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 approximate value of $I_{0} (x)$ at the nearest valid argument.

$ifail = - 99$: 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

and

ε

be the relative errors in the argument and result respectively.

δ

is somewhat larger than the machine precision (i.e., if

δ

is due to data errors etc.), then

ε

and

δ

are approximately related by:

ε ≃ |\frac{x I_{1} (x)}{I_{0} (x)}| δ .

Figure 1 shows the behaviour of the error amplification factor

|\frac{x I_{1} (x)}{I_{0} (x)}| .

Figure 1

However if

δ

is of the same order as machine precision, then rounding errors could make

ε

slightly larger than the above relation predicts.

For small

x

the amplification factor is approximately

\frac{x^{2}}{2}

, which implies strong attenuation of the error, but in general

ε

can never be less than the machine precision.

For large

x

ε ≃ x δ

and we have strong amplification of errors. However the function must fail for quite moderate values of

x

, because

I_{0} (x)

would overflow; hence in practice the loss of accuracy for large

x

is not excessive. Note that for large

x

the errors will be dominated by those of the standard function exp.

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.

Open in the MATLAB editor: s18ae_example

function s18ae_example


fprintf('s18ae example results\n\n');

x = [0    0.5    1    3    6    8    10    15    20   -1];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s18ae(x(j));
end

disp('      x          I_0(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s18ae_plot;



function s18ae_plot
  x = [0:0.2:4];
  for j = 1:numel(x)
    [I0(j), ifail] = s18ae(x(j));
  end

  fig1 = figure;
  plot(x,I0,'-r');
  xlabel('x');
  ylabel('I_0(x)');
  title('Bessel Function I_0(x)');
  axis([0 4 0 12]);

s18ae example results

      x          I_0(x)
   0.000e+00   1.000e+00
   5.000e-01   1.063e+00
   1.000e+00   1.266e+00
   3.000e+00   4.881e+00
   6.000e+00   6.723e+01
   8.000e+00   4.276e+02
   1.000e+01   2.816e+03
   1.500e+01   3.396e+05
   2.000e+01   4.356e+07
  -1.000e+00   1.266e+00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox