public static double s19ac(
	double x,
	out int ifail
)

Public Shared Function s19ac ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s19ac(
	double x, 
	[OutAttribute] int% ifail
)

static member s19ac : 
        x : float * 
        ifail : int byref -> float 

s19ac returns a value for the Kelvin function $\ker x$.

s19ac evaluates an approximation to the Kelvin function $\ker x$.

Note:  for $x<0$ the function is undefined and at $x=0$ it is infinite so we need only consider $x>0$.

The method is based on several Chebyshev expansions:

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

For $1<x\le 3$, where $q\left(t\right)$ 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 $t=\frac{6}{x}-1$.

When $x$ is sufficiently close to zero, the result is computed as and when $x$ is even closer to zero, simply as $\ker x=-\gamma -\log \left(\frac{x}{2}\right)$.

For large $x$, $\ker x$ 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, $\epsilon$ be the relative error in the result and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine precision, then we have: For very small $x$, the relative error amplification factor is approximately given by $\frac{1}{\left|\log \left(x\right)\right|}$, which implies a strong attenuation of relative error. However, $\epsilon$ in general cannot be less than the machine precision.

For small $x$, errors are damped 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 the absolute accuracy for 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, $\ker x$, which asymptotically behaves like $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$, becomes so small that it cannot be calculated without causing underflow, and 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 $\ker x$, 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.