NAG CL Interface
s19aqc (kelvin_​ker_​vector)

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1 Purpose

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

2 Specification

#include <nag.h>
void  s19aqc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s19aqc, nag_specfun_kelvin_ker_vector or nag_kelvin_ker_vector.

3 Description

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=-f(t)log(x)+π16x2g(t)+y(t)  
where f(t), g(t) and y(t) are expansions in the variable t=2x4-1.
For 1<x3,
kerx=exp(-1116x) q(t)  
where q(t) is an expansion in the variable t=x-2.
For x>3,
kerx=π 2x e-x/2 [(1+1xc(t))cosβ-1xd(t)sinβ]  
where β= x2+ π8 , and c(t) and d(t) are expansions in the variable t= 6x-1.
When x is sufficiently close to zero, the result is computed as
kerx=-γ-log(x2)+(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

NIST Digital Library of Mathematical Functions

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
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] double Output
On exit: kerxi, the function values.
4: ivalid[n] Integer Output
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 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: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 1|log(x)| , 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 Parallelism and Performance

s19aqc is not threaded in any implementation.

9 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.

10 Example

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

10.1 Program Text

Program Text (s19aqce.c)

10.2 Program Data

Program Data (s19aqce.d)

10.3 Program Results

Program Results (s19aqce.r)