s19ab returns a value for the Kelvin function beix.


public static double s19ab(
	double x,
	out int ifail
Visual Basic
Public Shared Function s19ab ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s19ab(
	double x, 
	[OutAttribute] int% ifail
static member s19ab : 
        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

s19ab returns a value for the Kelvin function beix.


s19ab evaluates an approximation to the Kelvin function beix.
Note:  bei-x=beix, so the approximation need only consider x0.0.
The method is based on several Chebyshev expansions:
For 0x5,
beix=x24r=0arTrt,   with ​t=2x54-1;
For x>5,
where α=x2-π8, β=x2+π8,
and at, bt, ct, and dt are expansions in the variable t=10x-1.
When x is sufficiently close to zero, the result is computed as beix=x24. If this result would underflow, the result returned is beix=0.0.
For large x, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the method must fail.


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, absx is too large for an accurate result to be returned. On failure, the method returns zero. See also the Users' Note for your implementation.
An error occured, see message report.


Since the function is oscillatory, the absolute error rather than the relative error is important. Let E be the absolute error in the function, and δ be the relative error in the argument. If δ is somewhat larger than the machine precision, then we have:
(provided E is within machine bounds).
For small x the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large x, the error behaviour is oscillatory and its amplitude grows like x2πex/2. Therefore it is impossible to calculate the functions with any accuracy when xex/2>2πδ. Note that this value of x is much smaller than the minimum value of x for which the function overflows.

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#): s19abe.cs

Example program data: s19abe.d

Example program results: s19abe.r

See Also