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NAG Library Manual

# NAG Library Routine DocumentS17ALF

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

S17ALF determines the leading ${\mathbf{N}}$ zeros of one of the Bessel functions ${J}_{\alpha }\left(x\right)$, ${Y}_{\alpha }\left(x\right)$, ${J}_{\alpha }^{\prime }\left(x\right)$ or ${Y}_{\alpha }^{\prime }\left(x\right)$ for real $x$ and non-negative $\alpha$.

## 2  Specification

 SUBROUTINE S17ALF ( A, N, MODE, REL, X, IFAIL)
 INTEGER N, MODE, IFAIL REAL (KIND=nag_wp) A, REL, X(N)

## 3  Description

S17ALF attempts to find the leading $N$ zeros of one of the Bessel functions ${J}_{\alpha }\left(x\right)$, ${Y}_{\alpha }\left(x\right)$, ${J}_{\alpha }^{\prime }\left(x\right)$ or ${Y}_{\alpha }^{\prime }\left(x\right)$, where $x$ is real. When $\alpha$ is real, these functions each have an infinite number of real zeros, all of which are simple with the possible exception of $x=0$. If $\alpha \ge 0$, the $\mathit{n}$th positive zero is denoted by ${j}_{\alpha ,\mathit{n}},{j}_{\alpha ,\mathit{n}}^{\prime },{y}_{\alpha ,\mathit{n}}$ and ${y}_{\alpha ,\mathit{n}}^{\prime }$, respectively, for $\mathit{n}=1,2,\dots ,N$, except that $x=0$ is counted as the first zero of ${J}_{\alpha }^{\prime }\left(x\right)$ when $\alpha =0$. Since ${J}_{0}^{\prime }\left(x\right)=-{J}_{1}\left(x\right)$, it therefore follows that ${j}_{0,1}^{\prime }=0$ and ${j}_{0,n}^{\prime }=-{j}_{1,n-1}$ for $n=2,3,\dots ,N-1$. Further details can be found in Section 9.5 of Abramowitz and Stegun (1972).
S17ALF is based on Algol 60 procedures given by Temme (1979). Initial approximations to the zeros are computed from asymptotic expansions. These are then improved by higher-order Newton iteration making use of the differential equation for the Bessel functions.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Temme N M (1976) On the numerical evaluation of the ordinary Bessel function of the second kind J. Comput. Phys. 21 343–350
Temme N M (1979) An algorithm with Algol 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives J. Comput. Phys. 32 270–279

## 5  Parameters

1:     A – REAL (KIND=nag_wp)Input
On entry: the order $\alpha$ of the function.
Constraint: $0.0\le {\mathbf{A}}\le 100000.0$.
2:     N – INTEGERInput
On entry: the number $N$ of zeros required.
Constraint: ${\mathbf{N}}\ge 1$.
3:     MODE – INTEGERInput
On entry: specifies the form of the function whose zeros are required.
${\mathbf{MODE}}=1$
The zeros of ${J}_{\alpha }\left(x\right)$ are required.
${\mathbf{MODE}}=2$
The zeros of ${Y}_{\alpha }\left(x\right)$ are required;
${\mathbf{MODE}}=3$
The zeros of ${J}_{\alpha }^{\prime }\left(x\right)$ are required;
${\mathbf{MODE}}=4$
The zeros of ${Y}_{\alpha }^{\prime }\left(x\right)$ are required.
Constraint: $1\le {\mathbf{MODE}}\le 4$.
4:     REL – REAL (KIND=nag_wp)Input
On entry: the relative accuracy to which the zeros are required.
Suggested value: the square root of the machine precision.
Constraint: ${\mathbf{REL}}>0.0$.
5:     X(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the $N$ required zeros of the function specified by MODE.
6:     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, ${\mathbf{A}}<0.0$, or ${\mathbf{A}}>100000.0$, or ${\mathbf{N}}\le 0$, or ${\mathbf{MODE}}<1$, or ${\mathbf{MODE}}>4$, or ${\mathbf{REL}}\le 0.0$.

## 7  Accuracy

If the value of REL is set to ${10}^{-d}$, then the required zeros should have approximately $d$ correct significant digits.

None.

## 9  Example

This example determines the leading five positive zeros of the Bessel function ${J}_{0}\left(x\right)$.

### 9.1  Program Text

Program Text (s17alfe.f90)

### 9.2  Program Data

Program Data (s17alfe.d)

### 9.3  Program Results

Program Results (s17alfe.r)