S19ARF (PDF version)
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NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

S19ARF returns an array of values for the Kelvin function keix.

2  Specification

REAL (KIND=nag_wp)  X(N), F(N)

3  Description

S19ARF evaluates an approximation to the Kelvin function keixi for an array of arguments xi, for i=1,2,,n.
Note:  for x<0 the function is undefined, so we need only consider x0.
The routine is based on several Chebyshev expansions:
For 0x1,
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 routine 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
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 routine fails.

4  References

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

5  Parameters

1:     N – INTEGERInput
On entry: n, the number of points.
Constraint: N0.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument xi of the function, for i=1,2,,N.
Constraint: Xi0.0, for i=1,2,,N.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: keixi, the function values.
4:     IVALID(N) – INTEGER arrayOutput
On exit: IVALIDi contains the error code for xi, for i=1,2,,N.
No error.
xi is too large, the result underflows. Fi contains zero. The threshold value is the same as for IFAIL=1 in S19ADF, as defined in the Users' Note for your implementation.
xi<0.0, the function is undefined. Fi contains 0.0.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, at least one value of X was invalid.
Check IVALID for more information.
On entry, N=value.
Constraint: N0.

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 routine 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 keix, below the limit which causes a failure with IFAIL=1.

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 (s19arfe.f90)

9.2  Program Data

Program Data (s19arfe.d)

9.3  Program Results

Program Results (s19arfe.r)

S19ARF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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