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NAG Library Function Document

nag_bessel_zeros (s17alc)

▸▿ 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

1

Purpose

nag_bessel_zeros (s17alc) 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

#include <nag.h>

#include <nags.h>

void	nag_bessel_zeros (double a, Integer n, Integer mode, double rel, double x[], NagError *fail)

3

Description

nag_bessel_zeros (s17alc) 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).

nag_bessel_zeros (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: $a$ – doubleInput

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: $rel$ – doubleInput

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: $x [n]$ – doubleOutput

On exit: the

N

required zeros of the function specified by mode.

6: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $mode = 〈value〉$ .
Constraint: $mode \leq 4$ .

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

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $a = 〈value〉$ .
Constraint: $a \leq 100000.0$ .

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

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

nag_bessel_zeros (s17alc) 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)

NAG Library Function Document

nag_bessel_zeros (s17alc)

▸▿ 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

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