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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_kelvin_bei_vector (s19ap)

## Purpose

nag_specfun_kelvin_bei_vector (s19ap) returns an array of values for the Kelvin function beix$\mathrm{bei}x$.

## Syntax

[f, ivalid, ifail] = s19ap(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_kelvin_bei_vector(x, 'n', n)

## Description

nag_specfun_kelvin_bei_vector (s19ap) evaluates an approximation to the Kelvin function beixi$\mathrm{bei}{x}_{i}$ for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Note:  bei(x) = beix$\mathrm{bei}\left(-x\right)=\mathrm{bei}x$, so the approximation need only consider x0.0$x\ge 0.0$.
The function is based on several Chebyshev expansions:
For 0x5$0\le x\le 5$,
 beix = (x2)/4 ∑′ arTr(t),   with ​t = 2(x/5)4 − 1; r = 0
$bei⁡x = x24 ∑′r=0 ar Tr (t) , with ​ t=2 (x5) 4 - 1 ;$
For x > 5$x>5$,
 beix = (ex / sqrt(2))/(sqrt(2πx)) [(1 + 1/xa(t))sinα − 1/xb(t)cosα] $bei⁡x = e x/2 2πx [ ( 1 + 1x a (t) ) sin⁡α - 1x b (t) cos⁡α ]$
 + (ex / sqrt(2))/(sqrt(2π x)) [(1 + 1/xc(t))cosβ − 1/xd(t)sinβ] $+ e x/2 2π x [ ( 1 + 1x c (t) ) cos⁡β - 1x d (t) sin⁡β ]$
where α = x/(sqrt(2))π/8 $\alpha =\frac{x}{\sqrt{2}}-\frac{\pi }{8}$, β = x/(sqrt(2)) + π/8 $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$,
and a(t)$a\left(t\right)$, b(t)$b\left(t\right)$, c(t)$c\left(t\right)$, and d(t)$d\left(t\right)$ are expansions in the variable t = 10/x1$t=\frac{10}{x}-1$.
When x$x$ is sufficiently close to zero, the result is computed as beix = (x2)/4 $\mathrm{bei}x=\frac{{x}^{2}}{4}$. If this result would underflow, the result returned is beix = 0.0$\mathrm{bei}x=0.0$.
For large x$x$, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the function must fail.

## References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## Parameters

### Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
The argument xi${x}_{\mathit{i}}$ of the function, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of points.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     f(n) – double array
beixi$\mathrm{bei}{x}_{i}$, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
abs(xi)$\mathrm{abs}\left({x}_{i}\right)$ is too large for an accurate result to be returned. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in nag_specfun_kelvin_bei (s19ab), as defined in the Users' Note for your implementation.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ 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 ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: n0${\mathbf{n}}\ge 0$.

## Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. Let E$E$ be the absolute error in the function, and δ$\delta$ be the relative error in the argument. If δ$\delta$ is somewhat larger than the machine precision, then we have:
 E ≃ |x/(sqrt(2))( − ber1x + bei1x)|δ $E≃ | x2 ( - ber1⁡x+ bei1⁡x ) |δ$
(provided E$E$ is within machine bounds).
For small x$x$ the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large x$x$, the error behaviour is oscillatory and its amplitude grows like sqrt(x/(2π))ex / sqrt(2)$\sqrt{\frac{x}{2\pi }}{e}^{x/\sqrt{2}}$. Therefore it is impossible to calculate the functions with any accuracy when sqrt(x)ex / sqrt(2) > (sqrt(2π))/δ $\sqrt{x}{e}^{x/\sqrt{2}}>\frac{\sqrt{2\pi }}{\delta }$. Note that this value of x$x$ is much smaller than the minimum value of x$x$ for which the function overflows.

None.

## Example

```function nag_specfun_kelvin_bei_vector_example
x = [0.1; 1; 2.5; 5; 10; 15; -1];
[f, ivalid, ifail] = nag_specfun_kelvin_bei_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
1.000e-01   2.500e-03    0
1.000e+00   2.496e-01    0
2.500e+00   1.457e+00    0
5.000e+00   1.160e-01    0
1.000e+01   5.637e+01    0
1.500e+01  -2.953e+03    0
-1.000e+00   2.496e-01    0

```
```function s19ap_example
x = [0.1; 1; 2.5; 5; 10; 15; -1];
[f, ivalid, ifail] = s19ap(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
1.000e-01   2.500e-03    0
1.000e+00   2.496e-01    0
2.500e+00   1.457e+00    0
5.000e+00   1.160e-01    0
1.000e+01   5.637e+01    0
1.500e+01  -2.953e+03    0
-1.000e+00   2.496e-01    0

```