C Library Functions - bes (3)

NAME

bes(3f) - [M_bessel::INTRO] calculate Bessel functions J(X), Y(X), I(X), K(X) for doubleprecision arguments and integer orders

Synopsis
Description
Options
Authors
Example

SYNOPSIS

subroutine bes(x,no,kode,rslt1,rslt2,t1,t2,ierr)

DESCRIPTION

This routine calculates the Bessel functions J(X),Y(X), I(X), or K(X) for doubleprecision arguments and integer orders. Backwards recurrence techniques are used for the J(X) and I(X) functions except for very small arguments, where double precision series evaluation is used. Forward recurrence is used for the Y(X) and K(X) functions with double precision Chebyshev approximations used for function initialization. Accuracy is between thirteen and fourteen significant figures. For specific functions J0(), J1(), Y0(), Y1(), See BESJ0(), BESJ1(), BESY0(), and BESY1().

OPTIONS

Description of parameters

   X     = input,doubleprecision argument of the Bessel function.
           The argument may be positive,zero, or negative
           (neg. arg. for Y(X) or K(X) produces error
           message since results may be complex.)
           Restriction on range is

              FOR J(X), -1100.0 .LE. X .LE. 1100.0
              FOR Y(X), 0.0 .LE. X .LE. 1100.0
              FOR I(X), -600.0 .LE. X .LE. 600.0
              FOR K(X), 0.0 .LE. X .LE. 600.0

   NO    = Input,integer order of function desired for a
           single value to be returned, or the maximum order
           desired (+ or -) if an array of values is to be
           returned.
           Let XX = abs(X). Then bounds on orders are

           1. For 0.0 .LE. XX .LE. 0.025,
                The absolute value of maximum order, ANO,
                and argument supplied (ABS(X)) must
                satisfy the relation

                log(GAMMA(ANO))-ANO*log(XX/2.0)
                + log(XX)/2.0  .LE. 679.0

                For a given argument and an order greater
                than that allowed by the above relation

                JN(X) = 0.0, N.NE.0, AND =1.0 FOR N=0
                YN(X) = -INF
                IN(X) = 0.0, N.NE.0, AND =1.0 FOR N=0
                KN(X) = INF
           2. FOR 0.025 .LT. XX .LE. 0.20,
                ABS(NO) .LE. INT(140.0*XX + 83.0)
           3. FOR 0.20 .LT. XX .LE. 1.0,
               ABS(NO) .LE. INT(42.0*XX + 102.0)
           4. FOR 1.0 .LT. XX .LE. 20.0,
                ABS(NO) .LE. INT(0.02*XX**3 - 0.86*XX**2 +
                                 17.15*XX + 124.0)
           5. FOR 20.0 .LT. XX .LE. 100.0,
                ABS(NO) .LE. INT(2.75*XX + 228.0)
           6. FOR 100.0 .LT. XX .LE. 400.0,
                ABS(NO) .LE. INT(1.67*XX + 336.0)
           7. FOR 400.0 .LT. XX .LE. 1100.0,
                ABS(NO) .LE. INT(1.33*XX + 470.0)

   KODE  = Input,integer indicator for the particular
           function to be computed.

             KODE = 10 -- FUNCTION J(X) ONLY
                  = 11 --          Y(X) ONLY
                  = 12 --          J(X) AND Y(X)

                  = 20 --          I(X) ONLY
                  = 21 --          K(X) ONLY
                  = 22 --          I(X) AND K(X)

   RSLT1 = output,contains the doubleprecision function value for J(X)
           or I(X) corresponding to the order and argument
           supplied, depending on the KODE value. This
           parameter would contain the result if only one
           function value is to be returned.

   RSLT2 = output,contains the doubleprecision function value for Y(X)
           or K(X) in a manner similar to RSLT1.

   T1    = output,a work area which will contain the array of
           doubleprecision function values for J(X) or I(X) of orders
           zero through NO, depending on KODE.
           T1 must be dimensioned in the calling program and
           must contain at least M cells of storage, where

               M = MAX(ABS(NO),INT(2*ABS(X))) + 51

           In using the array, T1(1) = function of order 0,
           --- T1(NO+1) = function of order NO.

   T2    = Output,similar to T1 for the functions Y(X) or
           K(X). an exception is that if only J(X) or I(X)
           are called, then T2 needs no dimension in the
           calling program, but the parameter must still
           appear in the calling sequence. Otherwise, T2
           must be dimensioned at least M.

   IERR  = Output,error flag for the conditions

            -- Normal code
               =0, Normal - No errors
            -- Abnormal codes
               =1, Argument out of range
               =2, Order too large for argument supplied
               =3, Argument too large for I(X) and K(X)
               =4, Negative arguments for Y(X) or K(X)
               =5, Incorrect parameter KODE

     BS is documented for the original version in SC-M-69-336

AUTHORS

Based on routines originally from

Sandia Mathematical Program Library Applied Mathematics Division 2642 Sandia Laboratories P. O. Box 5800 Albuquerque, New Mexico 87115 Control Data 6600 Version 5.1, 10 December 1973

Written by Ronald D. Halbgewachs, July, 1968. Modified by RDH for increased accuracy, May 8,1973.

EXAMPLE

Nemo Release 3.1

bes (3)

February 23, 2025

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