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NAG Toolbox: nag_specfun_kelvin_bei_vector (s19ap)

Purpose

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

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 beixibeixi for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  bei(x) = beixbei(-x)=beix, so the approximation need only consider x0.0x0.0.
The function is based on several Chebyshev expansions:
For 0x50x5,
beix = (x2)/4 arTr(t),   with ​t = 2(x/5)41;
r = 0
beix = x24 r=0 ar Tr (t) ,   with ​ t=2 (x5) 4 - 1 ;
For x > 5x>5,
beix = (ex / sqrt(2))/(sqrt(2πx)) [(1 + 1/xa(t))sinα1/xb(t)cosα]
beix = e x/2 2πx [ ( 1 + 1x a (t) ) sinα - 1x b (t) cosα ]
+ (ex / sqrt(2))/(sqrt(2π x)) [(1 + 1/xc(t))cosβ1/xd(t)sinβ]
+ e x/2 2π x [ ( 1 + 1x c (t) ) cosβ - 1x d (t) sinβ ]
where α = x/(sqrt(2))π/8 α= x2- π8 , β = x/(sqrt(2)) + π/8 β= x2+ π8 ,
and a(t)a(t), b(t)b(t), c(t)c(t), and d(t)d(t) are expansions in the variable t = 10/x1t= 10x-1.
When xx is sufficiently close to zero, the result is computed as beix = (x2)/4 beix= x24 . If this result would underflow, the result returned is beix = 0.0beix=0.0.
For large xx, 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
n, the dimension of the array, must satisfy the constraint n0n0.
The argument xixi of the function, for i = 1,2,,ni=1,2,,n.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
nn, the number of points.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     f(n) – double array
beixibeixi, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)ivalidi contains the error code for xixi, for i = 1,2,,ni=1,2,,n.
ivalid(i) = 0ivalidi=0
No error.
ivalid(i) = 1ivalidi=1
abs(xi)abs(xi) is too large for an accurate result to be returned. f(i)fi contains zero. The threshold value is the same as for ifail = 1ifail=1 in nag_specfun_kelvin_bei (s19ab), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: n0n0.

Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. Let EE be the absolute error in the function, and δδ be the relative error in the argument. If δδ is somewhat larger than the machine precision, then we have:
E |x/(sqrt(2))(ber1x + bei1x)|δ
E | x2 ( - ber1x+ bei1x ) |δ
(provided EE is within machine bounds).
For small xx the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large xx, the error behaviour is oscillatory and its amplitude grows like sqrt(x/(2π))ex / sqrt(2) x2π ex/2. Therefore it is impossible to calculate the functions with any accuracy when sqrt(x)ex / sqrt(2) > (sqrt(2π))/δ xex/2> 2πδ . Note that this value of xx is much smaller than the minimum value of xx for which the function overflows.

Further Comments

None.

Example

function nag_specfun_kelvin_bei_vector_example
x = [0.1; 1; 2.5; 5; 10; 15; -1];
[f, ivalid, ifail] = nag_specfun_kelvin_bei_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   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

function s19ap_example
x = [0.1; 1; 2.5; 5; 10; 15; -1];
[f, ivalid, ifail] = s19ap(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   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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