C345 Zeros of Bessel Functions J and Y

Subroutine subprograms RBZEJY and DBZEJY calculate, for real order $a\; \ge 0$ , the first N > 0 zeros

$j$_a,n, y_a,n, j'_a,n, y'_a,n(n=1,2,...,N)

of the Bessel functions $J$_a(x), Y_a(x), J'_a(x), Y'_a(x) , respectively. The prime denotes the derivative of the function with respect to x.

On CDC and Cray computers, the double-precision version DBZEJY is not available.

Structure:

SUBROUTINE subprograms
User Entry Names: RBZEJY, DBZEJY
Obsolete User Entry Names: BZEJY $\equiv$ RBZEJY
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    CALL tBZEJY(A,N,MODE,REL,X)

A

(type according to t) Order a.

N

( INTEGER) Number N of zeros wanted.

MODE

( INTEGER) defines the function for which the zeros are to be calculated:

1

zeros of $J$_a(x) ,

2

zeros of $Y$_a(x) ,

3

zeros of $J\text{'}$_a(x) ,

4

zeros of $Y\text{'}$_a(x) .

REL

(type according to t) The requested relative accuracy.

X

(type according to t) One-dimensional array of length N at least. On exit, X(n), ($n=1,2,...,N$ ) contains the first N positive (in the case $A=0$ and $MODE=3$ , non-negative) zeros of the function defined by MODE.

Method:

Initial approximations to the zeros are computed from asymptotic expansions. These values are improved by higher-order Newton iteration making use of the differential equation for the Bessel functions. (For details see Ref. 1).

Error handling:

Error C345.1:

A message is written on Unit 6, unless subroutine MTLSET (N002) has been called. The contents of X is left unchanged. $N\; \le 0$ acts as do nothing.

The subroutine is based on Algol procedures published in the References.

References:

1. N.M. Temme, An algorithm with Algol60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives, J. Comput. Phys. 32 (1979) 270--279.
2. N.M. Temme, On the numerical evaluation of the ordinary Bessel function of the second kind, J. Comput. Phys. 21 (1976) 343--350.

C346