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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_kelvin_bei (s19ab)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_kelvin_bei (s19ab) returns a value for the Kelvin function beix via the function name.


[result, ifail] = s19ab(x)
[result, ifail] = nag_specfun_kelvin_bei(x)


nag_specfun_kelvin_bei (s19ab) evaluates an approximation to the Kelvin function beix.
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=ex/22πx 1+1xat sinα-1xbtcosα  
+ex/22π x 1+1xct cosβ-1xdtsinβ  
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.


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


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.

Optional Input Parameters


Output Parameters

1:     result – double scalar
The result of the function.
2:     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:
On entry, absx is too large for an accurate result to be returned. On soft failure, the function returns zero.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s19ab_example

fprintf('s19ab example results\n\n');

x = [0.1   1    2.5   5   10   15   -1];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s19ab(x(j));

disp('      x          bei(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s19ab example results

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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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