# NAG FL Interfaces18gkf (bessel_​j_​seq_​complex)

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## 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 $n=1,2,\dots ,|N|+1$.

## 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)
C Header Interface
#include <nag.h>
 void s18gkf_ (const Complex *z, const double *a, const Integer *nl, Complex b[], Integer *ifail)
The routine may be called by the names s18gkf or nagf_specfun_bessel_j_seq_complex.

## 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 $\left(|N|+1\right)$-member sequence is generated for orders $\alpha ,\alpha +1,\dots ,\alpha +|N|$ 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
 $J-q(z)=cos(πq)Jq(z)-sin(πq)Yq(z)$
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.
NIST Digital Library of Mathematical Functions

## 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: $\mathbf{nl}$Integer Input
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) array Output
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, $|{\mathbf{nl}}|=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{nl}}|\le 101$.
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}<1.0$.
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}\ge 0.0$.
On entry, ${\mathbf{nl}}=⟨\mathit{\text{value}}⟩$.
Constraint: when ${\mathbf{nl}}<0$, ${\mathbf{z}}\ne \left(0.0,0.0\right)$.
${\mathbf{ifail}}=2$
Computation abandoned due to the likelihood of overflow.
${\mathbf{ifail}}=3$
Computation completed but some precision has been lost.
${\mathbf{ifail}}=4$
Computation abandoned due to total loss of precision.
${\mathbf{ifail}}=5$
Computation abandoned due to failure to satisfy the termination condition.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,|{\mathrm{log}}_{10}|z||,|{\mathrm{log}}_{10}|\alpha ||\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $|z|$ and $|\alpha |$, 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)