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 Arguments

1: $a$ – Real (Kind=nag_wp) Input

On entry: the order

α

of the function.

Constraint:

0.0 \leq a \leq 100000.0

2: $n$ – Integer Input

On entry: the number

N

of zeros required.

Constraint:

n \geq 1

3: $mode$ – Integer Input

On entry: 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

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:

rel > 0.0

5: $x (n)$ – Real (Kind=nag_wp) array Output

On exit: the

N

required zeros of the function specified by mode.

6: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 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 $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $a = 〈value〉$ .
Constraint: $a \leq 100000.0$ .

On entry, $a = 〈value〉$ .
Constraint: $a \geq 0.0$ .

On entry, $mode = 〈value〉$ .
Constraint: $mode \leq 4$ .

On entry, $mode = 〈value〉$ .
Constraint: $mode \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $rel = 〈value〉$ .
Constraint: $rel > 0.0$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

s17alf is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

J_{0} (x)

Interfaces: FL CL AD

NAG FL Interface Introduction

S (Specfun) Chapter Contents

S (Specfun) Chapter Introduction

s17al: FL CL AD

NAG FL Interfaces17alf (bessel_​zeros)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
s17alf (bessel_zeros)