SLATEC Routines --- ZBIRY ---
*DECK ZBIRY
SUBROUTINE ZBIRY (ZR, ZI, ID, KODE, BIR, BII, IERR)
C***BEGIN PROLOGUE ZBIRY
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 ***A DOUBLE PRECISION ROUTINE***
C On KODE=1, ZBIRY 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 ZR - DOUBLE PRECISION real part of argument Z
C ZI - DOUBLE PRECISION imag part of argument Z
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 BIR - DOUBLE PRECISION real part of result
C BII - DOUBLE PRECISION imag part of result
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=MAX(D1MACH(4),1.0D-18) is
C double precision unit roundoff limited to 18 digits precision.
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 D1MACH, I1MACH, ZABS, ZBINU, ZDIV, ZSQRT
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 930122 Added ZSQRT to EXTERNAL statement. (RWC)
C***END PROLOGUE ZBIRY