],_ + [0.26,1.068946539,h(0.26),h(0.26)-1.068946539],_ + [0.27,1.071764232,h(0.27),h(0.27)-1.071764232],_ + [0.28,1.074594879,h(0.28),h(0.28)-1.074594879],_ + [0.29,1.077438555,h(0.29),h(0.29)-1.077438555],_ + [0.30,1.080295334,h(0.30),h(0.30)-1.080295334],_ + [0.31,1.083165293,h(0.31),h(0.31)-1.083165293],_ + [0.32,1.086048507,h(0.32),h(0.32)-1.086048507],_ + [0.33,1.088945053,h(0.33),h(0.33)-1.088945053],_ + [0.34,1.091855008,h(0.34),h(0.34)-1.091855008],_ + [0.35,1.094778451,h(0.35),h(0.35)-1.094778451],_ + [0.36,1.097715458,h(0.36),h(0.36)-1.097715458],_ + [0.37,1.100666108,h(0.37),h(0.37)-1.100666108],_ + [0.38,1.103630481,h(0.38),h(0.38)-1.103630481],_ + [0.39,1.106608656,h(0.39),h(0.39)-1.106608656],_ + [0.40,1.109600714,h(0.40),h(0.40)-1.109600714],_ + [0.41,1.112606735,h(0.41),h(0.41)-1.112606735],_ + [0.42,1.115626800,h(0.42),h(0.42)-1.115626800],_ + [0.43,1.118660991,h(0.43),h(0.43)-1.118660991],_ + [0.44,1.121709391,h(0.44),h(0.44)-1.121709391],_ + [0.45,1.124772082,h(0.45),h(0.45)-1.124772082],_ + [0.46,1.127849147,h(0.46),h(0.46)-1.127849147],_ + [0.47,1.130940671,h(0.47),h(0.47)-1.130940671],_ + [0.48,1.134046738,h(0.48),h(0.48)-1.134046738],_ + [0.49,1.137167432,h(0.49),h(0.49)-1.137167432],_ + [0.50,1.140302841,h(0.50),h(0.50)-1.140302841]] +--R +--R Compiling function h with type DoubleFloat -> DoubleFloat +--R +--R (16) +--R [[0.,1.,1.,0.], +--R [1.0E-2,1.002505566,1.002505565988876,- 1.1123990617534218E-11], +--R [2.0E-2,1.0050223060000001,1.0050223058229502,- 1.7704993027223281E-10], +--R +--R [2.9999999999999999E-2, 1.007550283, 1.0075502826056368, +--R - 3.9436320875552155E-10] +--R , +--R +--R [4.0000000000000001E-2, 1.0100895599999999, 1.0100895598460362, +--R - 1.5396373065357238E-10] +--R , +--R +--R [5.0000000000000003E-2, 1.012640202, 1.0126402014616676, +--R - 5.3833248969681335E-10] +--R , +--R +--R [5.9999999999999998E-2, 1.015202272, 1.0152022717813329, +--R - 2.1866708443951666E-10] +--R , +--R +--R [7.0000000000000007E-2, 1.017775836, 1.0177758355479642, +--R - 4.5203574217111964E-10] +--R , +--R +--R [8.0000000000000002E-2, 1.0203609579999999, 1.0203609579215664, +--R - 7.8433481931483584E-11] +--R , +--R +--R [8.9999999999999997E-2, 1.0229577050000001, 1.0229577044820872, +--R - 5.1791282373869763E-10] +--R , +--R +--R [0.10000000000000001, 1.0255661410000001, 1.0255661412323602, +--R 2.3236013113603349E-10] +--R , +--R [0.11,1.028186335,1.0281863346010778,- 3.989222285838423E-10], +--R [0.12,1.030818352,1.0308183514457605,- 5.5423954314903767E-10], +--R [0.13,1.033462259,1.0334622590557541,5.5754068029045811E-11], +--R [0.14000000000000001,1.036118125,1.0361181251552536,1.5525358776358189E-10], +--R +--R [0.14999999999999999, 1.0387860179999999, 1.0387860179063644, +--R - 9.3635543763070928E-11] +--R , +--R [0.16,1.0414660060000001,1.0414660059121499,- 8.7850171581749237E-11], +--R +--R [0.17000000000000001, 1.0441581579999999, 1.0441581582197257, +--R 2.1972579311579921E-10] +--R , +--R +--R [0.17999999999999999, 1.0468625439999999, 1.0468625443233892, +--R 3.233893153264944E-10] +--R , +--R [0.19,1.0495792340000001,1.0495792341677359,1.6773582522944253E-10], +--R [0.20000000000000001,1.052308298,1.0523082981508358,1.5083578830399347E-10], +--R +--R [0.20999999999999999, 1.0550498070000001, 1.055049807127405, +--R 1.2740497545848939E-10] +--R , +--R [0.22,1.0578038329999999,1.0578038324120198,- 5.8798010904581588E-10], +--R +--R [0.23000000000000001, 1.0605704460000001, 1.0605704457823433, +--R - 2.1765678148710776E-10] +--R , +--R +--R [0.23999999999999999, 1.0633497190000001, 1.0633497194823853, +--R 4.8238524286148277E-10] +--R , +--R [0.25,1.0661417259999999,1.0661417262257755,2.2577562042158661E-10], +--R +--R [0.26000000000000001, 1.0689465389999999, 1.0689465391990731, +--R 1.9907320236711712E-10] +--R , +--R [0.27000000000000002,1.071764232,1.0717642320650853,6.5085270506415327E-11], +--R +--R [0.28000000000000003, 1.0745948789999999, 1.0745948789662336, +--R - 3.3766323070949511E-11] +--R , +--R +--R [0.28999999999999998, 1.0774385550000001, 1.0774385545279166, +--R - 4.7208348341598594E-10] +--R , +--R +--R [0.29999999999999999, 1.0802953340000001, 1.0802953338619241, +--R - 1.3807599508197654E-10] +--R , +--R [0.31,1.083165293,1.0831652925698594,- 4.3014058981327707E-10], +--R +--R [0.32000000000000001, 1.0860485070000001, 1.0860485067465939, +--R - 2.5340618492464273E-10] +--R , +--R +--R [0.33000000000000002, 1.088945053, 1.0889450529837443, +--R - 1.6255663481956617E-11] +--R , +--R [0.34000000000000002,1.091855008,1.0918550083731842,3.7318415024856222E-10], +--R +--R [0.34999999999999998, 1.094778451, 1.0947784505105673, +--R - 4.8943271657719833E-10] +--R , +--R +--R [0.35999999999999999, 1.0977154579999999, 1.0977154574988892, +--R - 5.0111070848402051E-10] +--R , +--R [0.37,1.100666108,1.1006661079520708,- 4.7929216151487708E-11], +--R [0.38,1.1036304809999999,1.1036304809985678,- 1.4321877017664519E-12], +--R +--R [0.39000000000000001, 1.1066086559999999, 1.1066086562850108, +--R 2.8501090376664706E-10] +--R , +--R +--R [0.40000000000000002, 1.1096007139999999, 1.1096007139798676, +--R - 2.0132340239342739E-11] +--R , +--R [0.40999999999999998,1.112606735,1.1126067347771349,- 2.228650597402293E-10] +--R , +--R [0.41999999999999998,1.1156268,1.1156267999000615,- 9.9938501918472866E-11], +--R [0.42999999999999999,1.118660991,1.1186609911048895,1.0488943047448629E-10], +--R [0.44,1.121709391,1.1217093906846374,- 3.1536262490305944E-10], +--R +--R [0.45000000000000001, 1.124772082, 