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NAG Library Manual

# NAG Library Routine DocumentS19APF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S19APF returns an array of values for the Kelvin function $\mathrm{bei}x$.

## 2  Specification

 SUBROUTINE S19APF ( N, X, F, IVALID, IFAIL)
 INTEGER N, IVALID(N), IFAIL REAL (KIND=nag_wp) X(N), F(N)

## 3  Description

S19APF evaluates an approximation to the Kelvin function $\mathrm{bei}{x}_{i}$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
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 $x>5$,
 $bei⁡x = e x/2 2πx 1 + 1x a t sin⁡α - 1x b t cos⁡α$
 $+ e x/2 2π x 1 + 1x c t cos⁡β - 1x d t sin⁡β$
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.

## 4  References

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

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{N}}\ge 0$.
2:     X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
3:     F(N) – REAL (KIND=nag_wp) arrayOutput
On exit: $\mathrm{bei}{x}_{i}$, the function values.
4:     IVALID(N) – INTEGER arrayOutput
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 S19ABF, as defined in the Users' Note for your implementation.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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: ${\mathbf{N}}\ge 0$.

## 7  Accuracy

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.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s19apfe.f90)

### 9.2  Program Data

Program Data (s19apfe.d)

### 9.3  Program Results

Program Results (s19apfe.r)