# NAG Library Routine Document

## 1Purpose

s19abf returns a value for the Kelvin function $\mathrm{bei}x$ via the function name.

## 2Specification

Fortran Interface
 Function s19abf ( x,
 Real (Kind=nag_wp) :: s19abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s19abf_ (const double *x, Integer *ifail)

## 3Description

s19abf evaluates an approximation to the Kelvin function $\mathrm{bei}x$.
Note:  $\mathrm{bei}\left(-x\right)=\mathrm{bei}x$, so the approximation need only consider $x\ge 0.0$.
The routine is based on several Chebyshev expansions:
For $0\le x\le 5$,
 $bei⁡x = x24 ∑ r=0 ′ ar Tr t , with ​ t=2 x5 4 - 1 ;$
For $x>5$,
 $bei⁡x=ex/22πx 1+1xat sin⁡α-1xbtcos⁡α$
 $+ex/22π x 1+1xct cos⁡β-1xdtsin⁡β$
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 computed as $\mathrm{bei}x=\frac{{x}^{2}}{4}$. If this result would underflow, the result returned is $\mathrm{bei}x=0.0$.
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

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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, $\mathrm{abs}\left({\mathbf{x}}\right)$ is too large for an accurate result to be returned. On soft failure, the routine returns zero. See also the Users' Note for your implementation.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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 function, 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 impossible to calculate the functions 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

s19abf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s19abfe.f90)

### 10.2Program Data

Program Data (s19abfe.d)

### 10.3Program Results

Program Results (s19abfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017