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# NAG Toolbox: nag_specfun_bessel_i1_real_vector (s18at)

## Purpose

nag_specfun_bessel_i1_real_vector (s18at) returns an array of values for the modified Bessel function ${I}_{1}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s18at(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_i1_real_vector(x, 'n', n)

## Description

nag_specfun_bessel_i1_real_vector (s18at) evaluates an approximation to the modified Bessel function of the first kind ${I}_{1}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
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:     $\mathrm{x}\left({\mathbf{n}}\right)$ – 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
${I}_{1}\left({x}_{i}\right)$, 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 ${I}_{1}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_bessel_i1_real (s18af), 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.
Check ivalid for more information.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
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, for quite moderate values of $x$ ($x>\stackrel{^}{x}$, the threshold value), the function must fail because ${I}_{1}\left(x\right)$ 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 s18at_example

fprintf('s18at example results\n\n');

x = [0; 0.5; 1; 3; 6; 8; 10; 15; 20; -1];

[f, ivalid, ifail] = s18at(x);

fprintf('     x           I_1(x)   ivalid\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end

```
```s18at example results

x           I_1(x)   ivalid
0.000e+00   0.000e+00    0
5.000e-01   2.579e-01    0
1.000e+00   5.652e-01    0
3.000e+00   3.953e+00    0
6.000e+00   6.134e+01    0
8.000e+00   3.999e+02    0
1.000e+01   2.671e+03    0
1.500e+01   3.281e+05    0
2.000e+01   4.245e+07    0
-1.000e+00  -5.652e-01    0
```

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