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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_i1_real (s18af)

## Purpose

nag_specfun_bessel_i1_real (s18af) returns a value for the modified Bessel function ${I}_{1}\left(x\right)$, via the function name.

## Syntax

[result, ifail] = s18af(x)
[result, ifail] = nag_specfun_bessel_i1_real(x)

## Description

nag_specfun_bessel_i1_real (s18af) evaluates an approximation to the modified Bessel function of the first kind ${I}_{1}\left(x\right)$.
Note:  ${I}_{1}\left(-x\right)=-{I}_{1}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The function is based on three Chebyshev expansions:
For $0,
 $I1x=x∑′r=0arTrt, where ​t=2 x4 2-1;$
For $4,
 $I1x=ex∑′r=0brTrt, where ​t=x-84;$
For $x>12$,
 $I1x=exx ∑′r=0crTrt, where ​t=2 12x -1.$
For small $x$, ${I}_{1}\left(x\right)\simeq x$. 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 ${I}_{1}\left(x\right)$ cannot be represented without overflow.

## 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:
${\mathbf{ifail}}=1$
x is too large. On soft failure the function returns the approximate value of ${I}_{1}\left(x\right)$ 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:
 $ε≃ xI0x- I1x I1 x δ.$
Figure 1 shows the behaviour of the error amplification factor
 $xI0x - I1x I1x .$ 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 $x$, $\epsilon \simeq \delta$ and there is no amplification of errors.
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 ${I}_{1}\left(x\right)$ 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 s18af_example

fprintf('s18af 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] = s18af(x(j));
end

disp('      x          I_1(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s18af_plot;

function s18af_plot
x = [0:0.2:4];
for j = 1:numel(x)
[I1(j), ifail] = s18af(x(j));
end

fig1 = figure;
plot(x,I1,'-r');
xlabel('x');
ylabel('I_1(x)');
title('Bessel Function I_1(x)');
axis([0 4 0 10]);

```
```s18af example results

x          I_1(x)
0.000e+00   0.000e+00
5.000e-01   2.579e-01
1.000e+00   5.652e-01
3.000e+00   3.953e+00
6.000e+00   6.134e+01
8.000e+00   3.999e+02
1.000e+01   2.671e+03
1.500e+01   3.281e+05
2.000e+01   4.245e+07
-1.000e+00  -5.652e-01
``` 