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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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

Not applicable.

9 Further Comments

None.

10 Example

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

J_{0} (x)

NAG Library Function Documentnag_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

NAG Library Function Document

nag_bessel_zeros (s17alc)