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NAG Toolbox

NAG Toolbox: nag_specfun_bessel_y0_real_vector (s17aq)

Purpose

nag_specfun_bessel_y0_real_vector (s17aq) returns an array of values of the Bessel function Y0(x)Y0(x).

Syntax

[f, ivalid, ifail] = s17aq(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_y0_real_vector(x, 'n', n)

Description

nag_specfun_bessel_y0_real_vector (s17aq) evaluates an approximation to the Bessel function of the second kind Y0(xi)Y0(xi) for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  Y0(x)Y0(x) is undefined for x0x0 and the function will fail for such arguments.
The function is based on four Chebyshev expansions:
For 0 < x80<x8,
Y0(x) = 2/πlnx arTr(t) +  brTr(t),   with ​t = 2(x/8)21.
r = 0 r = 0
Y0 (x) = 2π lnx r=0 ar Tr (t) + r=0 br Tr (t) ,   with ​ t = 2 (x8) 2 - 1 .
For x > 8x>8,
Y0 (x) = sqrt(2/(πx)) {P0(x)sin(xπ/4) + Q0(x)cos(xπ/4)}
Y0 (x) = 2πx { P0 (x) sin(x-π4) + Q0 (x) cos(x-π4) }
where P0(x) = r = 0 crTr(t)P0(x)=r=0crTr(t),
and Q0(x) = 8/xr = 0 drTr(t),with ​ t = 2 (8/x)21.Q0(x)= 8xr=0drTr(t),with ​ t=2 ( 8x) 2-1.
For xx near zero, Y0(x)2/π (ln(x/2) + γ) Y0(x)2π (ln(x2)+γ) , where γγ denotes Euler's constant. This approximation is used when xx is sufficiently small for the result to be correct to machine precision.
For very large xx, it becomes impossible to provide results with any reasonable accuracy (see Section [Accuracy]), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of Y0(x)Y0(x); only the amplitude, sqrt(2/(πn))2πn , can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the function will fail if x1 / machine precisionx1/machine precision.

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

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.
Constraint: x(i) > 0.0xi>0.0, 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
Y0(xi)Y0(xi), 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
On entry,xixi is too large. f(i)fi contains the amplitude of the Y0Y0 oscillation, sqrt(2/(πxi)) 2πxi .
ivalid(i) = 2ivalidi=2
On entry,xi0.0xi0.0, Y0Y0 is undefined. f(i)fi contains 0.00.0.
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

Let δδ be the relative error in the argument and EE be the absolute error in the result. (Since Y0(x)Y0(x) oscillates about zero, absolute error and not relative error is significant, except for very small xx.)
If δδ is somewhat larger than the machine representation error (e.g., if δδ is due to data errors etc.), then EE and δδ are approximately related by
E |xY1(x)| δ
E | x Y1 (x) | δ
(provided EE is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xY1(x)||xY1(x)|.
However, if δδ is of the same order as the machine representation errors, then rounding errors could make EE slightly larger than the above relation predicts.
For very small xx, the errors are essentially independent of δδ and the function should provide relative accuracy bounded by the machine precision.
For very large xx, the above relation ceases to apply. In this region, Y0(x)sqrt(2/(πx))sin(xπ/4)Y0(x) 2πx sin(x- π4). The amplitude sqrt(2/(πx)) 2πx  can be calculated with reasonable accuracy for all xx, but sin(xπ/4)sin(x-π4) cannot. If xπ/4 x- π4  is written as 2Nπ + θ2Nπ+θ where NN is an integer and 0θ < 2π0θ<2π, then sin(xπ/4)sin(x- π4) is determined by θθ only. If xδ1xδ-1, θθ cannot be determined with any accuracy at all. Thus if xx is greater than, or of the order of the inverse of machine precision, it is impossible to calculate the phase of Y0(x)Y0(x) and the function must fail.
Figure 1
Figure 1

Further Comments

None.

Example

function nag_specfun_bessel_y0_real_vector_example
x = [0.5; 1; 3; 6; 8; 10; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_y0_real_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
   5.000e-01  -4.445e-01    0
   1.000e+00   8.826e-02    0
   3.000e+00   3.769e-01    0
   6.000e+00  -2.882e-01    0
   8.000e+00   2.235e-01    0
   1.000e+01   5.567e-02    0
   1.000e+03   4.716e-03    0

function s17aq_example
x = [0.5; 1; 3; 6; 8; 10; 1000];
[f, ivalid, ifail] = s17aq(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
   5.000e-01  -4.445e-01    0
   1.000e+00   8.826e-02    0
   3.000e+00   3.769e-01    0
   6.000e+00  -2.882e-01    0
   8.000e+00   2.235e-01    0
   1.000e+01   5.567e-02    0
   1.000e+03   4.716e-03    0


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