NAG CL Interface
s17alc (bessel_​zeros)

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1 Purpose

s17alc determines the leading n zeros of one of the Bessel functions Jα(x), Yα(x), Jα(x) or Yα(x) for real x and non-negative α.

2 Specification

#include <nag.h>
void  s17alc (double a, Integer n, Integer mode, double rel, double x[], NagError *fail)
The function may be called by the names: s17alc, nag_specfun_bessel_zeros or nag_bessel_zeros.

3 Description

s17alc attempts to find the leading N zeros of one of the Bessel functions Jα(x), Yα(x), Jα(x) or Yα(x), where x is real. When α 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 α0, the nth positive zero is denoted by jα,n,jα,n,yα,n and yα,n, respectively, for n=1,2,,N, except that x=0 is counted as the first zero of Jα(x) when α=0. Since J0(x)=-J1(x), it, therefore, follows that j0,1=0 and j0,n=-j1,n-1 for n=2,3,,N-1. Further details can be found in Section 9.5 of Abramowitz and Stegun (1972).
s17alc 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 Arguments

1: a double Input
On entry: the order α of the function.
Constraint: 0.0a100000.0.
2: n Integer Input
On entry: the number N of zeros required.
Constraint: n1.
3: mode Integer Input
On entry: specifies the form of the function whose zeros are required.
The zeros of Jα(x) are required.
The zeros of Yα(x) are required;
The zeros of Jα(x) are required;
The zeros of Yα(x) are required.
Constraint: 1mode4.
4: rel double Input
On entry: the relative accuracy to which the zeros are required.
Suggested value: the square root of the machine precision.
Constraint: rel>0.0.
5: x[n] double Output
On exit: the N required zeros of the function specified by mode.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, mode=value.
Constraint: mode4.
On entry, mode=value.
Constraint: mode1.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, a=value.
Constraint: a100000.0.
On entry, a=value.
Constraint: a0.0.
On entry, rel=value.
Constraint: rel>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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17alc is not threaded in any implementation.

9 Further Comments


10 Example

This example determines the leading five positive zeros of the Bessel function J0(x).

10.1 Program Text

Program Text (s17alce.c)

10.2 Program Data

Program Data (s17alce.d)

10.3 Program Results

Program Results (s17alce.r)