public static double s19ad(
	double x,
	out int ifail
)

Public Shared Function s19ad ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s19ad(
	double x, 
	[OutAttribute] int% ifail
)

static member s19ad : 
        x : float * 
        ifail : int byref -> float 

s19ad returns a value for the Kelvin function $\mathrm{kei} x$.

s19ad evaluates an approximation to the Kelvin function $\mathrm{kei} x$.

Note:  for $x<0$ the function is undefined, so we need only consider $x\ge 0$.

The method is based on several Chebyshev expansions:

For $0\le x\le 1$, where $f\left(t\right)$, $g\left(t\right)$ and $v\left(t\right)$ are expansions in the variable $t=2x^4-1$;

For $1<x\le 3$, where $u\left(t\right)$ is an expansion in the variable $t=x-2$;

For $x>3$, where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.

For $x<0$, the function is undefined, and hence the method fails and returns zero.

When $x$ is sufficiently close to zero, the result is computed as and when $x$ is even closer to zero simply as For large $x$, $\mathrm{kei} x$ is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the method fails.

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

Errors or warnings detected by the method:

Let $E$ be the absolute error in the result, and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine representation error, then we have: For small $x$, errors are attenuated by the function and hence are limited by the machine precision.

For medium and large $x$, the error behaviour, like the function itself, is oscillatory and hence only absolute accuracy of the function can be maintained. For this range of $x$, the amplitude of the absolute error decays like $\sqrt{\frac{\pi x}{2}}{e}^{-x/\sqrt{2}}$, which implies a strong attenuation of error. Eventually, $\mathrm{kei} x$, which is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$,becomes so small that it cannot be calculated without causing underflow and therefore the method returns zero. Note that for large $x$, the errors are dominated by those of the standard function exp.

None.

Underflow may occur for a few values of $x$ close to the zeros of $\mathrm{kei} x$, below the limit which causes a failure with $\mathbf{ifail}=1$.

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