s19ad returns a value for the Kelvin function keix.


public static double s19ad(
	double x,
	out int ifail
Visual Basic
Public Shared Function s19ad ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s19ad(
	double x, 
	[OutAttribute] int% ifail
static member s19ad : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Constraint: x0.0.
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

s19ad returns a value for the Kelvin function keix.


s19ad evaluates an approximation to the Kelvin function keix.
Note:  for x<0 the function is undefined, so we need only consider x0.
The method is based on several Chebyshev expansions:
For 0x1,
where ft, gt and vt are expansions in the variable t=2x4-1;
For 1<x3,
where ut 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.
For x<0, the function is undefined, and hence the method fails and returns zero.
When x is sufficiently close to zero, the result is computed as
and when x is even closer to zero simply as
For large x, keix 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, x<0.0: 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, and δ be the relative error in the argument. If δ is somewhat larger than the machine representation error, then we have:
For small x, errors are attenuated 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 absolute accuracy of 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, keix, which is asymptotically given by π2xe-x/2,becomes so small that it cannot be calculated without causing underflow and therefore 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 keix, 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#): s19ade.cs

Example program data: s19ade.d

Example program results: s19ade.r

See Also