NAG CL Interface
s19apc (kelvin_​bei_​vector)

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1 Purpose

s19apc returns an array of values for the Kelvin function beix.

2 Specification

#include <nag.h>
void  s19apc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s19apc, nag_specfun_kelvin_bei_vector or nag_kelvin_bei_vector.

3 Description

s19apc evaluates an approximation to the Kelvin function beixi for an array of arguments xi, for i=1,2,,n.
Note:  bei(-x)=beix, so the approximation need only consider x0.0.
The function is based on several Chebyshev expansions:
For 0x5,
beix = x24 r=0 ar Tr (t) ,   with ​ t=2 (x5) 4 - 1 ;  
For x>5,
beix = e x/2 2πx [(1+1xa(t))sinα-1xb(t)cosα]  
+ e x/2 2π x [(1+1xc(t))cosβ-1xd(t)sinβ]  
where α= x2- π8 , β= x2+ π8 ,
and a(t), b(t), c(t), and d(t) are expansions in the variable t= 10x-1.
When x is sufficiently close to zero, the result is computed as beix= x24 . If this result would underflow, the result returned is beix=0.0.
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 function must fail.

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.
3: f[n] double Output
On exit: beixi, 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
abs(xi) is too large for an accurate result to be returned. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in s19abc , as defined in the Users' Note for your implementation.
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

Since the function is oscillatory, the absolute error rather than the relative error is important. Let E be the absolute error in the function, 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 x2π ex/2. Therefore, it is impossible to calculate the functions with any accuracy when xex/2> 2πδ . Note that this value of x is much smaller than the minimum value of x for which the function overflows.

8 Parallelism and Performance

s19apc is not threaded in any implementation.

9 Further Comments

None.

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 (s19apce.c)

10.2 Program Data

Program Data (s19apce.d)

10.3 Program Results

Program Results (s19apce.r)