nag_kelvin_kei (s19adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_kelvin_kei (s19adc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_kelvin_kei (s19adc) returns a value for the Kelvin function keix.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_kelvin_kei (double x, NagError *fail)

3  Description

nag_kelvin_kei (s19adc) evaluates an approximation to the Kelvin function keix.
Note:  for x<0 the function is undefined, so we need only consider x0.
The function is based on several Chebyshev expansions:
For 0x1,
keix=-π4ft+x24-gtlogx+vt
where ft, gt and vt are expansions in the variable t=2x4-1;
For 1<x3,
keix=exp-98x ut
where ut is an expansion in the variable t=x-2;
For x>3,
keix=π 2x e-x/2 1+1x ctsinβ+1xdtcosβ
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 function fails and returns zero.
When x is sufficiently close to zero, the result is computed as
keix=-π4+1-γ-logx2 x24
and when x is even closer to zero simply as
keix=-π4.
For large x, keix is asymptotically given by π 2x e-x/2 and this becomes so small that it cannot be computed without underflow and the function fails.

4  References

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

5  Arguments

1:     xdoubleInput
On entry: the argument x of the function.
Constraint: x0.0.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL_ARG_GT
On entry, x=value. The function returns zero.
Constraint: xvalue.
x is too large, the result underflows and the function returns zero.
NE_REAL_ARG_LT
On entry, x=value.
Constraint: x0.0.
The function is undefined and returns zero.

7  Accuracy

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:
E x2 - ker1x+ kei1x δ.
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 π2x e-x/2,becomes so small that it cannot be calculated without causing underflow and therefore the function returns zero. Note that for large x, the errors are dominated by those of the standard math library function exp.

8  Parallelism and Performance

Not applicable.

9  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 fail.code= NE_REAL_ARG_GT.

10  Example

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

10.1  Program Text

Program Text (s19adce.c)

10.2  Program Data

Program Data (s19adce.d)

10.3  Program Results

Program Results (s19adce.r)


nag_kelvin_kei (s19adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014