NAG Library Routine Document

S17ALF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S17ALF determines the leading

N

zeros of one of the Bessel functions

J_{α} (x)

Y_{α} (x)

J_{α}^{'} (x)

Y_{α}^{'} (x)

for real

x

and non-negative

α

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_{α} (x)

Y_{α} (x)

J_{α}^{'} (x)

Y_{α}^{'} (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

α \geq 0

, the

n

th positive zero is denoted by

j_{α, n}, j_{α, n}^{'}, y_{α, n}

and

y_{α, n}^{'}

, respectively, for

n = 1, 2, \dots, N

, except that

x = 0

is counted as the first zero of

J_{α}^{'} (x)

when

α = 0

. Since

J_{0}^{'} (x) = - J_{1} (x)

, it therefore follows that

j_{0, 1}^{'} = 0

and

j_{0, n}^{'} = - 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

α

of the function.

Constraint:

0.0 \leq A \leq 100000.0

2: N – INTEGERInput

On entry: the number

N

of zeros required.

Constraint:

N \geq 1

3: MODE – INTEGERInput

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) 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 ​ 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 ​ 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 $- 1 or 1$ 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,	$A < 0.0$ ,
or	$A > 100000.0$ ,
or	$N \leq 0$ ,
or	$MODE < 1$ ,
or	$MODE > 4$ ,
or	$REL \leq 0.0$ .

7 Accuracy

If the value of REL is set to

10^{- d}

, then the required zeros should have approximately

d

correct significant digits.

8 Further Comments

None.

9 Example

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

J_{0} (x)

NAG Library Routine DocumentS17ALF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

S17ALF