# cbesj.f

          SUBROUTINE CBESJ (Z, FNU, KODE, N, CY, NZ, IERR)
    C***BEGIN PROLOGUE  CBESJ
    C***PURPOSE  Compute a sequence of the Bessel functions J(a,z) for
    C            complex argument z and real nonnegative orders a=b,b+1,
    C            b+2,... where b>0.  A scaling option is available to
    C            help avoid overflow.
    C***LIBRARY   SLATEC
    C***CATEGORY  C10A4
    C***TYPE      COMPLEX (CBESJ-C, ZBESJ-C)
    C***KEYWORDS  BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
    C             BESSEL FUNCTIONS OF THE FIRST KIND, J BESSEL FUNCTIONS
    C***AUTHOR  Amos, D. E., (SNL)
    C***DESCRIPTION
    C
    C         On KODE=1, CBESJ computes an N member sequence of complex
    C         Bessel functions CY(L)=J(FNU+L-1,Z) for real nonnegative
    C         orders FNU+L-1, L=1,...,N and complex Z in the cut plane
    C         -pi=0
    C           KODE   - A parameter to indicate the scaling option
    C                    KODE=1  returns
    C                            CY(L)=J(FNU+L-1,Z), L=1,...,N
    C                        =2  returns
    C                            CY(L)=J(FNU+L-1,Z)*exp(-abs(Y)), L=1,...,N
    C                            where Y=Im(Z)
    C           N      - Number of terms in the sequence, N>=1
    C
    C         Output
    C           CY     - Result vector of type COMPLEX
    C           NZ     - Number of underflows set to zero
    C                    NZ=0    Normal return
    C                    NZ>0    CY(L)=0, L=N-NZ+1,...,N
    C           IERR   - Error flag
    C                    IERR=0  Normal return     - COMPUTATION COMPLETED
    C                    IERR=1  Input error       - NO COMPUTATION
    C                    IERR=2  Overflow          - NO COMPUTATION
    C                            (Im(Z) too large on KODE=1)
    C                    IERR=3  Precision warning - COMPUTATION COMPLETED
    C                            (Result has half precision or less
    C                            because abs(Z) or FNU+N-1 is large)
    C                    IERR=4  Precision error   - NO COMPUTATION
    C                            (Result has no precision because
    C                            abs(Z) or FNU+N-1 is too large)
    C                    IERR=5  Algorithmic error - NO COMPUTATION
    C                            (Termination condition not met)
    C
    C *Long Description:
    C
    C         The computation is carried out by the formulae
    C
    C            J(a,z) = exp( a*pi*i/2)*I(a,-i*z),  Im(z)>=0
    C
    C            J(a,z) = exp(-a*pi*i/2)*I(a, i*z),  Im(z)<0
    C
    C         where the I Bessel function is computed as described in the
    C         prologue to CBESI.
    C
    C         For negative orders, the formula
    C
    C            J(-a,z) = J(a,z)*cos(a*pi) - Y(a,z)*sin(a*pi)
    C
    C         can be used.  However, for large orders close to integers, the
    C         the function changes radically.  When a is a large positive
    C         integer, the magnitude of J(-a,z)=J(a,z)*cos(a*pi) is a
    C         large negative power of ten.  But when a is not an integer,
    C         Y(a,z) dominates in magnitude with a large positive power of
    C         ten and the most that the second term can be reduced is by
    C         unit roundoff from the coefficient.  Thus, wide changes can
    C         occur within unit roundoff of a large integer for a.  Here,
    C         large means a>abs(z).
    C
    C         In most complex variable computation, one must evaluate ele-
    C         mentary functions.  When the magnitude of Z or FNU+N-1 is
    C         large, losses of significance by argument reduction occur.
    C         Consequently, if either one exceeds U1=SQRT(0.5/UR), then
    C         losses exceeding half precision are likely and an error flag
    C         IERR=3 is triggered where UR=R1MACH(4)=UNIT ROUNDOFF.  Also,
    C         if either is larger than U2=0.5/UR, then all significance is
    C         lost and IERR=4.  In order to use the INT function, arguments
    C         must be further restricted not to exceed the largest machine
    C         integer, U3=I1MACH(9).  Thus, the magnitude of Z and FNU+N-1
    C         is restricted by MIN(U2,U3).  In IEEE arithmetic, U1,U2, and
    C         U3 approximate 2.0E+3, 4.2E+6, 2.1E+9 in single precision
    C         and 4.7E+7, 2.3E+15 and 2.1E+9 in double precision.  This
    C         makes U2 limiting in single precision and U3 limiting in
    C         double precision.  This means that one can expect to retain,
    C         in the worst cases on IEEE machines, no digits in single pre-
    C         cision and only 6 digits in double precision.  Similar con-
    C         siderations hold for other machines.
    C
    C         The approximate relative error in the magnitude of a complex
    C         Bessel function can be expressed as P*10**S where P=MAX(UNIT
    C         ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
    C         sents the increase in error due to argument reduction in the
    C         elementary functions.  Here, S=MAX(1,ABS(LOG10(ABS(Z))),
    C         ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
    C         ABS(Z),ABS(EXPONENT OF FNU)) ).  However, the phase angle may
    C         have only absolute accuracy.  This is most likely to occur
    C         when one component (in magnitude) is larger than the other by
    C         several orders of magnitude.  If one component is 10**K larger
    C         than the other, then one can expect only MAX(ABS(LOG10(P))-K,
    C         0) significant digits; or, stated another way, when K exceeds
    C         the exponent of P, no significant digits remain in the smaller
    C         component.  However, the phase angle retains absolute accuracy
    C         because, in complex arithmetic with precision P, the smaller
    C         component will not (as a rule) decrease below P times the
    C         magnitude of the larger component.  In these extreme cases,
    C         the principal phase angle is on the order of +P, -P, PI/2-P,
    C         or -PI/2+P.
    C
    C***REFERENCES  1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
    C                 matical Functions, National Bureau of Standards
    C                 Applied Mathematics Series 55, U. S. Department
    C                 of Commerce, Tenth Printing (1972) or later.
    C               2. D. E. Amos, Computation of Bessel Functions of
    C                 Complex Argument, Report SAND83-0086, Sandia National
    C                 Laboratories, Albuquerque, NM, May 1983.
    C               3. D. E. Amos, Computation of Bessel Functions of
    C                 Complex Argument and Large Order, Report SAND83-0643,
    C                 Sandia National Laboratories, Albuquerque, NM, May
    C                 1983.
    C               4. D. E. Amos, A Subroutine Package for Bessel Functions
    C                 of a Complex Argument and Nonnegative Order, Report
    C                 SAND85-1018, Sandia National Laboratory, Albuquerque,
    C                 NM, May 1985.
    C               5. D. E. Amos, A portable package for Bessel functions
    C                 of a complex argument and nonnegative order, ACM
    C                 Transactions on Mathematical Software, 12 (September
    C                 1986), pp. 265-273.
    C
    C***ROUTINES CALLED  CBINU, I1MACH, R1MACH
    C***REVISION HISTORY  (YYMMDD)
    C   830501  DATE WRITTEN
    C   890801  REVISION DATE from Version 3.2
    C   910415  Prologue converted to Version 4.0 format.  (BAB)
    C   920128  Category corrected.  (WRB)
    C   920811  Prologue revised.  (DWL)
    C***END PROLOGUE  CBESJ