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NAG Toolbox: nag_specfun_bessel_j0_real (s17ae)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_j0_real (s17ae) returns the value of the Bessel function J0x, via the function name.


[result, ifail] = s17ae(x)
[result, ifail] = nag_specfun_bessel_j0_real(x)


nag_specfun_bessel_j0_real (s17ae) evaluates an approximation to the Bessel function of the first kind J0x.
Note:  J0-x=J0x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x8,
J0x=r=0arTrt,   with ​t=2 x8 2 -1.  
For x>8,
J0x= 2πx P0xcosx-π4-Q0xsinx- π4 ,  
where P0x=r=0brTrt,
and Q0x= 8xr=0crTrt,
with t=2 8x 2-1.
For x near zero, J0x1. 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 J0x; 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.


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


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.

Optional Input Parameters


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:
x is too large. On soft failure the function returns the amplitude of the J0 oscillation, 2πx .
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Let δ be the relative error in the argument and E be the absolute error in the result. (Since J0x oscillates about zero, absolute error and not relative error is significant.)
If δ is somewhat larger than the machine precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xJ1x.
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, J0x 2πx cosx- π4. The amplitude 2πx  can be calculated with reasonable accuracy for all x, but cosx- π4 cannot. If x- π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then cosx- π4  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 inverse of the machine precision, it is impossible to calculate the phase of J0x and the function must fail.
Figure 1
Figure 1

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s17ae_example

fprintf('s17ae 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] = s17ae(x(j));

disp('      x         J_0(x)');
fprintf('%12.3e%12.3e\n',[x; result]);


function s17ae_plot
x = [-30:0.25:30];
  for j = 1:numel(x)
    [J0(j), ifail] = s17ae(x(j));

  fig1 = figure;
  title('Bessel Function J_0(x)');
  axis([-30 30 -0.5 1.1]);

s17ae example results

      x         J_0(x)
   0.000e+00   1.000e+00
   5.000e-01   9.385e-01
   1.000e+00   7.652e-01
   3.000e+00  -2.601e-01
   6.000e+00   1.506e-01
   8.000e+00   1.717e-01
   1.000e+01  -2.459e-01
  -1.000e+00   7.652e-01
   1.000e+03   2.479e-02

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Chapter Contents
Chapter Introduction
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