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NAG Toolbox: nag_specfun_bessel_y1_real_vector (s17ar)

Purpose

nag_specfun_bessel_y1_real_vector (s17ar) returns an array of values of the Bessel function Y1(x)Y1(x).

Syntax

[f, ivalid, ifail] = s17ar(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_y1_real_vector(x, 'n', n)

Description

nag_specfun_bessel_y1_real_vector (s17ar) evaluates an approximation to the Bessel function of the second kind Y1(xi)Y1(xi) for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  Y1(x)Y1(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,
Y1(x) = 2/πlnxx/8 arTr(t)2/(πx) + x/8 brTr(t),   with ​t = 2(x/8)21.
r = 0 r = 0
Y1 (x) = 2π lnx x8 r=0 ar Tr (t) - 2πx + x8 r=0 br Tr (t) ,   with ​ t = 2 (x8) 2 - 1 .
For x > 8x>8,
Y1 (x) = sqrt(2/(πx)) {P1(x)sin(x3π/4) + Q1(x)cos(x3π/4)}
Y1 (x) = 2πx { P1 (x) sin( x - 3 π4 ) + Q1 (x) cos( x - 3 π4 ) }
where P1 (x) = r = 0  cr Tr (t) P1 (x) = r=0 cr Tr (t) ,
and Q1 (x) = 8/x r = 0  dr Tr (t) Q1 (x) = 8x r=0 dr Tr (t) , with t = 2 (8/x)2 1 t = 2 (8x) 2 - 1 .
For xx near zero, Y1 (x) 2/(πx) Y1 (x) - 2 πx . This approximation is used when xx is sufficiently small for the result to be correct to machine precision. For extremely small xx, there is a danger of overflow in calculating 2/(πx) - 2 πx  and for such arguments the function will fail.
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 Y1(x)Y1(x); only the amplitude, sqrt(2/(πx))2πx , 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
Y1(xi)Y1(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 Y1Y1 oscillation, sqrt(2/(πxi)) 2πxi .
ivalid(i) = 2ivalidi=2
On entry,xi0.0xi0.0, Y1Y1 is undefined. f(i)fi contains 0.00.0.
ivalid(i) = 3ivalidi=3
xixi is too close to zero, there is a danger of overflow. On soft failure, f(i)fi contains the value of Y1(x)Y1(x) at the smallest valid argument.
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 Y1(x)Y1(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 precision (e.g., if δδ is due to data errors etc.), then EE and δδ are approximately related by:
E |xY0(x)Y1(x)| δ
E | x Y0 (x) - Y1 (x) | δ
(provided EE is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xY0(x)Y1(x)||xY0(x)-Y1(x)|.
However, if δδ is of the same order as machine precision, then rounding errors could make EE slightly larger than the above relation predicts.
For very small xx, absolute error becomes large, but the relative error in the result is of the same order as δδ.
For very large xx, the above relation ceases to apply. In this region, Y1 (x) sqrt( 2/(πx) ) sin(x(3π)/4) Y1 (x) 2 πx sin( x - 3π 4 ) . The amplitude sqrt( 2/(πx) ) 2 πx  can be calculated with reasonable accuracy for all xx, but sin(x(3π)/4) sin(x- 3π4)  cannot. If x(3π)/4 x- 3π4  is written as 2Nπ + θ2Nπ+θ where NN is an integer and 0θ < 2π0θ<2π, then sin(x(3π)/4)sin(x- 3π4) is determined by θθ only. If x > δ1x>δ-1, θθ cannot be determined with any accuracy at all. Thus if xx is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of Y1(x)Y1(x) and the function must fail.
Figure 1
Figure 1

Further Comments

None.

Example

function nag_specfun_bessel_y1_real_vector_example
x = [0.5; 1; 3; 6; 8; 10; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_y1_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  -1.471e+00    0
   1.000e+00  -7.812e-01    0
   3.000e+00   3.247e-01    0
   6.000e+00  -1.750e-01    0
   8.000e+00  -1.581e-01    0
   1.000e+01   2.490e-01    0
   1.000e+03  -2.478e-02    0

function s17ar_example
x = [0.5; 1; 3; 6; 8; 10; 1000];
[f, ivalid, ifail] = s17ar(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  -1.471e+00    0
   1.000e+00  -7.812e-01    0
   3.000e+00   3.247e-01    0
   6.000e+00  -1.750e-01    0
   8.000e+00  -1.581e-01    0
   1.000e+01   2.490e-01    0
   1.000e+03  -2.478e-02    0


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