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NAG Toolbox: nag_specfun_bessel_j1_real (s17af)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_specfun_bessel_j1_real (s17af) returns the value of the Bessel function J1x, via the function name.

Syntax

[result, ifail] = s17af(x)
[result, ifail] = nag_specfun_bessel_j1_real(x)

Description

nag_specfun_bessel_j1_real (s17af) evaluates an approximation to the Bessel function of the first kind J1x.
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:     x – double scalar
The argument x of the function.

Optional Input Parameters

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     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:
   ifail=1
x is too large. On soft failure the function returns the amplitude of the J1 oscillation, 2πx .
   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 the argument x from a file, evaluates the function at each value of x and prints the results.
function s17af_example


fprintf('s17af example results\n\n');

x = [0     0.5     1    3    6   8    10   -1  1000];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s17af(x(j));
end

disp('      x         J_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s17af_plot;



function s17af_plot
x = [-30:0.25:30];
  for j = 1:numel(x)
    [J1(j), ifail] = s17af(x(j));
  end

  fig1 = figure;
  plot(x,J1,'-r');
  xlabel('x');
  ylabel('J_1(x)');
  title('Bessel Function J_1(x)');
  axis([-30 30 -0.6 0.6]);

s17af example results

      x         J_1(x)
   0.000e+00   0.000e+00
   5.000e-01   2.423e-01
   1.000e+00   4.401e-01
   3.000e+00   3.391e-01
   6.000e+00  -2.767e-01
   8.000e+00   2.346e-01
   1.000e+01   4.347e-02
  -1.000e+00  -4.401e-01
   1.000e+03   4.728e-03
s17af_fig1.png

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