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# NAG Toolbox: nag_specfun_bessel_i0_real (s18ae)

## Purpose

nag_specfun_bessel_i0_real (s18ae) returns the value of the modified Bessel function ${I}_{0}\left(x\right)$, via the function name.

## Syntax

[result, ifail] = s18ae(x)
[result, ifail] = nag_specfun_bessel_i0_real(x)

## Description

nag_specfun_bessel_i0_real (s18ae) evaluates an approximation to the modified Bessel function of the first kind ${I}_{0}\left(x\right)$.
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:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
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:
${\mathbf{ifail}}=1$
$\left|{\mathbf{x}}\right|$ is too large. On soft failure the function returns the approximate value of ${I}_{0}\left(x\right)$ at the nearest valid argument.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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 the function must fail for quite moderate values of $x$, because ${I}_{0}\left(x\right)$ 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.

None.

## Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
```function s18ae_example

fprintf('s18ae 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] = s18ae(x(j));
end

disp('      x          I_0(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s18ae_plot;

function s18ae_plot
x = [0:0.2:4];
for j = 1:numel(x)
[I0(j), ifail] = s18ae(x(j));
end

fig1 = figure;
plot(x,I0,'-r');
xlabel('x');
ylabel('I_0(x)');
title('Bessel Function I_0(x)');
axis([0 4 0 12]);

```
```s18ae example results

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

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Chapter Introduction
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