S19ANF (PDF version)
S Chapter Contents
S Chapter Introduction
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

S19ANF returns an array of values for the Kelvin function berx.

2  Specification

REAL (KIND=nag_wp)  X(N), F(N)

3  Description

S19ANF evaluates an approximation to the Kelvin function berxi for an array of arguments xi, for i=1,2,,n.
Note:  ber-x=berx, so the approximation need only consider x0.0.
The routine is based on several Chebyshev expansions:
For 0x5,
berx=r=0arTrt,   with ​ t=2 x5 4-1.
For x>5,
berx= e x/2 2πx 1+ 1 x a t cosα + 1x b tsinα + e-x/22πx 1+ 1xct sinβ+ 1xdtcosβ ,
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 set directly to ber0=1.0.
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 routine must fail.

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.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: berxi, the function values.
4:     IVALID(N) – INTEGER arrayOutput
On exit: IVALIDi contains the error code for xi, for i=1,2,,N.
No error.
absxi is too large for an accurate result to be returned. Fi contains zero. The threshold value is the same as for IFAIL=1 in S19AAF, as defined in the Users' Note for your implementation.
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

Since the function is oscillatory, the absolute error rather than the relative error is important. 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 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 x 2π e x/2 . Therefore it is not possible to calculate the function with any accuracy when x e x/2 > 2π δ . 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 (s19anfe.f90)

9.2  Program Data

Program Data (s19anfe.d)

9.3  Program Results

Program Results (s19anfe.r)

S19ANF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual

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