1.1247720814728976, +--R - 5.2710236175812497E-10] +--R , +--R +--R [0.46000000000000002, 1.1278491470000001, 1.1278491468476701, +--R - 1.52329926450534E-10] +--R , +--R +--R [0.46999999999999997, 1.1309406710000001, 1.1309406707352239, +--R - 2.6477620096443388E-10] +--R , +--R +--R [0.47999999999999998, 1.1340467380000001, 1.134046737613986, +--R - 3.8601410956573545E-10] +--R , +--R [0.48999999999999999,1.137167432,1.1371674325184589,5.1845883142220828E-10], +--R [0.5,1.140302841,1.1403028410431715,4.3171466401759062E-11]] +--R Type: List List DoubleFloat +--E 15 + +@ +In the following table we show values returned by Ei(x). +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. pp 239-241 + +<<*>>= +--S 16 of 20 +[[0.50,0.454219905,Ei(0.50),Ei(0.50)-0.454219905],_ + [0.51,0.487032167,Ei(0.51),Ei(0.51)-0.487032167],_ + [0.52,0.519530633,Ei(0.52),Ei(0.52)-0.519530633],_ + [0.53,0.551730445,Ei(0.53),Ei(0.53)-0.551730445],_ + [0.54,0.583645931,Ei(0.54),Ei(0.54)-0.583645931],_ + [0.55,0.615290657,Ei(0.55),Ei(0.55)-0.615290657],_ + [0.56,0.646677490,Ei(0.56),Ei(0.56)-0.646677490],_ + [0.57,0.677818642,Ei(0.57),Ei(0.57)-0.677818642],_ + [0.58,0.708725720,Ei(0.58),Ei(0.58)-0.708725720],_ + [0.59,0.739409764,Ei(0.59),Ei(0.59)-0.739409764],_ + [0.60,0.769881290,Ei(0.60),Ei(0.60)-0.769881290],_ + [0.61,0.800150320,Ei(0.61),Ei(0.61)-0.800150320],_ + [0.62,0.830226417,Ei(0.62),Ei(0.62)-0.830226417],_ + [0.63,0.860118716,Ei(0.63),Ei(0.63)-0.860118716],_ + [0.64,0.889835949,Ei(0.64),Ei(0.64)-0.889835949],_ + [0.65,0.919386468,Ei(0.65),Ei(0.65)-0.919386468],_ + [0.66,0.948778277,Ei(0.66),Ei(0.66)-0.948778277],_ + [0.67,0.978019042,Ei(0.67),Ei(0.67)-0.978019042],_ + [0.68,1.007116121,Ei(0.68),Ei(0.68)-1.007116121],_ + [0.69,1.036076576,Ei(0.69),Ei(0.69)-1.036076576],_ + [0.70,1.064907195,Ei(0.70),Ei(0.70)-1.064907195],_ + [0.71,1.093614501,Ei(0.71),Ei(0.71)-1.093614501],_ + [0.72,1.122204777,Ei(0.72),Ei(0.72)-1.122204777],_ + [0.73,1.150684069,Ei(0.73),Ei(0.73)-1.150684069],_ + [0.74,1.179058208,Ei(0.74),Ei(0.74)-1.179058208],_ + [0.75,1.207332816,Ei(0.75),Ei(0.75)-1.207332816],_ + [0.76,1.235513319,Ei(0.76),Ei(0.76)-1.235513319],_ + [0.77,1.263604960,Ei(0.77),Ei(0.77)-1.263604960],_ + [0.78,1.291612805,Ei(0.78),Ei(0.78)-1.291612805],_ + [0.79,1.319541753,Ei(0.79),Ei(0.79)-1.319541753],_ + [0.80,1.347396548,Ei(0.80),Ei(0.80)-1.347396548],_ + [0.81,1.375181783,Ei(0.81),Ei(0.81)-1.375181783],_ + [0.82,1.402901910,Ei(0.82),Ei(0.82)-1.402901910],_ + [0.83,1.430561245,Ei(0.83),Ei(0.83)-1.430561245],_ + [0.84,1.458163978,Ei(0.84),Ei(0.84)-1.458163978],_ + [0.85,1.485714176,Ei(0.85),Ei(0.85)-1.485714176],_ + [0.86,1.513215791,Ei(0.86),Ei(0.86)-1.513215791],_ + [0.87,1.540672664,Ei(0.87),Ei(0.87)-1.540672664],_ + [0.88,1.568088534,Ei(0.88),Ei(0.88)-1.568088534],_ + [0.89,1.595467036,Ei(0.89),Ei(0.89)-1.595467036],_ + [0.90,1.622811714,Ei(0.90),Ei(0.90)-1.622811714],_ + [0.91,1.650126019,Ei(0.91),Ei(0.91)-1.650126019],_ + [0.92,1.677413317,Ei(0.92),Ei(0.92)-1.677413317],_ + [0.93,1.704676891,Ei(0.93),Ei(0.93)-1.704676891],_ + [0.94,1.731919946,Ei(0.94),Ei(0.94)-1.731919946],_ + [0.95,1.759145612,Ei(0.95),Ei(0.95)-1.759145612],_ + [0.96,1.786356947,Ei(0.96),Ei(0.96)-1.786356947],_ + [0.97,1.813556941,Ei(0.97),Ei(0.97)-1.813556941],_ + [0.98,1.840748519,Ei(0.98),Ei(0.98)-1.840748519],_ + [0.99,1.867934543,Ei(0.99),Ei(0.99)-1.867934543],_ + [1.00,1.895117816,Ei(1.00),Ei(1.00)-1.895117816],_ + [1.01,1.922301085,Ei(1.01),Ei(1.01)-1.922301085],_ + [1.02,1.949487042,Ei(1.02),Ei(1.02)-1.949487042],_ + [1.03,1.976678325,Ei(1.03),Ei(1.03)-1.976678325],_ + [1.04,2.003877525,Ei(1.04),Ei(1.04)-2.003877525],_ + [1.05,2.031087184,Ei(1.05),Ei(1.05)-2.031087184],_ + [1.06,2.058309800,Ei(1.06),Ei(1.06)-2.058309800],_ + [1.07,2.085547825,Ei(1.07),Ei(1.07)-2.085547825],_ + [1.08,2.112803672,Ei(1.08),Ei(1.08)-2.112803672],_ + [1.09,2.140079712,Ei(1.09),Ei(1.09)-2.140079712],_ + [1.10,2.167378280,Ei(1.10),Ei(1.10)-2.167378280],_ + [1.11,2.194701672,Ei(1.11),Ei(1.11)-2.194701672],_ + [1.12,2.222052152,Ei(1.12),Ei(1.12)-2.222052152],_ + [1.13,2.249431949,Ei(1.13),Ei(1.13)-2.249431949],_ + [1.14,2.276843260,Ei(1.14),Ei(1.14)-2.276843260],_ + [1.15,2.304288252,Ei(1.15),Ei(1.15)-2.304288252],_ + [1.16,2.331769062,Ei(1.16),Ei(1.16)-2.331769062],_ + [1.17,2.359287800,Ei(1.17),Ei(1.17)-2.359287800],_ + [1.18,2.386846549,Ei(1.18),Ei(1.18)-2.386846549],_ + [1.19,2.414447367,Ei(1.19),Ei(1.19)-2.414447367],_ + [1.20,2.442092285,Ei(1.20),Ei(1.20)-2.442092285],_ + [1.21,2.469783315,Ei(1.21),Ei(1.21)-2.469783315],_ + [1.22,2.497522442,Ei(1.22),Ei(1.22)-2.497522442],_ + [1.23,2.525311634,Ei(1.23),Ei(1.23)-2.525311634],_ + [1.24,2.553152836,Ei(1.24),Ei(1.24)-2.553152836],_ + [1.25,2.581047974,Ei(1.25),Ei(1.25)-2.581047974],_ + [1.26,2.608998956,Ei(1.26),Ei(1.26)-2.608998956],_ + [1.27,2.637007673,Ei(1.27),Ei(1.27)-2.637007673],_ + [1.28,2.665075997,Ei(1.28),Ei(1.28)-2.665075997],_ + [1.29,2.693205785,Ei(1.29),Ei(1.29)-2.693205785],_ + [1.30,2.721398880,Ei(1.30),Ei(1.30)-2.721398880],_ + [1.31,2.749657110,Ei(1.31),Ei(1.31)-2.749657110],_ + [1.32,2.777982287,Ei(1.32),Ei(1.32)-2.777982287],_ + [1.33,2.806376214,Ei(1.33),Ei(1.33)-2.806376214],_ + [1.34,2.834840677,Ei(1.34),Ei(1.34)-2.834840677],_ + [1.35,2.863377453,Ei(1.35),Ei(1.35)-2.863377453],_ + [1.36,2.891988308,Ei(1.36),Ei(1.36)-2.891988308],_ + [1.37,2.920674997,Ei(1.37),Ei(1.37)-2.920674997],_ + [1.38,2.949439263,Ei(1.38),Ei(1.38)-2.949439263],_ + [1.39,2.978282844,Ei(1.39),Ei(1.39)-2.978282844],_ + [1.40,3.007207464,Ei(1.40),Ei(1.40)-3.007207464],_ + [1.41,3.036214843,Ei(1.41),Ei(1.41)-3.036214843],_ + [1.42,3.065306691,Ei(1.42),Ei(1.42)-3.065306691],_ + [1.43,3.094484712,Ei(1.43),Ei(1.43)-3.094484712],_ + [1.44,3.123750601,Ei(1.44),Ei(1.44)-3.123750601],_ + [1.45,3.153106049,Ei(1.45),Ei(1.45)-3.153106049],_ + [1.46,3.182552741,Ei(1.46),Ei(1.46)-3.182552741],_ + [1.47,3.212092355,Ei(1.47),Ei(1.47)-3.212092355],_ + [1.48,3.241726566,Ei(1.48),Ei(1.48)-3.241726566],_ + [1.49,3.271457042,Ei(1.49),Ei(1.49)-3.271457042],_ + [1.50,3.301285449,Ei(1.50),Ei(1.50)-3.301285449],_ + [1.51,3.331213449,Ei(1.51),Ei(1.51)-3.331213449],_ + [1.52,3.361242701,Ei(1.52),Ei(1.52)-3.361242701],_ + [1.53,3.391374858,Ei(1.53),Ei(1.53)-3.391374858],_ + [1.54,3.421611576,Ei(1.54),Ei(1.54)-3.421611576],_ + [1.55,3.451954503,Ei(1.55),Ei(1.55)-3.451954503],_ + [1.56,3.482405289,Ei(1.56),Ei(1.56)-3.482405289],_ + [1.57,3.512965580,Ei(1.57),Ei(1.57)-3.512965580],_ + [1.58,3.543637024,Ei(1.58),Ei(1.58)-3.543637024],_ + [1.59,3.574421266,Ei(1.59),Ei(1.59)-3.574421266],_ + [1.60,3.605319949,Ei(1.60),Ei(1.60)-3.605319949],_ + [1.61,3.636334719,Ei(1.61),Ei(1.61)-3.636334719],_ + [1.62,3.667467221,Ei(1.62),Ei(1.62)-3.667467221],_ + [1.63,3.698719099,Ei(1.63),Ei(1.63)-3.698719099],_ + [1.64,3.730091999,Ei(1.64),Ei(1.64)-3.730091999],_ + [1.65,3.761587569,Ei(1.65),Ei(1.65)-3.761587569],_ + [1.66,3.793207456,Ei(1.66),Ei(1.66)-3.793207456],_ + [1.67,3.824953310,Ei(1.67),Ei(1.67)-3.824953310],_ + [1.68,3.856826783,Ei(1.68),Ei(1.68)-3.856826783],_ + [1.69,3.888829528,Ei(1.69),Ei(1.69)-3.888829528],_ + [1.70,3.920963201,Ei(1.70),Ei(1.70)-3.920963201],_ + [1.71,3.953229462,Ei(1.71),Ei(1.71)-3.953229462],_ + [1.72,3.985629972,Ei(1.72),Ei(1.72)-3.985629972],_ + [1.73,4.018166395,Ei(1.73),Ei(1.73)-4.018166395],_ + [1.74,4.050840400,Ei(1.74),Ei(1.74)-4.050840400],_ + [1.75,4.083653659,Ei(1.75),Ei(1.75)-4.083653659],_ + [1.76,4.116607847,Ei(1.76),Ei(1.76)-4.116607847],_ + [1.77,4.149704645,Ei(1.77),Ei(1.77)-4.149704645],_ + [1.78,4.182945736,Ei(1.78),Ei(1.78)-4.182945736],_ + [1.79,4.216332809,Ei(1.79),Ei(1.79)-4.216332809],_ + [1.80,4.249867557,Ei(1.80),Ei(1.80)-4.249867557],_ + [1.81,4.283551681,Ei(1.81),Ei(1.81)-4.283551681],_ + [1.82,4.317386883,Ei(1.82),Ei(1.82)-4.317386883],_ + [1.83,4.351374872,Ei(1.83),Ei(1.83)-4.351374872],_ + [1.84,4.385517364,Ei(1.84),Ei(1.84)-4.385517364],_ + [1.85,4.419816080,Ei(1.85),Ei(1.85)-4.419816080],_ + [1.86,4.454272746,Ei(1.86),Ei(1.86)-4.454272746],_ + [1.87,4.488889097,Ei(1.87),Ei(1.87)-4.488889097],_ + [1.88,4.523666872,Ei(1.88),Ei(1.88)-4.523666872],_ + [1.89,4.558607817,Ei(1.89),Ei(1.89)-4.558607817],_ + [1.90,4.593713687,Ei(1.90),Ei(1.90)-4.593713687],_ + [1.91,4.628986242,Ei(1.91),Ei(1.91)-4.628986242],_ + [1.92,4.664427249,Ei(1.92),Ei(1.92)-4.664427249],_ + [1.93,4.700038485,Ei(1.93),Ei(1.93)-4.700038485],_ + [1.94,4.735821734,Ei(1.94),Ei(1.94)-4.735821734],_ + [1.95,4.771778785,Ei(1.95),Ei(1.95)-4.771778785],_ + [1.96,4.807911438,Ei(1.96),Ei(1.96)-4.807911438],_ + [1.97,4.844221501,Ei(1.97),Ei(1.97)-4.844221501],_ + [1.98,4.880710791,Ei(1.98),Ei(1.98)-4.880710791],_ + [1.99,4.917381131,Ei(1.99),Ei(1.99)-4.917381131],_ + [2.00,4.954234356,Ei(2.00),Ei(2.00)-4.954234356]] +--R +--R +--R (17) +--R [[0.5,0.45421990499999998,0.45421990486317332,- 1.3682666111236585E-10], +--R +--R [0.51000000000000001, 0.48703216700000002, 0.48703216680456007, +--R - 1.9543994200788006E-10] +--R , +--R +--R [0.52000000000000002, 0.51953063300000002, 0.51953063245569719, +--R - 5.443028250340376E-10] +--R , +--R +--R [0.53000000000000003, 0.55173044500000001, 0.55173044523266401, +--R 2.3266399917787339E-10] +--R , +--R +--R [0.54000000000000004, 0.58364593099999995, 0.58364593072977955, +--R - 2.7022040161028826E-10] +--R , +--R +--R [0.55000000000000004, 0.61529065699999996, 0.61529065706218644, +--R 6.2186478189119043E-11] +--R , +--R +--R [0.56000000000000005, 0.64667748999999997, 0.64667748977430584, +--R - 2.2569413005157912E-10] +--R , +--R +--R [0.56999999999999995, 0.67781864199999997, 0.67781864189137597, +--R - 1.0862399868472039E-10] +--R , +--R +--R [0.57999999999999996, 0.70872572, 0.70872571962101083, +--R - 3.7898917337741977E-10] +--R , +--R +--R [0.58999999999999997, 0.73940976400000002, 0.73940976415103654, +--R 1.510365166268457E-10] +--R , +--R +--R [0.59999999999999998, 0.76988129000000005, 0.76988128993735938, +--R - 6.2640670428493195E-11] +--R , +--R +--R [0.60999999999999999, 0.80015031999999997, 0.80015031983004981, +--R - 1.699501650520574E-10] +--R , +--R [0.62,0.83022641699999999,0.83022641734618519,3.4618519162421535E-10], +--R [0.63,0.86011871600000001,0.86011871636343917,3.6343916764991491E-10], +--R +--R [0.64000000000000001, 0.88983594899999996, 0.88983594847818637, +--R - 5.2181359233571811E-10] +--R , +--R +--R [0.65000000000000002, 0.91938646800000001, 0.91938646824544334, +--R 2.4544333232512372E-10] +--R , +--R +--R [0.66000000000000003, 0.94877827699999995, 0.94877827649472835, +--R - 5.0527160233571067E-10] +--R , +--R +--R [0.67000000000000004, 0.97801904200000001, 0.97801904189549682, +--R - 1.045031838842192E-10] +--R , +--R +--R [0.68000000000000005, 1.0071161209999999, 1.0071161209277915, +--R - 7.2208461432410331E-11] +--R , +--R +--R [0.68999999999999995, 1.0360765759999999, 1.0360765763978435, +--R 3.978435358931165E-10] +--R , +--R [0.69999999999999996,1.064907195,1.0649071946242905,- 3.757094635403746E-10] +--R , +--R [0.70999999999999996,1.093614501,1.0936145014081782,4.0817815794014223E-10], +--R +--R [0.71999999999999997, 1.1222047770000001, 