NAG CL Interfaces13aac (integral_​exp)

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1Purpose

s13aac returns the value of the exponential integral ${E}_{1}\left(x\right)$.

2Specification

 #include
 double s13aac (double x, NagError *fail)
The function may be called by the names: s13aac, nag_specfun_integral_exp or nag_exp_integral.

3Description

s13aac calculates an approximate value for
 $E1 (x) = -Ei (-x) = ∫x∞ e-u u du .$
using Chebyshev expansions, where $x$ is real. For $x<0$, the real part of the principal value of the integral is taken. The value ${E}_{1}\left(0\right)$ is infinite, and so, when $x=0$, s13aac exits with an error and returns the largest representable machine number.
For $0,
 $E1(x)=y(t)-ln⁡x=∑′rarTr(t)-ln⁡x,$
where $t=\frac{1}{2}x-1$.
For $x>4$,
 $E1(x)=e-xxy(t)=e-xx∑′rarTr(t),$
where $t=-1.0+\frac{14.5}{\left(x+3.25\right)}=\frac{11.25-x}{3.25+x}$.
In both cases, $-1\le t\le +1$.
For $x<0$, the approximation is based on expansions proposed by Cody and Thatcher Jr. (1969). Precautions are taken to maintain good relative accuracy in the vicinity of ${x}_{0}\approx -0.372507\dots \text{}$, which corresponds to a simple zero of Ei($-x$).
s13aac guards against producing underflows and overflows by using the argument ${x}_{\mathrm{hi}}$ , see the Users' Note for your implementation for the value of ${x}_{\mathrm{hi}}$. To guard against overflow, if $x<-{x}_{\mathrm{hi}}$ the function terminates and returns the negative of the largest representable machine number. To guard against underflow, if $x>{x}_{\mathrm{hi}}$ the result is set directly to zero.

4References

NIST Digital Library of Mathematical Functions
Cody W J and Thatcher Jr. H C (1969) Rational Chebyshev approximations for the exponential integral Ei$\left(x\right)$ Math. Comp. 23 289–303

5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
Constraint: $-{x}_{\mathrm{hi}}\le {\mathbf{x}}<0.0$ or ${\mathbf{x}}>0.0$.
2: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_LE
On entry, ${\mathbf{x}}=0.0$ and the function is infinite.
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$ and the constant ${x}_{\mathrm{hi}}=⟨\mathit{\text{value}}⟩$. The evaluation has been abandoned due to the likelihood of overflow.
Constraint: ${\mathbf{x}}\ge -{x}_{\mathrm{hi}}$.

7Accuracy

Unless stated otherwise, it is assumed that $x>0$.
If $\delta$ and $\epsilon$ are the relative errors in argument and result respectively, then in principle,
 $|ε|≃ | e-x E1 (x) ×δ|$
so the relative error in the argument is amplified in the result by at least a factor ${e}^{-x}/{E}_{1}\left(x\right)$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is simply a result of round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation and round-off.
The behaviour of this amplification factor is shown in the following graph:
It should be noted that, for absolutely small $x$, the amplification factor tends to zero and eventually the error in the result will be limited by machine precision.
For absolutely large $x$,
 $ε∼xδ=Δ,$
the absolute error in the argument.
For $x<0$, empirical tests have shown that the maximum relative error is a loss of approximately $1$ decimal place.

8Parallelism and Performance

s13aac is not threaded in any implementation.

None.

10Example

The following program reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

10.1Program Text

Program Text (s13aace.c)

10.2Program Data

Program Data (s13aace.d)

10.3Program Results

Program Results (s13aace.r)