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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_kelvin_kei_vector (s19ar)

## Purpose

nag_specfun_kelvin_kei_vector (s19ar) returns an array of values for the Kelvin function $\mathrm{kei}x$.

## Syntax

[f, ivalid, ifail] = s19ar(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_kelvin_kei_vector(x, 'n', n)

## Description

nag_specfun_kelvin_kei_vector (s19ar) evaluates an approximation to the Kelvin function $\mathrm{kei}{x}_{i}$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  for $x<0$ the function is undefined, so we need only consider $x\ge 0$.
The function is based on several Chebyshev expansions:
For $0\le x\le 1$,
 $kei⁡x=-π4ft+x24-gtlogx+vt$
where $f\left(t\right)$, $g\left(t\right)$ and $v\left(t\right)$ are expansions in the variable $t=2{x}^{4}-1$;
For $1,
 $kei⁡x=exp-98x ut$
where $u\left(t\right)$ is an expansion in the variable $t=x-2$;
For $x>3$,
 $kei⁡x=π 2x e-x/2 1+1x ctsin⁡β+1xdtcos⁡β$
where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.
For $x<0$, the function is undefined, and hence the function fails and returns zero.
When $x$ is sufficiently close to zero, the result is computed as
 $kei⁡x=-π4+1-γ-logx2 x24$
and when $x$ is even closer to zero simply as
 $kei⁡x=-π4.$
For large $x$, $\mathrm{kei}x$ is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the function fails.

## 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 $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\ge 0.0$, 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
$\mathrm{kei}{x}_{i}$, 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, the result underflows. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_kelvin_kei (s19ad), as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}<0.0$, the function is undefined. ${\mathbf{f}}\left(\mathit{i}\right)$ contains $0.0$.
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 $E$ be the absolute error in the result, and $\delta$ be the relative error in the argument. If $\delta$ is somewhat larger than the machine representation error, then we have:
 $E≃ x2 - ker1⁡x+ kei1⁡x δ.$
For small $x$, errors are attenuated by the function and hence are limited by the machine precision.
For medium and large $x$, the error behaviour, like the function itself, is oscillatory and hence only absolute accuracy of the function can be maintained. For this range of $x$, the amplitude of the absolute error decays like $\sqrt{\frac{\pi x}{2}}{e}^{-x/\sqrt{2}}$, which implies a strong attenuation of error. Eventually, $\mathrm{kei}x$, which is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$, becomes so small that it cannot be calculated without causing underflow and therefore the function returns zero. Note that for large $x$, the errors are dominated by those of the standard function exp.

Underflow may occur for a few values of $x$ close to the zeros of $\mathrm{kei}x$, below the limit which causes a failure with ${\mathbf{ifail}}={\mathbf{1}}$.

## Example

This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.
```function s19ar_example

fprintf('s19ar example results\n\n');

x = [0; 0.1; 1; 2.5; 5; 10; 15];

[f, ivalid, ifail] = s19ar(x);

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

```
```s19ar example results

x           kei(x)   ivalid
0.000e+00  -7.854e-01    0
1.000e-01  -7.769e-01    0
1.000e+00  -4.950e-01    0
2.500e+00  -1.107e-01    0
5.000e+00   1.119e-02    0
1.000e+01  -3.075e-04    0
1.500e+01   7.963e-06    0
```