public static double s18ae(
	double x,
	out int ifail
)

Public Shared Function s18ae ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s18ae(
	double x, 
	[OutAttribute] int% ifail
)

static member s18ae : 
        x : float * 
        ifail : int byref -> float 

s18ae returns the value of the modified Bessel function $I_0\left(x\right)$.

s18ae evaluates an approximation to the modified Bessel function of the first kind $I_0\left(x\right)$.

Note:  $I_0\left(-x\right)=I_0\left(x\right)$, 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 $x$, $I_0\left(x\right)\simeq 1$. 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 of the danger of overflow in calculating $e^x$.

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 $\frac{x^2}{2}$, 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 errors. However the method must fail for quite moderate values of $x$, because $I_0\left(x\right)$ 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.