public static double s18ad(
	double x,
	out int ifail
)

Public Shared Function s18ad ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s18ad(
	double x, 
	[OutAttribute] int% ifail
)

static member s18ad : 
        x : float * 
        ifail : int byref -> float 

s18ad returns the value of the modified Bessel function $K_1\left(x\right)$.

s18ad evaluates an approximation to the modified Bessel function of the second kind $K_1\left(x\right)$.

Note:  $K_1\left(x\right)$ 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, $K_1\left(x\right)\simeq \frac{1}{x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For very small $x$ on some machines, it is impossible to calculate $\frac{1}{x}$ without overflow and the method must fail.

For large $x$, where there is a danger of underflow due to the smallness of $K_1$, 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 the machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.

For small $x$, $\epsilon \simeq \delta$ and there is no amplification of errors.

For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of the relative error. Eventually $K_1$, 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.