public static double s18af(
	double x,
	out int ifail
)

Public Shared Function s18af ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s18af(
	double x, 
	[OutAttribute] int% ifail
)

static member s18af : 
        x : float * 
        ifail : int byref -> float 

s18af returns a value for the modified Bessel function $I_1\left(x\right)$.

s18af evaluates an approximation to the modified Bessel function of the first kind $I_1\left(x\right)$.

Note:  $I_1\left(-x\right)=-I_1\left(x\right)$, so the approximation need only consider $x\ge 0$.

The method is based on three Chebyshev expansions:

For $0<x\le 4$, For $4<x\le 12$, For $x>12$, For small $x$, $I_1\left(x\right)\simeq x$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For large $x$, the method must fail because $I_1\left(x\right)$ cannot be represented without overflow.

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

For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of errors. However the method must fail for quite moderate values of $x$ because $I_1\left(x\right)$ would overflow; hence in practice the loss of accuracy for large $x$ is not excessive. 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.