nag_kelvin_bei_vector (s19apc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_kelvin_bei_vector (s19apc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_kelvin_bei_vector (s19apc) returns an array of values for the Kelvin function beix.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_kelvin_bei_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_kelvin_bei_vector (s19apc) evaluates an approximation to the Kelvin function beixi for an array of arguments xi, for i=1,2,,n.
Note:  bei-x=beix, so the approximation need only consider x0.0.
The function is based on several Chebyshev expansions:
For 0x5,
beix = x24 r=0 ar Tr t ,   with ​ t=2 x5 4 - 1 ;
For x>5,
beix = e x/2 2πx 1 + 1x a t sinα - 1x b t cosα
+ e x/2 2π x 1 + 1x c t cosβ - 1x d t sinβ
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 function must fail.

4  References

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

5  Arguments

1:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
On entry: the argument xi of the function, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: beixi, the function values.
4:     ivalid[n]IntegerOutput
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
No error.
absxi is too large for an accurate result to be returned. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in nag_kelvin_bei (s19abc), as defined in the Users' Note for your implementation.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
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.
On entry, at least one value of x was invalid.
Check ivalid for more information.

7  Accuracy

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:
E x2 - ber1x+ bei1x δ
(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 x2π ex/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.

8  Further Comments


9  Example

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

9.1  Program Text

Program Text (s19apce.c)

9.2  Program Data

Program Data (s19apce.d)

9.3  Program Results

Program Results (s19apce.r)

nag_kelvin_bei_vector (s19apc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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