# cbiry.f

          SUBROUTINE CBIRY (Z, ID, KODE, BI, IERR)
    C***BEGIN PROLOGUE  CBIRY
    C***PURPOSE  Compute the Airy function Bi(z) or its derivative dBi/dz
    C            for complex argument z.  A scaling option is available
    C            to help avoid overflow.
    C***LIBRARY   SLATEC
    C***CATEGORY  C10D
    C***TYPE      COMPLEX (CBIRY-C, ZBIRY-C)
    C***KEYWORDS  AIRY FUNCTION, BESSEL FUNCTION OF ORDER ONE THIRD,
    C             BESSEL FUNCTION OF ORDER TWO THIRDS
    C***AUTHOR  Amos, D. E., (SNL)
    C***DESCRIPTION
    C
    C         On KODE=1, CBIRY computes the complex Airy function Bi(z)
    C         or its derivative dBi/dz on ID=0 or ID=1 respectively.
    C         On KODE=2, a scaling option exp(abs(Re(zeta)))*Bi(z) or
    C         exp(abs(Re(zeta)))*dBi/dz is provided to remove the
    C         exponential behavior in both the left and right half planes
    C         where zeta=(2/3)*z**(3/2).
    C
    C         The Airy functions Bi(z) and dBi/dz are analytic in the
    C         whole z-plane, and the scaling option does not destroy this
    C         property.
    C
    C         Input
    C           Z      - Argument of type COMPLEX
    C           ID     - Order of derivative, ID=0 or ID=1
    C           KODE   - A parameter to indicate the scaling option
    C                    KODE=1  returns
    C                            BI=Bi(z)  on ID=0
    C                            BI=dBi/dz on ID=1
    C                            at z=Z
    C                        =2  returns
    C                            BI=exp(abs(Re(zeta)))*Bi(z)  on ID=0
    C                            BI=exp(abs(Re(zeta)))*dBi/dz on ID=1
    C                            at z=Z where zeta=(2/3)*z**(3/2)
    C
    C         Output
    C           BI     - Result of type COMPLEX
    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                            (Re(Z) too large with KODE=1)
    C                    IERR=3  Precision warning - COMPUTATION COMPLETED
    C                            (Result has less than half precision)
    C                    IERR=4  Precision error   - NO COMPUTATION
    C                            (Result has no precision)
    C                    IERR=5  Algorithmic error - NO COMPUTATION
    C                            (Termination condition not met)
    C
    C *Long Description:
    C
    C         Bi(z) and dBi/dz are computed from I Bessel functions by
    C
    C                Bi(z) =  c*sqrt(z)*( I(-1/3,zeta) + I(1/3,zeta) )
    C               dBi/dz =  c*   z   *( I(-2/3,zeta) + I(2/3,zeta) )
    C                    c =  1/sqrt(3)
    C                 zeta =  (2/3)*z**(3/2)
    C
    C         when abs(z)>1 and from power series when abs(z)<=1.
    C
    C         In most complex variable computation, one must evaluate ele-
    C         mentary functions.  When the magnitude of Z is large, losses
    C         of significance by argument reduction occur.  Consequently, if
    C         the magnitude of ZETA=(2/3)*Z**(3/2) exceeds U1=SQRT(0.5/UR),
    C         then losses exceeding half precision are likely and an error
    C         flag IERR=3 is triggered where UR=R1MACH(4)=UNIT ROUNDOFF.
    C         Also, if the magnitude of ZETA is larger than U2=0.5/UR, then
    C         all significance is lost and IERR=4.  In order to use the INT
    C         function, ZETA must be further restricted not to exceed
    C         U3=I1MACH(9)=LARGEST INTEGER.  Thus, the magnitude of ZETA
    C         must be restricted by MIN(U2,U3).  In IEEE arithmetic, U1,U2,
    C         and U3 are approximately 2.0E+3, 4.2E+6, 2.1E+9 in single
    C         precision and 4.7E+7, 2.3E+15, 2.1E+9 in double precision.
    C         This makes U2 limiting is single precision and U3 limiting
    C         in double precision.  This means that the magnitude of Z
    C         cannot exceed approximately 3.4E+4 in single precision and
    C         2.1E+6 in double precision.  This also means that one can
    C         expect to retain, in the worst cases on 32-bit machines,
    C         no digits in single precision and only 6 digits in double
    C         precision.
    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 and Large Order, Report SAND83-0643,
    C                 Sandia National Laboratories, Albuquerque, NM, May
    C                 1983.
    C               3. 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               4. 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  CBIRY