s19ac returns a value for the Kelvin function kerx.


public static double s19ac(
	double x,
	out int ifail
Visual Basic
Public Shared Function s19ac ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s19ac(
	double x, 
	[OutAttribute] int% ifail
static member s19ac : 
        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

s19ac returns a value for the Kelvin function kerx.


s19ac evaluates an approximation to the Kelvin function kerx.
Note:  for x<0 the function is undefined and at x=0 it is infinite so we need only consider x>0.
The method is based on several Chebyshev expansions:
For 0<x1,
where ft, gt and yt are expansions in the variable t=2x4-1.
For 1<x3,
where qt is an expansion in the variable t=x-2.
For x>3,
where β=x2+π8, and ct and dt are expansions in the variable t=6x-1.
When x is sufficiently close to zero, the result is computed as
and when x is even closer to zero, simply as kerx=-γ-logx2.
For large x, kerx is asymptotically given by π2xe-x/2 and this becomes so small that it cannot be computed without underflow and the method fails.


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:
On entry, x is too large: the result underflows. On failure, the method returns zero. See also the Users' Note for your implementation.
On entry, x0.0: the function is undefined. On failure the method returns zero.
An error occured, see message report.


Let E be the absolute error in the result, ε be the relative error in the result and δ be the relative error in the argument. If δ is somewhat larger than the machine precision, then we have:
For very small x, the relative error amplification factor is approximately given by 1logx, which implies a strong attenuation of relative error. However, ε in general cannot be less than the machine precision.
For small x, errors are damped by the function and hence are limited by the machine precision.
For medium and large x, the error behaviour, like the function itself, is oscillatory, and hence only the absolute accuracy for the function can be maintained. For this range of x, the amplitude of the absolute error decays like πx2e-x/2 which implies a strong attenuation of error. Eventually, kerx, which asymptotically behaves like π2xe-x/2, becomes so small that it cannot be calculated without causing underflow, and the method returns zero. Note that for large x the errors are dominated by those of the standard function exp.

Parallelism and Performance


Further Comments

Underflow may occur for a few values of x close to the zeros of kerx, below the limit which causes a failure with ifail=1.


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#): s19ace.cs

Example program data: s19ace.d

Example program results: s19ace.r

See Also