SLATEC Routines --- CBIRY ---

```*DECK CBIRY
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
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