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NAG Toolbox: nag_specfun_bessel_i0_real_vector (s18as)

Purpose

nag_specfun_bessel_i0_real_vector (s18as) returns an array of values of the modified Bessel function I0(x)I0(x).

Syntax

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

Description

nag_specfun_bessel_i0_real_vector (s18as) evaluates an approximation to the modified Bessel function of the first kind I0(xi)I0(xi) for an array of arguments xixi, for i = 1,2,,ni=1,2,,n.
Note:  I0(x) = I0(x)I0(-x)=I0(x), so the approximation need only consider x0x0.
The function is based on three Chebyshev expansions:
For 0 < x40<x4,
I0(x) = ex arTr(t),   where ​t = 2(x/4)1.
r = 0
I0(x)=exr=0arTr(t),   where ​ t=2 (x4) -1.
For 4 < x124<x12,
I0(x) = ex brTr(t),   where ​t = (x8)/4.
r = 0
I0(x)=exr=0brTr(t),   where ​ t=x-84.
For x > 12x>12,
I0(x) = (ex)/(sqrt(x)) crTr(t),   where ​t = 2(12/x)1.
r = 0
I0(x)=exx r=0crTr(t),   where ​ t=2 (12x) -1.
For small xx, I0(x)1I0(x)1. This approximation is used when xx is sufficiently small for the result to be correct to machine precision.
For large xx, the function must fail because of the danger of overflow in calculating exex.

References

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

Parameters

Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0n0.
The argument xixi of the function, for i = 1,2,,ni=1,2,,n.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
nn, the number of points.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     f(n) – double array
I0(xi)I0(xi), the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)ivalidi contains the error code for xixi, for i = 1,2,,ni=1,2,,n.
ivalid(i) = 0ivalidi=0
No error.
ivalid(i) = 1ivalidi=1
xixi is too large. f(i)fi contains the approximate value of I0(xi)I0(xi) at the nearest valid argument. The threshold value is the same as for ifail = 1ifail=1 in nag_specfun_bessel_i0_real (s18ae), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1
On entry, at least one value of x was invalid.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: n0n0.

Accuracy

Let δδ and εε be the relative errors in the argument and result respectively.
If δδ is somewhat larger than the machine precision (i.e., if δδ is due to data errors etc.), then εε and δδ are approximately related by:
ε |( x I1(x) )/( I0 (x) )|δ.
ε | x I1(x) I0 (x) |δ.
Figure 1 shows the behaviour of the error amplification factor
|( xI1(x))/(I0(x))|.
| xI1(x) I0(x) |.
Figure 1
Figure 1
However if δδ is of the same order as machine precision, then rounding errors could make εε slightly larger than the above relation predicts.
For small xx the amplification factor is approximately (x2)/2 x22 , which implies strong attenuation of the error, but in general εε can never be less than the machine precision.
For large xx, εxδεxδ and we have strong amplification of errors. However, for quite moderate values of xx (x > x>x^, the threshold value), the function must fail because I0(x)I0(x) would overflow; hence in practice the loss of accuracy for xx close to x^ is not excessive and the errors will be dominated by those of the standard function exp.

Further Comments

None.

Example

function nag_specfun_bessel_i0_real_vector_example
x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];
[f, ivalid, ifail] = nag_specfun_bessel_i0_real_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   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

function s18as_example
x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];
[f, ivalid, ifail] = s18as(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
  fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
 

    X           Y
   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


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