C343: Bessel Functions J and I with Positive Argument and Non-Integer Order

Author(s): K.S. Kölbig	Library: MATHLIB
Submitter:	Submitted: 24.01.1986
Language: Fortran	Revised: 15.03.1993

Subroutine subprograms BSJA, BSIA, DBSJA, DBSIA and QBSJA, QBSIA calculate the sequences of Bessel functions

for real argument x>0, , and .

The quadruple-precision versions QBSJA and QBSIA are available only on computers which support a REAL*16 Fortran data type.

Structure:

SUBROUTINE subprograms
User Entry Names: BSJA, BSIA, DBSJA, DBSIA, QBSJA, QBSIA
Files Referenced: Unit 6
External References: GAMMA, DGAMMA, QGAMMA, MTLMTR, ABEND

Usage:

Single-precision version:
CALL BSJA(X,A,NL,ND,B) or CALL BSIA(X,A,NL,ND,B)

X

(REAL) Argument x.

A

(REAL) Order a of the first Bessel function in the computed sequence.

NL

(INTEGER) Specifies the order of the last Bessel function in the computed sequence. It is permissible for NL to be negative.

ND

(INTEGER) Requested number of correct significant decimal digits.

B

(REAL) One-dimensional array with dimension (0:d) where .
On exit, , , contains , , or , the plus sign of the subscript being taken if , the minus sign if .

Double-precision version:
CALL DBSJA(X,A,NL,ND,B) or CALL DBSIA(X,A,NL,ND,B)
where X, A and B are of type DOUBLE PRECISION.
Quadruple-precision version:
CALL QBSJA(X,A,NL,ND,B) or CALL QBSIA(X,A,NL,ND,B)
where X, A and B are of type REAL*16.

Method:

For , the method of computation is a variant of Miller's backwards recurrence algorithm (see Ref. 1). The requested accuracy is obtained, when possible, by a judicious choice of the initial value of the recursion index, together with at least one repetition of the recursion with this index increased by 5. For , only the first two functions in the sequence are computed in this way. The remaining functions are computed by the standard Bessel function recurrence relation.

Restrictions:

, , .

Accuracy:

If X is close to a zero of one of the functions , the accuracy of that particular function will be less than ND significant digits. There may also be a loss of accuracy in any of the computed functions if A is close to 0 or 1, and in other special situations.

Error handling:

Error C343.1: .
Error C343.2: or .
Error C343.3: .
Error C343.4: Difficulties of convergence. Try smaller .
In all cases, a message is written on Unit 6, unless subroutine MTLSET (N002) has been called. If Error 1 to 3 occurs, the initial contents of array B is left unchanged. If the requested accuracy cannot be obtained after the initial recursion index has been increased fifty times (Error 4), the contents of array B is undefined.

The subprogram is based on Algol procedures described in Ref. 1.

References:

1. W. Gautschi, Algorithm 236, Bessel functions of the first kind, Collected Algorithms from CACM (1972)