s18ad returns the value of the modified Bessel function K1x.


public static double s18ad(
	double x,
	out int ifail
Visual Basic
Public Shared Function s18ad ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s18ad(
	double x, 
	[OutAttribute] int% ifail
static member s18ad : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Constraint: x>0.0.
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

s18ad returns the value of the modified Bessel function K1x.


s18ad evaluates an approximation to the modified Bessel function of the second kind K1x.
Note:  K1x is undefined for x0 and the method will fail for such arguments.
The method is based on five Chebyshev expansions:
For 0<x1,
K1x=1x+xlnxr=0arTrt-xr=0brTrt,   where ​t=2x2-1.
For 1<x2,
K1x=e-xr=0crTrt,   where ​t=2x-3.
For 2<x4,
K1x=e-xr=0drTrt,   where ​t=x-3.
For x>4,
K1x=e-xxr=0erTrt,   where ​t=9-x1+x.
For x near zero, K1x1x. 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 1x without overflow and the method must fail.
For large x, where there is a danger of underflow due to the smallness of K1, the result is set exactly to zero.


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:
x0.0, K1 is undefined. On failure the method returns zero.
x is too small, there is a danger of overflow. On failure the method returns approximately the largest representable value. (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
However if δ is of the same order as the machine precision, then rounding errors could make ε slightly larger than the above relation predicts.
For small x, εδ and there is no amplification of errors.
For large x, εxδ and we have strong amplification of the relative error. Eventually K1, which is asymptotically given by e-xx, 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.
Figure 1
Figure 1

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#): s18ade.cs

Example program data: s18ade.d

Example program results: s18ade.r

See Also