, 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: $x (n)$ – double array: The argument $x_{i}$ of the function, for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of points.

Constraint: $n \geq 0$ .

Output Parameters

1: $f (n)$ – double array

bei x_{i}

, the function values.

2: $ivalid (n)$ – int64int32nag_int array

ivalid (i)

contains the error code for

x_{i}

, for

i = 1, 2, \dots, n

$ivalid (i) = 0$: No error.
$ivalid (i) = 1$: $abs (x_{i})$ is too large for an accurate result to be returned. $f (i)$ 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: $n \geq 0$ .

$ifail = - 99$: 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 ≃ |\frac{x}{\sqrt{2}} (- {ber}_{1} x + {bei}_{1} x)| 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

\sqrt{\frac{x}{2 π}} e^{x / \sqrt{2}}

. Therefore it is impossible to calculate the functions with any accuracy when

\sqrt{x} e^{x / \sqrt{2}} > \frac{\sqrt{2 π}}{δ}

. 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

x_{i}

and prints the results.

Open in the MATLAB editor: s19ap_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox