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Chapter Contents
Chapter Introduction
NAG Toolbox

# 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 ${I}_{0}\left(x\right)$.

## 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 ${I}_{0}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  ${I}_{0}\left(-x\right)={I}_{0}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The function is based on three Chebyshev expansions:
For $0,
 $I0x=ex∑′r=0arTrt, where ​ t=2 x4 -1.$
For $4,
 $I0x=ex∑′r=0brTrt, where ​ t=x-84.$
For $x>12$,
 $I0x=exx ∑′r=0crTrt, where ​ t=2 12x -1.$
For small $x$, ${I}_{0}\left(x\right)\simeq 1$. 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 ${e}^{x}$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{f}\left({\mathbf{n}}\right)$ – double array
${I}_{0}\left({x}_{i}\right)$, the function values.
2:     $\mathrm{ivalid}\left({\mathbf{n}}\right)$int64int32nag_int array
${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of ${I}_{0}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_bessel_i0_real (s18ae), as defined in the Users' Note for your implementation.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ 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  ${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃ x I1x I0 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xI1x I0x .$
Figure 1
However if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ 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 $\epsilon$ can never be less than the machine precision.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of errors. However, for quite moderate values of $x$ ($x>\stackrel{^}{x}$, the threshold value), the function must fail because ${I}_{0}\left(x\right)$ would overflow; hence in practice the loss of accuracy for $x$ close to $\stackrel{^}{x}$ is not excessive and the errors will be dominated by those of the standard function exp.

None.

## Example

This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ 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));
end

```
```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
```