s17al 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

Syntax

C#
public static void s17al( double a, int n, int mode, double rel, double[] x, out int ifail )

Visual Basic
Public Shared Sub s17al ( _ a As Double, _ n As Integer, _ mode As Integer, _ rel As Double, _ x As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void s17al( double a, int n, int mode, double rel, array<double>^ x, [OutAttribute] int% ifail )

F#
static member s17al : a : float * n : int * mode : int * rel : float * x : float[] * ifail : int byref -> unit

Parameters

a: Type: System..::..Double
On entry: the order $α$ of the function.

Constraint: $0.0 \leq a \leq 100000.0$ .

n: Type: System..::..Int32
On entry: the number $N$ of zeros required.

Constraint: $n \geq 1$ .

mode

Type: System..::..Int32

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

rel: Type: System..::..Double
On entry: the relative accuracy to which the zeros are required.
Suggested value: the square root of the machine precision.

Constraint: $rel > 0.0$ .

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the $N$ required zeros of the function specified by mode.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

s17al 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).

s17al 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.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$a < 0.0$ ,
or	$a > 100000.0$ ,
or	$n \leq 0$ ,
or	$mode < 1$ ,
or	$mode > 4$ ,
or	$rel \leq 0.0$ .

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

If the value of rel is set to

10^{- d}

, then the required zeros should have approximately

d

correct significant digits.

Parallelism and Performance

None.

Further Comments

None.

Example

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

J_{0} (x)

Example program (C#): s17ale.cs

Example program data: s17ale.d

Example program results: s17ale.r