s18ae returns the value of the modified Bessel function I0x.


public static double s18ae(
	double x,
	out int ifail
Visual Basic
Public Shared Function s18ae ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s18ae(
	double x, 
	[OutAttribute] int% ifail
static member s18ae : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s18ae returns the value of the modified Bessel function I0x.


s18ae evaluates an approximation to the modified Bessel function of the first kind I0x.
Note:  I0-x=I0x, so the approximation need only consider x0.
The method is based on three Chebyshev expansions:
For 0<x4,
I0x=exr=0arTrt,   where ​t=2x4-1.
For 4<x12,
I0x=exr=0brTrt,   where ​t=x-84.
For x>12,
I0x=exxr=0crTrt,   where ​t=212x-1.
For small x, I0x1. 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 ex.


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

Error Indicators and Warnings

Errors or warnings detected by the method:
x is too large. On failure the method returns the approximate value of I0x at the nearest valid argument. (see the Users' Note for your implementation for details)
An error occured, see message report.


Let δ and ε be the relative errors in the argument and result respectively.
If δ is somewhat larger than the machine precision (i.e., if δ is due to data errors etc.), then ε and δ are approximately related by:
Figure 1 shows the behaviour of the error amplification factor
Figure 1
Figure 1
However if δ is of the same order as machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x the amplification factor is approximately x22, which implies strong attenuation of the error, but in general ε can never be less than the machine precision.
For large x, εxδ and we have strong amplification of errors. However the method must fail for quite moderate values of x, because I0x 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.

Parallelism and Performance


Further Comments



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

Example program (C#): s18aee.cs

Example program data: s18aee.d

Example program results: s18aee.r

See Also