public static double s18ac(
	double x,
	out int ifail
)

Public Shared Function s18ac ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s18ac(
	double x, 
	[OutAttribute] int% ifail
)

static member s18ac : 
        x : float * 
        ifail : int byref -> float 

s18ac returns the value of the modified Bessel function $K_0\left(x\right)$.

s18ac evaluates an approximation to the modified Bessel function of the second kind $K_0\left(x\right)$.

Note:  $K_0\left(x\right)$ is undefined for $x\le 0$ and the method will fail for such arguments.

The method is based on five Chebyshev expansions:

For $0<x\le 1$, For $1<x\le 2$, For $2<x\le 4$, For $x>4$, For $x$ near zero, $K_0\left(x\right)\simeq -\gamma -\ln \left(\frac{x}{2}\right)$, where $\gamma$ denotes Euler's constant. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For large $x$, where there is a danger of underflow due to the smallness of $K_0$, the result is set exactly to zero.

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

Errors or warnings detected by the method:

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.

If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by: Figure 1 shows the behaviour of the error amplification factor

However, if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.

For small $x$, the amplification factor is approximately $\left|\frac{1}{\ln x}\right|$, which implies strong attenuation of the error, but in general $\epsilon$ can never be less than the machine precision.

For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually $K_0$, which is asymptotically given by $\frac{e^{-x}}{\sqrt{x}}$, becomes so small that it cannot be calculated without underflow and hence the method will return zero. Note that for large $x$ the errors will be dominated by those of the standard function exp.

None.

None.

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