# NAG Library Routine Document

## 1Purpose

s18gkf returns a sequence of values for the Bessel functions ${J}_{\alpha +n-1}\left(z\right)$ or ${J}_{\alpha -n+1}\left(z\right)$ for complex $z$, non-negative $\alpha <1$ and $n=1,2,\dots ,\left|N\right|+1$.

## 2Specification

Fortran Interface
 Subroutine s18gkf ( z, a, nl, b,
 Integer, Intent (In) :: nl Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a Complex (Kind=nag_wp), Intent (In) :: z Complex (Kind=nag_wp), Intent (Out) :: b(abs(nl)+1)
#include nagmk26.h
 void s18gkf_ (const Complex *z, const double *a, const Integer *nl, Complex b[], Integer *ifail)

## 3Description

s18gkf evaluates a sequence of values for the Bessel function of the first kind ${J}_{\alpha }\left(z\right)$, where $z$ is complex and nonzero and $\alpha$ is the order with $0\le \alpha <1$. The $\left(\left|N\right|+1\right)$-member sequence is generated for orders $\alpha ,\alpha +1,\dots ,\alpha +\left|N\right|$ when $N\ge 0$. Note that $+$ is replaced by $-$ when $N<0$. For positive orders the routine may also be called with $z=0$, since ${J}_{q}\left(0\right)=0$ when $q>0$. For negative orders the formula
 $J-qz=cosπqJqz-sinπqYqz$
is used to generate the required sequence. The appropriate values of ${J}_{q}\left(z\right)$ and ${Y}_{q}\left(z\right)$ are obtained by calls to s17dcf and s17def.

## 4References

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

## 5Arguments

1:     $\mathbf{z}$ – Complex (Kind=nag_wp)Input
On entry: the argument $z$ of the function.
Constraint: ${\mathbf{z}}\ne \left(0.0,0.0\right)$ when ${\mathbf{nl}}<0$.
2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: the order $\alpha$ of the first member in the required sequence of function values.
Constraint: $0.0\le {\mathbf{a}}<1.0$.
3:     $\mathbf{nl}$ – IntegerInput
On entry: the value of $N$.
Constraint: $\mathrm{abs}\left({\mathbf{nl}}\right)\le 101$.
4:     $\mathbf{b}\left(\mathrm{abs}\left({\mathbf{nl}}\right)+1\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: with ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{3}}$, the required sequence of function values: ${\mathbf{b}}\left(\mathit{n}\right)$ contains ${J}_{\alpha +\mathit{n}-1}\left(z\right)$ if ${\mathbf{nl}}\ge 0$ and ${J}_{\alpha -\mathit{n}+1}\left(z\right)$ otherwise, for $\mathit{n}=1,2,\dots ,\mathrm{abs}\left({\mathbf{nl}}\right)+1$.
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, ${\mathbf{z}}=\left(0.0,0.0\right)$ when ${\mathbf{nl}}<0$, or ${\mathbf{a}}<0.0$, or ${\mathbf{a}}\ge 1.0$, or $\mathrm{abs}\left({\mathbf{nl}}\right)>101$.
${\mathbf{ifail}}=2$
The computation has been abandoned due to the likelihood of overflow.
${\mathbf{ifail}}=3$
The computation has been completed but some precision has been lost.
${\mathbf{ifail}}=4$
The computation has been abandoned due to total loss of precision.
${\mathbf{ifail}}=5$
The computation has been abandoned due to failure to satisfy the termination condition.
${\mathbf{ifail}}=-99$
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

All constants in s17dcf and s17def are specified to approximately $18$ digits of precision. If $t$ denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number of correct digits in the results obtained is limited by $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. Because of errors in argument reduction when computing elementary functions inside s17dcf and s17def, the actual number of correct digits is limited, in general, by $p-s$, where $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left|{\mathrm{log}}_{10}\left|z\right|\right|,\left|{\mathrm{log}}_{10}\left|\alpha \right|\right|\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $\left|z\right|$ and $\left|\alpha \right|$, the less the precision in the result.

## 8Parallelism and Performance

s18gkf is not threaded in any implementation.

None.

## 10Example

This example evaluates ${J}_{0}\left(z\right),{J}_{1}\left(z\right),{J}_{2}\left(z\right)$ and ${J}_{3}\left(z\right)$ at $z=0.6-0.8i$, and prints the results.

### 10.1Program Text

Program Text (s18gkfe.f90)

### 10.2Program Data

Program Data (s18gkfe.d)

### 10.3Program Results

Program Results (s18gkfe.r)

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