1.1222047768888612, +--R - 1.111388758801013E-10] +--R , +--R [0.72999999999999998,1.150684069,1.1506840693780345,3.780344925985446E-10], +--R +--R [0.73999999999999999, 1.1790582080000001, 1.1790582082553465, +--R 2.5534641068247765E-10] +--R , +--R [0.75,1.2073328160000001,1.2073328160012218,1.2216894162975223E-12], +--R +--R [0.76000000000000001, 1.2355133190000001, 1.2355133194354742, +--R 4.3547410122357633E-10] +--R , +--R +--R [0.77000000000000002, 1.2636049600000001, 1.2636049602240513, +--R 2.2405122201973882E-10] +--R , +--R +--R [0.78000000000000003, 1.291612805, 1.2916128047105979, +--R - 2.8940205787364448E-10] +--R , +--R [0.79000000000000004,1.319541753,1.3195417531244753,1.2447531894110853E-10], +--R +--R [0.80000000000000004, 1.3473965480000001, 1.3473965482123258, +--R 2.1232571256746269E-10] +--R , +--R +--R [0.81000000000000005, 1.3751817829999999, 1.3751817833361941, +--R 3.361941836033111E-10] +--R , +--R [0.81999999999999995,1.40290191,1.4029019100774811,7.7481132620960125E-11], +--R [0.82999999999999996,1.430561245,1.4305612453827297,3.8272962576968439E-10], +--R [0.83999999999999997,1.458163978,1.4581639782841678,2.8416780040174672E-10], +--R +--R [0.84999999999999998, 1.4857141760000001, 1.4857141762252541, +--R 2.2525403764461771E-10] +--R , +--R +--R [0.85999999999999999, 1.5132157909999999, 1.5132157910189581, +--R 1.8958168368499173E-11] +--R , +--R [0.87,1.5406726639999999,1.5406726644642923,4.6429238231837644E-10], +--R [0.88,1.5680885339999999,1.5680885336445423,- 3.5545766330358219E-10], +--R +--R [0.89000000000000001, 1.5954670360000001, 1.5954670359288246, +--R - 7.1175509930299086E-11] +--R , +--R +--R [0.90000000000000002, 1.622811714, 1.6228117136968674, +--R - 3.0313263010839364E-10] +--R , +--R +--R [0.91000000000000003, 1.650126019, 1.6501260188054063, +--R - 1.9459367450735954E-10] +--R , +--R +--R [0.92000000000000004, 1.6774133170000001, 1.677413316813162, +--R - 1.8683810054653804E-10] +--R , +--R +--R [0.93000000000000005, 1.7046768910000001, 1.7046768909800791, +--R - 1.9920953775454109E-11] +--R , +--R +--R [0.93999999999999995, 1.7319199460000001, 1.7319199460553549, +--R 5.5354831829390605E-11] +--R , +--R +--R [0.94999999999999996, 1.759145612, 1.7591456118676905, +--R - 1.3230949669207348E-10] +--R , +--R +--R [0.95999999999999996, 1.786356947, 1.7863569467301943, +--R - 2.6980573331059077E-10] +--R , +--R +--R [0.96999999999999997, 1.8135569410000001, 1.8135569406715355, +--R - 3.2846458886126584E-10] +--R , +--R +--R [0.97999999999999998, 1.8407485189999999, 1.8407485185040211, +--R - 4.9597881357499318E-10] +--R , +--R +--R [0.98999999999999999, 1.8679345430000001, 1.8679345427385856, +--R - 2.6141444564586891E-10] +--R , +--R [1.,1.895117816,1.8951178163559361,3.5593616942719564E-10], +--R [1.01,1.922301085,1.9223010854424856,4.4248560371329404E-10], +--R [1.02,1.9494870419999999,1.9494870416990668,- 3.0093305625200628E-10], +--R [1.03,1.976678325,1.976678324829928,- 1.7007195651785878E-10], +--R [1.04,2.003877525,2.0038775248189595,- 1.8104051591194548E-10], +--R [1.05,2.031087184,2.0310871840996643,9.9664276831390453E-11], +--R [1.0600000000000001,2.0583098,2.0583097996249284,- 3.7507152939042498E-10], +--R +--R [1.0700000000000001, 2.0855478249999999, 2.085547824842283, +--R - 1.5771695061062019E-10] +--R , +--R +--R [1.0800000000000001, 2.1128036720000001, 2.1128036715799325, +--R - 4.2006753631085303E-10] +--R , +--R +--R [1.0900000000000001, 2.1400797119999999, 2.1400797118485424, +--R - 1.5145751319778356E-10] +--R , +--R +--R [1.1000000000000001, 2.1673782799999999, 2.1673782795634038, +--R - 4.3659609261226251E-10] +--R , +--R +--R [1.1100000000000001, 2.1947016719999999, 2.1947016721913277, +--R 1.9132784245812218E-10] +--R , +--R +--R [1.1200000000000001, 2.2220521519999998, 2.2220521523263717, +--R 3.2637181845984742E-10] +--R , +--R +--R [1.1299999999999999, 2.2494319489999999, 2.2494319491981756, +--R 1.9817569807401014E-10] +--R , +--R +--R [1.1399999999999999, 2.2768432600000001, 2.2768432601165496, +--R 1.1654943676830953E-10] +--R , +--R +--R [1.1499999999999999, 2.3042882520000001, 2.304288251855628, +--R - 1.4437206985462581E-10] +--R , +--R +--R [1.1599999999999999, 2.3317690619999998, 2.3317690619808027, +--R - 1.9197088363398507E-11] +--R , +--R +--R [1.1699999999999999, 2.3592878000000002, 2.3592878001213737, +--R 1.2137357785491076E-10] +--R , +--R +--R [1.1799999999999999, 2.3868465489999999, 2.3868465491917359, +--R 1.9173596044197438E-10] +--R , +--R +--R [1.1899999999999999, 2.4144473670000002, 2.4144473665637345, +--R - 4.3626569024013406E-10] +--R , +--R [1.2,2.4420922850000002,2.4420922851926514,1.9265122830347536E-10], +--R [1.21,2.4697833149999999,2.4697833146991774,- 3.0082247803875362E-10], +--R [1.22,2.4975224420000002,2.497522442409561,4.095608296950104E-10], +--R [1.23,2.5253116339999999,2.5253116343560089,3.5600900005761105E-10], +--R [1.24,2.5531528360000002,2.5531528362393039,2.3930368797664414E-10], +--R [1.25,2.5810479740000001,2.5810479743554762,3.5547609300579097E-10], +--R [1.26,2.6089989560000002,2.6089989564882825,4.8828230347908175E-10], +--R [1.27,2.6370076729999998,2.6370076727691485,- 2.308513380455679E-10], +--R [1.28,2.6650759970000002,2.6650759965061086,- 4.9389159428869789E-10], +--R [1.29,2.693205785,2.6932057849832494,- 1.6750600906334512E-11], +--R [1.3,2.7213988800000002,2.7213988802320226,2.3202240129194251E-10], +--R +--R [1.3100000000000001, 2.7496571099999998, 2.7496571097757787, +--R - 2.2422108614250646E-10] +--R , +--R +--R [1.3200000000000001, 2.7779822869999999, 2.777982287348725, +--R 3.4872504883765032E-10] +--R , +--R +--R [1.3300000000000001, 2.8063762140000001, 2.8063762135905539, +--R - 4.0944625467886908E-10] +--R , +--R +--R [1.3400000000000001, 2.8348406769999999, 2.8348406767178056, +--R - 2.8219426795317304E-10] +--R , +--R [1.3500000000000001,2.863377453,2.8633774531730753,1.7307533184407475E-10], +--R +--R [1.3600000000000001, 2.8919883080000002, 2.8919883082530298, +--R 2.5302959727468988E-10] +--R , +--R +--R [1.3700000000000001, 2.9206749969999999, 2.9206749967162473, +--R - 2.8375257699053691E-10] +--R , +--R +--R [1.3799999999999999, 2.9494392629999999, 2.9494392633717355, +--R 3.7173553124603131E-10] +--R , +--R +--R [1.3899999999999999, 2.9782828440000002, 2.9782828436490232, +--R - 3.5097702522079999E-10] +--R , +--R +--R [1.3999999999999999, 3.0072074639999999, 3.0072074641506457, +--R 1.5064571812217764E-10] +--R , +--R +--R [1.4099999999999999, 3.0362148430000002, 3.0362148431877847, +--R 1.8778445465272853E-10] +--R , +--R [1.4199999999999999,3.065306691,3.065306691299837,2.9983704408209633E-10], +--R +--R [1.4299999999999999, 3.0944847119999999, 3.0944847117585681, +--R - 2.4143176347024564E-10] +--R , +--R +--R [1.4399999999999999, 3.1237506009999998, 3.1237506010575933, +--R 5.7593485536244771E-11] +--R , +--R [1.45,3.1531060489999998,3.1531060493877443,3.8774450317191622E-10], +--R [1.46,3.1825527409999999,3.1825527410990038,9.9003916176343409E-11], +--R [1.47,3.2120923549999998,3.2120923551495331,1.4953327465150323E-10], +--R [1.48,3.2417265660000001,3.2417265655423857,- 4.5761439082525612E-10], +--R [1.49,3.2714570420000002,3.2714570417503985,- 2.4960167266385724E-10], +--R [1.5,3.3012854489999999,3.3012854491297974,1.297975060765566E-10], +--R [1.51,3.3312134489999998,3.3312134493229739,3.2297409191528459E-10], +--R [1.52,3.3612427010000001,3.3612427006508958,- 3.4910430102286227E-10], +--R [1.53,3.3913748579999998,3.3913748584955847,4.9558490644585618E-10], +--R [1.54,3.4216115760000001,3.4216115756731122,- 3.2688785012169319E-10], +--R [1.55,3.4519545030000001,3.4519545027974381,- 2.0256196719969921E-10], +--R +--R [1.5600000000000001, 3.4824052889999999, 3.4824052886355643, +--R - 3.6443559281451599E-10] +--R , +--R +--R [1.5700000000000001, 3.5129655799999999, 3.5129655804542947, +--R 4.5429482398162691E-10] +--R , +--R +--R [1.5800000000000001, 3.5436370240000001, 3.5436370243589819, +--R 3.5898173322834737E-10] +--R , +--R +--R [1.5900000000000001, 3.5744212659999999, 3.5744212656246064, +--R - 3.7539349406756628E-10] +--R , +--R +--R [1.6000000000000001, 3.6053199490000001, 3.6053199490194707, +--R 1.9470647316666145E-11] +--R , +--R +--R [1.6100000000000001, 3.6363347190000002, 3.6363347191218383, +--R 1.2183809516841393E-10] +--R , +--R +--R [1.6200000000000001, 3.6674672209999999, 3.6674672206298222, +--R - 3.7017766629787729E-10] +--R , +--R +--R [1.6299999999999999, 3.6987190989999998, 3.6987190986647667, +--R - 3.3523317455319557E-10] +--R , +--R +--R [1.6399999999999999, 3.7300919989999999, 3.7300919990684158, +--R 6.8415939580290797E-11] +--R , +--R [1.6499999999999999,3.761587569,3.7615875686941349,- 3.0586511101660108E-10] +--R , +--R +--R [1.6599999999999999, 3.7932074560000002, 3.7932074556923925, +--R - 3.0760771707605272E-10] +--R , +--R +--R [1.6699999999999999, 3.8249533100000002, 3.824953309790788, +--R - 2.092122031172039E-10] +--R , +--R +--R [1.6799999999999999, 3.8568267829999998, 3.8568267825688243, +--R - 4.3117553971683265E-10] +--R , +--R [1.6899999999999999,3.888829528,3.8888295277276339,- 2.723661296499813E-10], +--R [1.7,3.9209632010000002,3.9209632013549038,3.5490366201429424E-10], +--R [1.71,3.9532294619999999,3.953229462185158,1.8515811106567526E-10], +--R [1.72,3.9856299719999999,3.985629971855627,- 1.4437295803304551E-10], +--R [1.73,4.0181663949999997,4.0181663951578672,1.5786749685275936E-10], +--R [1.74,4.0508404000000002,4.0508404002853169,2.8531665918762883E-10], +--R [1.75,4.0836536590000003,4.0836536590769557,7.6955330996497651E-11], +--R [1.76,4.116607847,4.1166078472572494,2.5724933294668517E-10], +--R [1.77,4.1497046449999999,4.1497046446724992,- 3.2750069323128628E-10], +--R [1.78,4.1829457359999997,4.1829457355238073,- 4.7619241883012364E-10], +--R [1.79,4.2163328089999998,4.2163328085967509,- 4.0324898975541146E-10], +--R [1.8,4.249867557,4.2498675574879341,4.879341375385593E-10], +--R +--R [1.8100000000000001, 4.2835516809999996, 4.2835516808285554, +--R - 1.7144419217629547E-10] +--R , +--R +--R [1.8200000000000001, 4.3173868830000002, 4.3173868825051116, +--R - 4.9488857456481128E-10] +--R , +--R +--R [1.8300000000000001, 4.3513748720000001, 4.3513748718773684, +--R - 1.2263168258641599E-10] +--R , +--R [1.8400000000000001,4.385517364,4.3855173639937215,- 6.2785332488601853E-12] +--R , +--R +--R [1.8500000000000001, 4.4198160800000004, 4.4198160798040753, +--R - 1.9592505395849003E-10] +--R , +--R [1.8600000000000001,4.454272746,4.4542727463703349,3.7033487387816422E-10], +--R +--R [1.8700000000000001, 4.4888890970000004, 4.4888890970746314, +--R 7.4630968072142423E-11] +--R , +--R +--R [1.8799999999999999, 4.5236668719999997, 4.523666871825391, +--R - 1.7460877188568702E-10] +--R , +--R +--R [1.8899999999999999, 4.5586078170000004, 4.5586078172613478, +--R 2.6134738817518155E-10] +--R , +--R +--R [1.8999999999999999, 4.5937136870000002, 4.5937136869535857, +--R - 4.6414427856689144E-11] +--R , +--R +--R [1.9099999999999999, 4.6289862419999999, 