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NAG Toolbox

NAG Toolbox: nag_specfun_bessel_j1_real_vector (s17at)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_bessel_j1_real_vector (s17at) returns an array of values of the Bessel function J1x.

Syntax

[f, ivalid, ifail] = s17at(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_j1_real_vector(x, 'n', n)

Description

nag_specfun_bessel_j1_real_vector (s17at) evaluates an approximation to the Bessel function of the first kind J1xi for an array of arguments xi, for i=1,2,,n.
Note:  J1-x=-J1x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x8,
J1x=x8r=0arTrt,   with ​t=2 x8 2-1.  
For x>8,
J1x=2πx P1xcosx-3π4-Q1xsinx-3π4  
where P1x=r=0brTrt,
and Q1x= 8xr=0crTrt,
with t=2 8x 2-1.
For x near zero, J1x x2 . This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For very large x, it becomes impossible to provide results with any reasonable accuracy (see Accuracy), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of J1x; only the amplitude, 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 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:     xn – double array
The argument xi of the function, for i=1,2,,n.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of points.
Constraint: n0.

Output Parameters

1:     fn – double array
J1xi, the function values.
2:     ivalidn int64int32nag_int array
ivalidi contains the error code for xi, for i=1,2,,n.
ivalidi=0
No error.
ivalidi=1
On entry,xi is too large. fi contains the amplitude of the J1 oscillation, 2πxi .
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: n0.
   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

Let δ be the relative error in the argument and E be the absolute error in the result. (Since J1x oscillates about zero, absolute error and not relative error is significant.)
If δ is somewhat larger than machine precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
ExJ0x-J1xδ  
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xJ0x-J1x.
However, if δ is of the same order as machine precision, then rounding errors could make E slightly larger than the above relation predicts.
For very large x, the above relation ceases to apply. In this region, J1x 2πx cosx- 3π4. The amplitude 2πx  can be calculated with reasonable accuracy for all x, but cosx- 3π4 cannot. If x- 3π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then cosx- 3π4 is determined by θ only. If xδ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of, the reciprocal of machine precision, it is impossible to calculate the phase of J1x and the function must fail.
Figure 1
Figure 1

Further Comments

None.

Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s17at_example


fprintf('s17at example results\n\n');

x = [0; 0.5; 1; 3; 6; 8; 10; -1; 1000];

[f, ivalid, ifail] = s17at(x);

fprintf('     x           J_1(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end


s17at example results

     x           J_1(x)   ivalid
   0.000e+00   0.000e+00    0
   5.000e-01   2.423e-01    0
   1.000e+00   4.401e-01    0
   3.000e+00   3.391e-01    0
   6.000e+00  -2.767e-01    0
   8.000e+00   2.346e-01    0
   1.000e+01   4.347e-02    0
  -1.000e+00  -4.401e-01    0
   1.000e+03   4.728e-03    0

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