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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_bessel_i1_real (s18af)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_i1_real (s18af) returns a value for the modified Bessel function I1x, via the function name.


[result, ifail] = s18af(x)
[result, ifail] = nag_specfun_bessel_i1_real(x)


nag_specfun_bessel_i1_real (s18af) evaluates an approximation to the modified Bessel function of the first kind I1x.
Note:  I1-x=-I1x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x4,
I1x=xr=0arTrt,   where ​t=2 x4 2-1;  
For 4<x12,
I1x=exr=0brTrt,   where ​t=x-84;  
For x>12,
I1x=exx r=0crTrt,   where ​t=2 12x -1.  
For small x, I1xx. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For large x, the function must fail because I1x cannot be represented without overflow.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications


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 approximate value of I1x at the nearest valid argument.
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 and ε be the relative errors in the argument and result respectively.
If 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 xI0x- I1x I1 x δ.  
Figure 1 shows the behaviour of the error amplification factor
xI0x - I1x I1x .  
Figure 1
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, εδ and there is no amplification of errors.
For large x, εxδ and we have strong amplification of errors. However the function must fail for quite moderate values of x because I1x 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



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
function s18af_example

fprintf('s18af 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] = s18af(x(j));

disp('      x          I_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);


function s18af_plot
  x = [0:0.2:4];
  for j = 1:numel(x)
    [I1(j), ifail] = s18af(x(j));

  fig1 = figure;
  title('Bessel Function I_1(x)');
  axis([0 4 0 10]);

s18af example results

      x          I_1(x)
   0.000e+00   0.000e+00
   5.000e-01   2.579e-01
   1.000e+00   5.652e-01
   3.000e+00   3.953e+00
   6.000e+00   6.134e+01
   8.000e+00   3.999e+02
   1.000e+01   2.671e+03
   1.500e+01   3.281e+05
   2.000e+01   4.245e+07
  -1.000e+00  -5.652e-01

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

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