nag_kelvin_ker_vector (s19aqc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_kelvin_ker_vector (s19aqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_kelvin_ker_vector (s19aqc) returns an array of values for the Kelvin function kerx.

2  Specification

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

3  Description

nag_kelvin_ker_vector (s19aqc) evaluates an approximation to the Kelvin function kerxi for an array of arguments xi, for i=1,2,,n.
Note:  for x<0 the function is undefined and at x=0 it is infinite so we need only consider x>0.
The function is based on several Chebyshev expansions:
For 0<x1,
kerx=-ftlogx+π16x2gt+yt
where ft, gt and yt are expansions in the variable t=2x4-1.
For 1<x3,
kerx=exp-1116x qt
where qt is an expansion in the variable t=x-2.
For x>3,
kerx=π 2x e-x/2 1+1xct cosβ-1xdtsinβ
where β= x2+ π8 , and ct and dt are expansions in the variable t= 6x-1.
When x is sufficiently close to zero, the result is computed as
kerx=-γ-logx2+π-38x2 x216
and when x is even closer to zero, simply as kerx=-γ-log x2 .
For large x, kerx 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:     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.
Constraint: x[i-1]>0.0, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: kerxi, the function values.
4:     ivalid[n]IntegerOutput
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large, the result underflows. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in nag_kelvin_ker (s19acc), as defined in the Users' Note for your implementation.
ivalid[i-1]=2
xi0.0, the function is undefined. f[i-1] contains 0.0.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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

7  Accuracy

Let E be the absolute error in the result, ε be the relative error in the result and δ be the relative error in the argument. If δ is somewhat larger than the machine precision, then we have:
E x2 ker1x+ kei1x δ,
ε x2 ker1x + kei1x kerx δ.
For very small x, the relative error amplification factor is approximately given by 1logx , which implies a strong attenuation of relative error. However, ε in general cannot be less than the machine precision.
For small x, errors are damped 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 the absolute accuracy for 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, kerx, which asymptotically behaves like π2x e-x/2, becomes so small that it cannot be calculated without causing underflow, and the function returns zero. Note that for large x the errors are dominated by those of the standard function exp.

8  Further Comments

Underflow may occur for a few values of x close to the zeros of kerx, below the limit which causes a failure with fail.code= NW_IVALID.

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 (s19aqce.c)

9.2  Program Data

Program Data (s19aqce.d)

9.3  Program Results

Program Results (s19aqce.r)


nag_kelvin_ker_vector (s19aqc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

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