NAG Library Function Document

nag_kelvin_kei_vector (s19arc)


    1  Purpose
    7  Accuracy


nag_kelvin_kei_vector (s19arc) returns an array of values for the Kelvin function keix.


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


nag_kelvin_kei_vector (s19arc) 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 function 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 function 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 function fails.


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


1:     n IntegerInput
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: keixi, 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.
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_kei (s19adc), as defined in the Users' Note for your implementation.
xi<0.0, the function is undefined. f[i-1] contains 0.0.
5:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, at least one value of x was invalid.
Check ivalid for more information.


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 function returns zero. Note that for large x, the errors are dominated by those of the standard function exp.

Parallelism and Performance

nag_kelvin_kei_vector (s19arc) is not threaded in any implementation.

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 fail.code= NW_IVALID.


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

Program Text

Program Text (s19arce.c)

Program Data

Program Data (s19arce.d)

Program Results

Program Results (s19arce.r)

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