# NAG CL Interfaces19anc (kelvin_​ber_​vector)

## 1Purpose

s19anc returns an array of values for the Kelvin function $\mathrm{ber}x$.

## 2Specification

 #include
 void s19anc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s19anc, nag_specfun_kelvin_ber_vector or nag_kelvin_ber_vector.

## 3Description

s19anc evaluates an approximation to the Kelvin function $\mathrm{ber}{x}_{i}$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  $\mathrm{ber}\left(-x\right)=\mathrm{ber}x$, so the approximation need only consider $x\ge 0.0$.
The function is based on several Chebyshev expansions:
For $0\le x\le 5$,
 $ber⁡x=∑′r=0arTrt, with ​ t=2 x5 4-1.$
For $x>5$,
 $ber⁡x= e x/2 2πx 1+ 1 x a t cos⁡α + 1x b tsin⁡α + e-x/22πx 1+ 1xct sin⁡β+ 1xdtcos⁡β ,$
where $\alpha =\frac{x}{\sqrt{2}}-\frac{\pi }{8}$, $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$,
and $a\left(t\right)$, $b\left(t\right)$, $c\left(t\right)$, and $d\left(t\right)$ are expansions in the variable $t=\frac{10}{x}-1$.
When $x$ is sufficiently close to zero, the result is set directly to $\mathrm{ber}0=1.0$.
For large $x$, 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.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $\mathrm{ber}{x}_{i}$, the function values.
4: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
$\mathrm{abs}\left({x}_{i}\right)$ is too large for an accurate result to be returned. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT in s19aac, as defined in the Users' Note for your implementation.
5: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 7Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. 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 precision, then we have:
 $E≃ x2 ber1⁡x+ bei1⁡x δ$
(provided $E$ is within machine bounds).
For small $x$ the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.
For medium and large $x$, the error behaviour is oscillatory and its amplitude grows like $\sqrt{\frac{x}{2\pi }}{e}^{x/\sqrt{2}}$. Therefore it is not possible to calculate the function with any accuracy when $\sqrt{x}{e}^{x/\sqrt{2}}>\frac{\sqrt{2\pi }}{\delta }$. Note that this value of $x$ is much smaller than the minimum value of $x$ for which the function overflows.

## 8Parallelism and Performance

s19anc is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s19ance.c)

### 10.2Program Data

Program Data (s19ance.d)

### 10.3Program Results

Program Results (s19ance.r)