*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