*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