4.6289862416057304, +--R - 3.9426950593224319E-10] +--R , +--R +--R [1.9199999999999999, 4.6644272490000001, 4.6644272492523706, +--R 2.5237056888727238E-10] +--R , +--R +--R [1.9299999999999999, 4.7000384850000003, 4.7000384854557851, +--R 4.5578474328067387E-10] +--R , +--R +--R [1.9399999999999999, 4.7358217339999999, 4.7358217335010897, +--R - 4.9891024644921345E-10] +--R , +--R [1.95,4.7717787850000004,4.7717787845898787,- 4.1012171436705103E-10], +--R [1.96,4.8079114379999996,4.8079114380324146,3.241495960537577E-11], +--R [1.97,4.8442215009999998,4.8442215014384944,4.3849457398437153E-10], +--R [1.98,4.8807107910000003,4.8807107909070337,- 9.2966523368431808E-11], +--R [1.99,4.917381131,4.9173811312144435,2.1444357400923764E-10], +--R [2.,4.9542343559999997,4.9542343560018924,1.8927082123809669E-12]] +--R Type: List List OnePointCompletion DoubleFloat +--E 16 + +@ +In the following table we show values returned by +$xe^{-x}Ei(x)$, chosen to keep the values in a reasonable range. +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. pp 242-243 + +<<*>>= +--S 17 of 20 +f(x)==x/10.0*exp(-x/10.0)*Ei(x/10.0) +--R +--R Type: Void +--E 17 + +--S 18 of 20 +[[2.0,1.340965420,f(2.0),f(2.0)-1.340965420],_ + [2.1,1.371486802,f(2.1),f(2.1)-1.371486802],_ + [2.2,1.397421992,f(2.2),f(2.2)-1.397421992],_ + [2.3,1.419171534,f(2.3),f(2.3)-1.419171534],_ + [2.4,1.437118315,f(2.4),f(2.4)-1.437118315],_ + [2.5,1.451625159,f(2.5),f(2.5)-1.451625159],_ + [2.6,1.463033397,f(2.6),f(2.6)-1.463033397],_ + [2.7,1.471662153,f(2.7),f(2.7)-1.471662153],_ + [2.8,1.477808187,f(2.8),f(2.8)-1.477808187],_ + [2.9,1.481746162,f(2.9),f(2.9)-1.481746162],_ + [3.0,1.483729204,f(3.0),f(3.0)-1.483729204],_ + [3.1,1.483989691,f(3.1),f(3.1)-1.483989691],_ + [3.2,1.482740191,f(3.2),f(3.2)-1.482740191],_ + [3.3,1.480174491,f(3.3),f(3.3)-1.480174491],_ + [3.4,1.476468706,f(3.4),f(3.4)-1.476468706],_ + [3.5,1.471782389,f(3.5),f(3.5)-1.471782389],_ + [3.6,1.466259659,f(3.6),f(3.6)-1.466259659],_ + [3.7,1.460030313,f(3.7),f(3.7)-1.460030313],_ + [3.8,1.453210902,f(3.8),f(3.8)-1.453210902],_ + [3.9,1.445905765,f(3.9),f(3.9)-1.445905765],_ + [4.0,1.438208032,f(4.0),f(4.0)-1.438208032],_ + [4.1,1.430200557,f(4.1),f(4.1)-1.430200557],_ + [4.2,1.421956813,f(4.2),f(4.2)-1.421956813],_ + [4.3,1.413541719,f(4.3),f(4.3)-1.413541719],_ + [4.4,1.405012424,f(4.4),f(4.4)-1.405012424],_ + [4.5,1.396419030,f(4.5),f(4.5)-1.396419030],_ + [4.6,1.387805263,f(4.6),f(4.6)-1.387805263],_ + [4.7,1.379209093,f(4.7),f(4.7)-1.379209093],_ + [4.8,1.370663313,f(4.8),f(4.8)-1.370663313],_ + [4.9,1.362196054,f(4.9),f(4.9)-1.362196054],_ + [5.0,1.353831278,f(5.0),f(5.0)-1.353831278],_ + [5.1,1.345589212,f(5.1),f(5.1)-1.345589212],_ + [5.2,1.337486755,f(5.2),f(5.2)-1.337486755],_ + [5.3,1.329537845,f(5.3),f(5.3)-1.329537845],_ + [5.4,1.321753788,f(5.4),f(5.4)-1.321753788],_ + [5.5,1.314143566,f(5.5),f(5.5)-1.314143566],_ + [5.6,1.306714107,f(5.6),f(5.6)-1.306714107],_ + [5.7,1.299470536,f(5.7),f(5.7)-1.299470536],_ + [5.8,1.292416395,f(5.8),f(5.8)-1.292416395],_ + [5.9,1.285553849,f(5.9),f(5.9)-1.285553849],_ + [6.0,1.278883860,f(6.0),f(6.0)-1.278883860],_ + [6.1,1.272406357,f(6.1),f(6.1)-1.272406357],_ + [6.2,1.266120373,f(6.2),f(6.2)-1.266120373],_ + [6.3,1.260024184,f(6.3),f(6.3)-1.260024184],_ + [6.4,1.254115417,f(6.4),f(6.4)-1.254115417],_ + [6.5,1.248391155,f(6.5),f(6.5)-1.248391155],_ + [6.6,1.242848032,f(6.6),f(6.6)-1.242848032],_ + [6.7,1.237482309,f(6.7),f(6.7)-1.237482309],_ + [6.8,1.232289952,f(6.8),f(6.8)-1.232289952],_ + [6.9,1.227266684,f(6.9),f(6.9)-1.227266684],_ + [7.0,1.222408053,f(7.0),f(7.0)-1.222408053],_ + [7.1,1.217709472,f(7.1),f(7.1)-1.217709472],_ + [7.2,1.213166264,f(7.2),f(7.2)-1.213166264],_ + [7.3,1.208773699,f(7.3),f(7.3)-1.208773699],_ + [7.4,1.204527026,f(7.4),f(7.4)-1.204527026],_ + [7.5,1.200421500,f(7.5),f(7.5)-1.200421500],_ + [7.6,1.196452401,f(7.6),f(7.6)-1.196452401],_ + [7.7,1.192615063,f(7.7),f(7.7)-1.192615063],_ + [7.8,1.188904881,f(7.8),f(7.8)-1.188904881],_ + [7.9,1.185317334,f(7.9),f(7.9)-1.185317334],_ + [8.0,1.181847987,f(8.0),f(8.0)-1.181847987],_ + [8.1,1.178492509,f(8.1),f(8.1)-1.178492509],_ + [8.2,1.175246676,f(8.2),f(8.2)-1.175246676],_ + [8.3,1.172106376,f(8.3),f(8.3)-1.172106376],_ + [8.4,1.169067617,f(8.4),f(8.4)-1.169067617],_ + [8.5,1.166126526,f(8.5),f(8.5)-1.166126526],_ + [8.6,1.163279354,f(8.6),f(8.6)-1.163279354],_ + [8.7,1.160522476,f(8.7),f(8.7)-1.160522476],_ + [8.8,1.157852390,f(8.8),f(8.8)-1.157852390],_ + [8.9,1.155265719,f(8.9),f(8.9)-1.155265719],_ + [9.0,1.152759209,f(9.0),f(9.0)-1.152759209],_ + [9.1,1.150329724,f(9.1),f(9.1)-1.150329724],_ + [9.2,1.147974251,f(9.2),f(9.2)-1.147974251],_ + [9.3,1.145689889,f(9.3),f(9.3)-1.145689889],_ + [9.4,1.143473855,f(9.4),f(9.4)-1.143473855],_ + [9.5,1.141323476,f(9.5),f(9.5)-1.141323476],_ + [9.6,1.139236185,f(9.6),f(9.6)-1.139236185],_ + [9.7,1.137209523,f(9.7),f(9.7)-1.137209523],_ + [9.8,1.135241130,f(9.8),f(9.8)-1.135241130],_ + [9.9,1.133328746,f(9.9),f(9.9)-1.133328746],_ + [10.0,1.131470205,f(10.0),f(10.0)-1.131470205]] +--R +--R Compiling function f with type Float -> OnePointCompletion +--R DoubleFloat +--R +--R (19) +--R [[2.,1.3409654200000001,- 0.13456013299662745,- 1.4755255529966276], +--R +--R [2.1000000000000001, 1.3714868019999999, - 0.12968783850914051, +--R - 1.5011746405091404] +--R , +--R +--R [2.2000000000000002, 1.3974219919999999, - 0.12432857913849607, +--R - 1.521750571138496] +--R , +--R [2.2999999999999998,1.419171534,- 0.11851397777493734,- 1.5376855117749373], +--R [2.3999999999999999,1.437118315,- 0.1122732093067646,- 1.5493915243067646], +--R [2.5,1.451625159,- 0.10563327984220373,- 1.5572584388422037], +--R +--R [2.6000000000000001, 1.463033397, - 9.8619263183169451E-2, +--R - 1.5616526601831695] +--R , +--R +--R [2.7000000000000002, 1.471662153, - 9.1254502584207586E-2, +--R - 1.5629166555842076] +--R , +--R +--R [2.7999999999999998, 1.4778081869999999, - 8.3560784069182492E-2, +--R - 1.5613689710691825] +--R , +--R +--R [2.8999999999999999, 1.4817461620000001, - 7.5558486253840138E-2, +--R - 1.5573046482538402] +--R , +--R [3.,1.4837292040000001,- 6.7266710614573164E-2,- 1.5509959146145733], +--R +--R [3.1000000000000001, 1.4839896910000001, - 5.8703395368669309E-2, +--R - 1.5426930863686694] +--R , +--R +--R [3.2000000000000002, 1.482740191, - 4.9885415529372513E-2, +--R - 1.5326256065293724] +--R , +--R +--R [3.2999999999999998, 1.4801744910000001, - 4.0828671227296824E-2, +--R - 1.5210031622272968] +--R , +--R +--R [3.3999999999999999, 1.4764687059999999, - 3.1548166016793916E-2, +--R - 1.5080168720167939] +--R , +--R [3.5,1.4717823889999999,- 2.2058076588733416E-2,- 1.4938404655887334], +--R +--R [3.6000000000000001, 1.4662596590000001, - 1.2371815072632724E-2, +--R - 1.4786314740726327] +--R , +--R +--R [3.7000000000000002, 1.4600303130000001, - 2.5020849182828334E-3, +--R - 1.4625323979182829] +--R , +--R +--R [3.7999999999999998, 1.4532109019999999, 7.5390688098497787E-3, +--R - 1.4456718331901501] +--R , +--R [3.8999999999999999,1.445905765,1.774021402057336E-2,- 1.4281655509794267], +--R [4.,1.4382080319999999,2.8090490467135878E-2,- 1.4101175415328639], +--R [4.0999999999999996,1.430200557,3.8579572390008463E-2,- 1.3916209846099916], +--R [4.2000000000000002,1.421956813,4.9197634492545148E-2,- 1.3727591785074549], +--R +--R [4.2999999999999998, 1.4135417189999999, 5.9935320871702981E-2, +--R - 1.3536063981282969] +--R , +--R +--R [4.4000000000000004, 1.4050124239999999, 7.0783716577210318E-2, +--R - 1.3342287074227897] +--R , +--R [4.5,1.3964190299999999,8.1734321515770605E-2,- 1.3146847084842292], +--R [4.5999999999999996,1.387805263,9.2779026453518668E-2,- 1.2950262365464813], +--R +--R [4.7000000000000002, 1.3792090930000001, 0.10391009090110491, +--R - 1.2752990020988952] +--R , +--R +--R [4.7999999999999998, 1.3706633130000001, 0.11512012269240564, +--R - 1.2555431903075944] +--R , +--R [4.9000000000000004,1.362196054,0.12640205909068003,- 1.23579399490932], +--R [5.,1.3538312779999999,0.13774914927563506,- 1.2160821287243648], +--R +--R [5.0999999999999996, 1.3455892119999999, 0.14915493808180313, +--R - 1.1964342739181968] +--R , +--R [5.2000000000000002,1.337486755,0.16061325087332862,- 1.1768735041266714], +--R +--R [5.2999999999999998, 1.3295378449999999, 0.17211817945300276, +--R - 1.1574196655469973] +--R , +--R +--R [5.4000000000000004, 1.3217537880000001, 0.1836640689145036, +--R - 1.1380897190854964] +--R , +--R [5.5,1.314143566,0.19524550535650245,- 1.1188980606434975], +--R +--R [5.5999999999999996, 1.3067141069999999, 0.20685730438580638, +--R - 1.0998568026141935] +--R , +--R +--R [5.7000000000000002, 1.2994705360000001, 0.21849450034417656, +--R - 1.0809760356558236] +--R , +--R +--R [5.7999999999999998, 1.2924163950000001, 0.23015233620004197, +--R - 1.0622640587999581] +--R , +--R [5.9000000000000004,1.285553849,0.24182625405213551,- 1.0437275949478646], +--R [6.,1.2788838600000001,0.25351188619722104,- 1.0253719738027791], +--R +--R [6.0999999999999996, 1.2724063569999999, 0.26520504671863687, +--R - 1.0072013102813631] +--R , +--R +--R [6.2000000000000002, 1.2661203729999999, 0.27690172355643178, +--R - 0.98921864944356819] +--R , +--R +--R [6.2999999999999998, 1.2600241839999999, 0.28859807102347912, +--R - 0.97142611297652082] +--R , +--R [6.4000000000000004,1.254115417,0.30029040273517321,- 0.9538250142648268], +--R [6.5,1.248391155,0.31197518492319382,- 0.93641597007680621], +--R +--R [6.5999999999999996, 1.2428480319999999, 0.32364903010640295, +--R - 0.91919900189359693] +--R , +--R [6.7000000000000002,1.237482309,0.33530869109425609,- 0.90217361790574391], +--R +--R [6.7999999999999998, 1.2322899519999999, 0.34695105530018927, +--R - 0.88533889669981058] +--R , +--R +--R [6.9000000000000004, 1.2272666839999999, 0.3585731393443225, +--R - 0.86869354465567739] +--R , +--R [7.,1.2224080530000001,0.37017208392651269,- 0.85223596907348742], +--R +--R [7.0999999999999996, 1.2177094719999999, 0.38174514895231304, +--R - 0.83596432304768686] +--R , +--R [7.2000000000000002,1.213166264,0.39328970889579112,- 0.81987655510420887], +--R [7.2999999999999998,1.208773699,0.40480324838440268,- 0.80397045061559735], +--R +--R [7.4000000000000004, 1.2045270260000001, 0.4162833579922634, +--R - 0.78824366800773671] +--R , +--R [7.5,1.2004215,0.42772773022919919,- 0.77269376977080084], +--R +--R [7.5999999999999996, 1.1964524009999999, 0.43913415571389103, +--R - 0.75731824528610892] +--R , +--R +--R [7.7000000000000002, 1.1926150630000001, 0.45050051952030151, +--R - 0.74211454347969852] +--R , +--R [7.7999999999999998,1.188904881,0.46182479768734747,- 0.7270800833126525], +--R +--R [7.9000000000000004, 1.1853173340000001, 0.47310505388250573, +--R - 0.71221228011749438] +--R , +--R [8.,1.181847987,0.48433943621069148,- 0.69750855078930862], +--R [8.0999999999999996,1.178492509,0.49552617416036354,- 0.68296633483963642], +--R [8.1999999999999993,1.175246676,0.50666357567934228,- 0.66858310032065771], +--R +--R [8.3000000000000007, 1.1721063759999999, 