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NAG Library Manual

# NAG Library Routine DocumentS18CQF

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

S18CQF returns an array of values of the scaled modified Bessel function ${e}^{x}{K}_{0}\left(x\right)$.

## 2  Specification

 SUBROUTINE S18CQF ( N, X, F, IVALID, IFAIL)
 INTEGER N, IVALID(N), IFAIL REAL (KIND=nag_wp) X(N), F(N)

## 3  Description

S18CQF evaluates an approximation to ${e}^{{x}_{i}}{K}_{0}\left({x}_{i}\right)$, where ${K}_{0}$ 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 ${K}_{0}\left(x\right)$.
The routine uses the same Chebyshev expansions as S18AQF, which returns an array of the unscaled values of ${K}_{0}\left(x\right)$.

## 4  References

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

## 5  Parameters

1:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{N}}\ge 0$.
2:     $\mathrm{X}\left({\mathbf{N}}\right)$ – 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:     $\mathrm{F}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput
On exit: ${e}^{{x}_{i}}{K}_{0}\left({x}_{i}\right)$, the function values.
4:     $\mathrm{IVALID}\left({\mathbf{N}}\right)$ – 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, ${x}_{i}\le 0.0$, ${K}_{0}\left({x}_{i}\right)$ is undefined. ${\mathbf{F}}\left(\mathit{i}\right)$ contains $0.0$.
5:     $\mathrm{IFAIL}$ – 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: ${\mathbf{N}}\ge 0$.
${\mathbf{IFAIL}}=-99$
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

## 7  Accuracy

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.

Not applicable.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (s18cqfe.f90)

### 10.2  Program Data

Program Data (s18cqfe.d)

### 10.3  Program Results

Program Results (s18cqfe.r)