nag_kelvin_kei (s19adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C 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 .
The function is based on several Chebyshev expansions.
For large x , keix  is 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 .
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_REAL_ARG_GT
On entry, x=value.
Constraint: xvalue.
x is too large, and the result underflows and the function returns zero.
NE_REAL_ARG_LT
On entry, x must not be less than 0.0: x=value .
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 x - ker 1 x + kei 1 x / 2 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 π x / 2 e - x / 2 , which implies a strong attenuation of error. Eventually, keix , which is asymptotically given by π / 2 x 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 math library function exp.

8  Further Comments

Underflow may occur for a few values of x  close to the zeros of keix , which causes failure NE_REAL_ARG_GT.

9  Example

The following program reads values of the argument x  from a file, evaluates the function at each value of x  and prints the results.

9.1  Program Text

Program Text (s19adce.c)

9.2  Program Data

Program Data (s19adce.d)

9.3  Program Results

Program Results (s19adce.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012