# NAG FL Interfaces19anf (kelvin_​ber_​vector)

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## 1Purpose

s19anf returns an array of values for the Kelvin function $\mathrm{ber}x$.

## 2Specification

Fortran Interface
 Subroutine s19anf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s19anf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s19anf or nagf_specfun_kelvin_ber_vector.

## 3Description

s19anf 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 routine is based on several Chebyshev expansions:
For $0\le x\le 5$,
 $ber⁡x=∑′r=0arTr(t), with ​ t=2 (x5) 4-1.$
For $x>5$,
 $ber⁡x= e x/2 2πx [(1+ 1 x a(t))cos⁡α+1xb(t)sin⁡α] + e-x/22πx [(1+ 1xc(t))sin⁡β+ 1xd(t)cos⁡β] ,$
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 routine 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: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{ber}{x}_{i}$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
$\mathrm{abs}\left({x}_{i}\right)$ is too large for an accurate result to be returned. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s19aaf , as defined in the Users' Note for your implementation.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 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

s19anf 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 (s19anfe.f90)

### 10.2Program Data

Program Data (s19anfe.d)

### 10.3Program Results

Program Results (s19anfe.r)