public static double s13aa(
	double x,
	out int ifail
)

Public Shared Function s13aa ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s13aa(
	double x, 
	[OutAttribute] int% ifail
)

static member s13aa : 
        x : float * 
        ifail : int byref -> float 

s13aa returns the value of the exponential integral $E_1\left(x\right)$.

s13aa calculates an approximate value for 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 $x=0$, s13aa exits with an error and returns the largest representable machine number.

For $0<x\le 4$, where $t=\frac{1}{2}x-1$.

For $x>4$, 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 $x_0\approx -0.372507\dots$, which corresponds to a simple zero of Ei($-x$).

s13aa guards against producing underflows and overflows by using the parameter $x_{\mathrm{hi}}$; see the Users' Note for your implementation for the value of $x_{\mathrm{hi}}$. To guard against overflow, if $x<-x_{\mathrm{hi}}$ the method 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.

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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

Errors or warnings detected by the method:

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, 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 $x$, 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.

None.

None.

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