SLATEC Routines --- ZBESH ---
*DECK ZBESH
SUBROUTINE ZBESH (ZR, ZI, FNU, KODE, M, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE ZBESH
C***PURPOSE Compute a sequence of the Hankel functions H(m,a,z)
C for superscript m=1 or 2, real nonnegative orders a=b,
C b+1,... where b>0, and nonzero complex argument z. A
C scaling option is available to help avoid overflow.
C***LIBRARY SLATEC
C***CATEGORY C10A4
C***TYPE COMPLEX (CBESH-C, ZBESH-C)
C***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
C BESSEL FUNCTIONS OF THE THIRD KIND, H BESSEL FUNCTIONS,
C HANKEL FUNCTIONS
C***AUTHOR Amos, D. E., (SNL)
C***DESCRIPTION
C
C ***A DOUBLE PRECISION ROUTINE***
C On KODE=1, ZBESH computes an N member sequence of complex
C Hankel (Bessel) functions CY(L)=H(M,FNU+L-1,Z) for super-
C script M=1 or 2, real nonnegative orders FNU+L-1, L=1,...,
C N, and complex nonzero Z in the cut plane -piCBESH returns the scaled
C functions
C
C CY(L) = H(M,FNU+L-1,Z)*exp(-(3-2*M)*Z*i), i**2=-1
C
C which removes the exponential behavior in both the upper
C and lower half planes. Definitions and notation are found
C in the NBS Handbook of Mathematical Functions (Ref. 1).
C
C Input
C ZR - DOUBLE PRECISION real part of nonzero argument Z
C ZI - DOUBLE PRECISION imag part of nonzero 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)=H(M,FNU+L-1,Z), L=1,...,N
C =2 returns
C CY(L)=H(M,FNU+L-1,Z)*exp(-(3-2M)*Z*i),
C L=1,...,N
C M - Superscript of Hankel function, M=1 or 2
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 for NZ values of L (if M=1 and
C Im(Z)>0 or if M=2 and Im(Z)<0, then
C CY(L)=0 for L=1,...,NZ; in the com-
C plementary half planes, the underflows
C may not be in an uninterrupted sequence)
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 (abs(Z) too small and/or FNU+N-1
C too large)
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 formula
C
C H(m,a,z) = (1/t)*exp(-a*t)*K(a,z*exp(-t))
C t = (3-2*m)*i*pi/2
C
C where the K Bessel function is computed as described in the
C prologue to CBESK.
C
C Exponential decay of H(m,a,z) occurs in the upper half z
C plane for m=1 and the lower half z plane for m=2. Exponential
C growth occurs in the complementary half planes. Scaling
C by exp(-(3-2*m)*z*i) removes the exponential behavior in the
C whole z plane as z goes to infinity.
C
C For negative orders, the formula
C
C H(m,-a,z) = H(m,a,z)*exp((3-2*m)*a*pi*i)
C
C can be used.
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, ZACON, ZBKNU, ZBUNK, ZUOIK
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 ZBESH