nag_specfun_bessel_zeros (s17al) 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.

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

Parameters

Compulsory Input Parameters

1: $a$ – double scalar

The order

α

of the function.

Constraint:

0.0 \leq a \leq 100000.0

2: $n$ – int64int32nag_int scalar

The number

N

of zeros required.

Constraint:

n \geq 1

3: $mode$ – int64int32nag_int scalar

Specifies the form of the function whose zeros are required.

$mode = 1$: The zeros of $J_{α} (x)$ are required.
$mode = 2$: The zeros of $Y_{α} (x)$ are required;
$mode = 3$: The zeros of $J_{α}^{'} (x)$ are required;
$mode = 4$: The zeros of $Y_{α}^{'} (x)$ are required.

Constraint:

1 \leq mode \leq 4

Optional Input Parameters

1: $rel$ – double scalar: Suggested value: the square root of the machine precision.
Default: $\sqrt{machine precision}$
The relative accuracy to which the zeros are required.

Constraint: $rel > 0.0$ .

Output Parameters

1: $x (n)$ – double array: The $N$ required zeros of the function specified by mode.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$a < 0.0$ ,
or	$a > 100000.0$ ,
or	$n \leq 0$ ,
or	$mode < 1$ ,
or	$mode > 4$ ,
or	$rel \leq 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

If the value of rel is set to

10^{- d}

, then the required zeros should have approximately

d

correct significant digits.

Further Comments

None.

Example

This example determines the leading five positive zeros of the Bessel function

J_{0} (x)

Open in the MATLAB editor: s17al_example

function s17al_example


fprintf('s17al example results\n\n');

a = 0;
n = int64(5);
mode = int64(1);

[result, ifail] = s17al(a, n, mode);

fprintf('Leading %2d zeros of J_0(x)\n',n);
fprintf('%12.4f\n',result);

s17al example results

Leading  5 zeros of J_0(x)
      2.4048
      5.5201
      8.6537
     11.7915
     14.9309

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox