# SLATEC Routines --- ZBESJ ---

```*DECK ZBESJ
SUBROUTINE ZBESJ (ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE  ZBESJ
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                      ***A DOUBLE PRECISION ROUTINE***
C         On KODE=1, ZBESJ 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         -piCBESJ returns
C         the scaled functions
C
C            CY(L) = exp(-abs(Y))*J(FNU+L-1,Z),  L=1,...,N and Y=Im(Z)
C
C         which remove the exponential growth in both the upper and
C         lower half planes as Z goes to infinity.  Definitions and
C         notation are found in the NBS Handbook of Mathematical
C         Functions (Ref. 1).
C
C         Input
C           ZR     - DOUBLE PRECISION real part of argument Z
C           ZI     - DOUBLE PRECISION imag part of argument Z
C           FNU    - DOUBLE PRECISION initial order, FNU>=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           CYR    - DOUBLE PRECISION real part of result vector
C           CYI    - DOUBLE PRECISION imag part of result vector
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=MAX(D1MACH(4),1.0D-18) is double
C         precision unit roundoff limited to 18 digits precision.  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  D1MACH, I1MACH, ZABS, ZBINU
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  ZBESJ
```