NAG Library Routine Document

s19adf  (kelvin_kei)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

s19adf returns a value for the Kelvin function keix via the function name.

2
Specification

Fortran Interface
Function s19adf ( x, ifail)
Real (Kind=nag_wp):: s19adf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include nagmk26.h
double  s19adf_ ( const double *x, Integer *ifail)

3
Description

s19adf evaluates an approximation to the Kelvin function keix.
Note:  for x<0 the function is undefined, so we need only consider x0.
The routine is based on several Chebyshev expansions:
For 0x1,
keix=-π4ft+x24-gtlogx+vt  
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
keix=-π4.  
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
Arguments

1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
Constraint: x0.0.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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:
ifail=1
On entry, x is too large: the result underflows. On soft failure, the routine returns zero. See also the Users' Note for your implementation.
ifail=2
On entry, x<0.0: the function is undefined. On soft failure the routine returns zero.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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
Parallelism and Performance

s19adf is not threaded in any implementation.

9
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.

10
Example

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

10.1
Program Text

Program Text (s19adfe.f90)

10.2
Program Data

Program Data (s19adfe.d)

10.3
Program Results

Program Results (s19adfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017