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NAG Toolbox: nag_specfun_kelvin_bei_vector (s19ap)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_kelvin_bei_vector (s19ap) returns an array of values for the Kelvin function beix.

Syntax

[f, ivalid, ifail] = s19ap(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_kelvin_bei_vector(x, 'n', n)

Description

nag_specfun_kelvin_bei_vector (s19ap) 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 + 1x a t sinα - 1x b t cosα  
+ e x/2 2π x 1 + 1x c t cosβ - 1x d t sinβ  
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 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.

References

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

Parameters

Compulsory Input Parameters

1:     xn – double array
The argument xi of the function, for i=1,2,,n.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of points.
Constraint: n0.

Output Parameters

1:     fn – double array
beixi, the function values.
2:     ivalidn int64int32nag_int array
ivalidi contains the error code for xi, for i=1,2,,n.
ivalidi=0
No error.
ivalidi=1
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 nag_specfun_kelvin_bei (s19ab), as defined in the Users' Note for your implementation.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
   ifail=2
Constraint: n0.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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.

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s19ap_example


fprintf('s19ap example results\n\n');

x = [0.1; 1; 2.5; 5; 10; 15; -1];

[f, ivalid, ifail] = s19ap(x);

fprintf('     x           bei(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end


s19ap example results

     x           bei(x)   ivalid
   1.000e-01   2.500e-03    0
   1.000e+00   2.496e-01    0
   2.500e+00   1.457e+00    0
   5.000e+00   1.160e-01    0
   1.000e+01   5.637e+01    0
   1.500e+01  -2.953e+03    0
  -1.000e+00   2.496e-01    0

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