# NAG Library Routine Document

## 1Purpose

s18crf returns an array of values of the scaled modified Bessel function ${e}^{x}{K}_{1}\left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s18crf ( 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 nagmk26.h
 void s18crf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)

## 3Description

s18crf evaluates an approximation to ${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, where ${K}_{1}$ is a modified Bessel function of the second kind for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The scaling factor ${e}^{x}$ removes most of the variation in ${K}_{1}\left(x\right)$.
The routine uses the same Chebyshev expansions as s18arf, which returns an array of the unscaled values of ${K}_{1}\left(x\right)$.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     $\mathbf{f}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${e}^{{x}_{i}}{K}_{1}\left({x}_{i}\right)$, the function values.
4:     $\mathbf{ivalid}\left({\mathbf{n}}\right)$ – 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.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, ${x}_{i}\le 0.0$, ${K}_{1}\left({x}_{i}\right)$ is undefined. ${\mathbf{f}}\left(\mathit{i}\right)$ contains $0.0$.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}$ is too close to zero, as determined by the value of the safe-range parameter x02amf: there is a danger of causing overflow. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the reciprocal of the safe-range parameter.
5:     $\mathbf{ifail}$ – 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, 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 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

Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.

## 8Parallelism and Performance

s18crf 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 (s18crfe.f90)

### 10.2Program Data

Program Data (s18crfe.d)

### 10.3Program Results

Program Results (s18crfe.r)

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