0.51775002437336082, +--R - 0.65435635162663908] +--R , +--R +--R [8.4000000000000004, 1.1690676170000001, 0.52878397682080924, +--R - 0.64028364017919082] +--R , +--R [8.5,1.166126526,0.53976395999758919,- 0.6263625660024108], +--R +--R [8.5999999999999996, 1.1632793539999999, 0.5506885688063663, +--R - 0.61259078519363364] +--R , +--R +--R [8.6999999999999993, 1.1605224759999999, 0.56155646370489876, +--R - 0.59896601229510116] +--R , +--R [8.8000000000000007,1.15785239,0.57236636842843447,- 0.58548602157156548], +--R [8.9000000000000004,1.155265719,0.58311706780150541,- 0.57214865119849456], +--R [9.,1.1527592090000001,0.59380740563471446,- 0.5589518033652856], +--R +--R [9.0999999999999996, 1.1503297240000001, 0.60443628270239691, +--R - 0.5458934412976032] +--R , +--R +--R [9.1999999999999993, 1.1479742509999999, 0.61500265479727367, +--R - 0.53297159620272627] +--R , +--R [9.3000000000000007,1.145689889,0.62550553085845151,- 0.52018435814154851], +--R +--R [9.4000000000000004, 1.1434738550000001, 0.63594397116933887, +--R - 0.50752988383066122] +--R , +--R [9.5,1.1413234759999999,0.6463170856222451,- 0.49500639037775485], +--R +--R [9.5999999999999996, 1.1392361849999999, 0.65662403204660502, +--R - 0.48261215295339488] +--R , +--R +--R [9.6999999999999993, 1.1372095230000001, 0.66686401459797295, +--R - 0.47034550840202716] +--R , +--R [9.8000000000000007,1.13524113,0.6770362822050533,- 0.45820484779494675], +--R +--R [9.9000000000000004, 1.1333287460000001, 0.68714012707221583, +--R - 0.44618861892778428] +--R , +--R [10.,1.1314702050000001,0.69717488323506582,- 0.43429532176493424]] +--R Type: List List OnePointCompletion DoubleFloat +--E 18 + +@ +In the following table we show the values returned for large +arguments to the Ei function. See +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. p 243 + +<<*>>= +--S 19 of 20 +g(y)==(y=0 => 1 ; (x:DFLOAT:=y^-1) ; x*exp(-x)*Ei(x)) +--R +--R Type: Void +--E 19 + +--S 20 of 20 +[[0.100,1.13147021,g(0.100),g(0.100)-1.13147021],_ + [0.095,1.12249671,g(0.095),g(0.095)-1.12249671],_ + [0.090,1.11389377,g(0.090),g(0.090)-1.11389377],_ + [0.085,1.10564739,g(0.085),g(0.085)-1.10564739],_ + [0.080,1.09773775,g(0.080),g(0.080)-1.09773775],_ + [0.075,1.09014087,g(0.075),g(0.075)-1.09014087],_ + [0.070,1.08283054,g(0.070),g(0.070)-1.08283054],_ + [0.065,1.07578038,g(0.065),g(0.065)-1.07578038],_ + [0.060,1.06896548,g(0.060),g(0.060)-1.06896548],_ + [0.055,1.06236365,g(0.055),g(0.055)-1.06236365],_ + [0.050,1.05595591,g(0.050),g(0.050)-1.05595591],_ + [0.045,1.04972640,g(0.045),g(0.045)-1.04972640],_ + [0.040,1.04366194,g(0.040),g(0.040)-1.04366194],_ + [0.035,1.03775135,g(0.035),g(0.035)-1.03775135],_ + [0.030,1.03198503,g(0.030),g(0.030)-1.03198503],_ + [0.025,1.02635451,g(0.025),g(0.025)-1.02635451],_ + [0.020,1.02085228,g(0.020),g(0.020)-1.02085228],_ + [0.015,1.01547157,g(0.015),g(0.015)-1.01547157],_ + [0.010,1.01020625,g(0.010),g(0.010)-1.01020625],_ + [0.005,1.00505077,g(0.005),g(0.005)-1.00505077],_ + [0.000,1.00000000,g(0.000),g(0.000)-1.00000000]] +--R +--R Compiling function g with type Float -> OnePointCompletion +--R DoubleFloat +--R +--R (21) +--R [[0.10000000000000001,1.13147021,1.1314702047341079,- 5.2658921667614322E-9], +--R +--R [9.5000000000000001E-2, 1.1224967100000001, 1.1224967463528539, +--R 3.6352853838295118E-8] +--R , +--R [8.9999999999999997E-2,1.11389377,1.1138937808537757,1.0853775656016751E-8], +--R +--R [8.5000000000000006E-2, 1.1056473899999999, 1.1056473901733923, +--R 1.733924115399077E-10] +--R , +--R +--R [8.0000000000000002E-2, 1.0977377500000001, 1.0977377526473173, +--R 2.647317254300674E-9] +--R , +--R +--R [7.4999999999999997E-2, 1.0901408699999999, 1.0901408684282585, +--R - 1.5717414036942046E-9] +--R , +--R [7.0000000000000007E-2,1.08283054,1.0828305423224371,2.3224371314967129E-9], +--R +--R [6.5000000000000002E-2, 1.0757803800000001, 1.0757803749062493, +--R - 5.0937507545256722E-9] +--R , +--R +--R [5.9999999999999998E-2, 1.0689654799999999, 1.0689654755715123, +--R - 4.4284875766464893E-9] +--R , +--R [5.5E-2,1.06236365,1.0623636462639567,- 3.7360432525446186E-9], +--R +--R [5.0000000000000003E-2, 1.05595591, 1.0559559055929626, +--R - 4.4070374016769165E-9] +--R , +--R +--R [4.4999999999999998E-2, 1.0497263999999999, 1.0497264028491122, +--R 2.8491122794349621E-9] +--R , +--R +--R [4.0000000000000001E-2, 1.04366194, 1.0436619362666135, +--R - 3.7333864888466906E-9] +--R , +--R [3.5000000000000003E-2,1.03775135,1.0377513519241477,1.924147730036907E-9], +--R +--R [2.9999999999999999E-2, 1.03198503, 1.0319850279857541, +--R - 2.0142458811989172E-9] +--R , +--R [2.5000000000000001E-2,1.02635451,1.026354511439006,1.4390060254498849E-9], +--R [2.0E-2,1.0208522799999999,1.0208522777971993,- 2.2028006085861307E-9], +--R +--R [1.4999999999999999E-2, 1.0154715700000001, 1.0154715653071829, +--R - 4.6928172459104189E-9] +--R , +--R [1.0E-2,1.01020625,1.0102062527748354,2.7748354725076751E-9], +--R +--R [5.0000000000000001E-3, 1.00505077, 1.0050507653866605, +--R - 4.6133394882019729E-9] +--R , +--R [0.,1.,1.,0.]] +--R Type: List List OnePointCompletion DoubleFloat +--E 20 + +)spool +)lisp (bye) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Lee, K.L.,``High-precision Chebyshev series approximation +to the exponential integral'', NASA-TN-D-5953, A-3571, No Copyright +Doc. ID=19700026648, Accession ID=70N35964, Aug 1970 +\bibitem{2} Abramowitz and Stegun,``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. pp238-243 +\end{thebibliography} +\end{document} +