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: $a$ – double 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: $rel$ – double 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: $x [n]$ – double Output

On exit: the

N

required zeros of the function specified by mode.

6: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

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)

s17al: FL CL CPP AD PY MB

NAG CL Interfaces17alc (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 CL Interface
s17alc (bessel_zeros)