Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_i0_scaled_vector (s18cs)

## Purpose

nag_specfun_bessel_i0_scaled_vector (s18cs) returns an array of values of the scaled modified Bessel function ${e}^{-\left|x\right|}{I}_{0}\left(x\right)$.

## Syntax

[f, ifail] = s18cs(x, 'n', n)
[f, ifail] = nag_specfun_bessel_i0_scaled_vector(x, 'n', n)

## Description

nag_specfun_bessel_i0_scaled_vector (s18cs) evaluates an approximation to ${e}^{-\left|{x}_{i}\right|}{I}_{0}\left({x}_{i}\right)$, where ${I}_{0}$ is a modified Bessel function of the first kind for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The scaling factor ${e}^{-\left|x\right|}$ removes most of the variation in ${I}_{0}\left(x\right)$.
The function uses the same Chebyshev expansions as nag_specfun_bessel_i0_real_vector (s18as), which returns an array of the unscaled values of ${I}_{0}\left(x\right)$.

## 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
${e}^{-\left|{x}_{i}\right|}{I}_{0}\left({x}_{i}\right)$, the function values.
2:     $\mathrm{ifail}$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  ${\mathbf{ifail}}=1$
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

Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.

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 s18cs_example

fprintf('s18cs example results\n\n');

x = [0; 0.5; 1; 3; 6; 10; 1000; -1];

[f, ifail] = s18cs(x);

fprintf('     x        e^-|x| I_0(x)\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e\n', x(i), f(i));
end

```
```s18cs example results

x        e^-|x| I_0(x)
0.000e+00   1.000e+00
5.000e-01   6.450e-01
1.000e+00   4.658e-01
3.000e+00   2.430e-01
6.000e+00   1.667e-01
1.000e+01   1.278e-01
1.000e+03   1.262e-02
-1.000e+00   4.658e-01
```