hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_specfun_bessel_i0_real_vector (s18as)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_bessel_i0_real_vector (s18as) returns an array of values of the modified Bessel function I0x.


[f, ivalid, ifail] = s18as(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_i0_real_vector(x, 'n', n)


nag_specfun_bessel_i0_real_vector (s18as) evaluates an approximation to the modified Bessel function of the first kind I0xi for an array of arguments xi, for i=1,2,,n.
Note:  I0-x=I0x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x4,
I0x=exr=0arTrt,   where ​ t=2 x4 -1.  
For 4<x12,
I0x=exr=0brTrt,   where ​ t=x-84.  
For x>12,
I0x=exx r=0crTrt,   where ​ t=2 12x -1.  
For small x, I0x1. 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 of the danger of overflow in calculating ex.


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


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
I0xi, the function values.
2:     ivalidn int64int32nag_int array
ivalidi contains the error code for xi, for i=1,2,,n.
No error.
xi is too large. fi contains the approximate value of I0xi at the nearest valid argument. The threshold value is the same as for ifail=1 in nag_specfun_bessel_i0_real (s18ae), as defined in the Users' Note for your implementation.
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.
Constraint: n0.
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 x I1x I0 x δ.  
Figure 1 shows the behaviour of the error amplification factor
xI1x I0x .  
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 the amplification factor is approximately x22 , which implies strong attenuation of the error, but in general ε can never be less than the machine precision.
For large x, εxδ and we have strong amplification of errors. However, for quite moderate values of x (x>x^, the threshold value), the function must fail because I0x would overflow; hence in practice the loss of accuracy for x close to x^ is not excessive and the errors will be dominated by those of the standard function exp.

Further Comments



This example reads values of x from a file, evaluates the function at each value of xi and prints the results.
function s18as_example

fprintf('s18as example results\n\n');

x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];

[f, ivalid, ifail] = s18as(x);

fprintf('     x           I_0(x)   ivalid\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));

s18as example results

     x           I_0(x)   ivalid
   0.000e+00   1.000e+00    0
   5.000e-01   1.063e+00    0
   1.000e+00   1.266e+00    0
   3.000e+00   4.881e+00    0
   6.000e+00   6.723e+01    0
   8.000e+00   4.276e+02    0
   1.000e+01   2.816e+03    0
   1.500e+01   3.396e+05    0
   2.000e+01   4.356e+07    0
  -1.000e+00   1.266e+00    0

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015