*DECK CBESH SUBROUTINE CBESH (Z, FNU, KODE, M, N, CY, NZ, IERR) C***BEGIN PROLOGUE CBESH 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 On KODE=1, CBESH 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 -pi=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 CY - Result vector of type COMPLEX 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=R1MACH(4)=UNIT ROUNDOFF. 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 CACON, CBKNU, CBUNK, CUOIK, 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 CBESH C COMPLEX CY, Z, ZN, ZT, CSGN REAL AA, ALIM, ALN, ARG, AZ, CPN, DIG, ELIM, FMM, FN, FNU, FNUL, * HPI, RHPI, RL, R1M5, SGN, SPN, TOL, UFL, XN, XX, YN, YY, R1MACH, * BB, ASCLE, RTOL, ATOL INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, M, * MM, MR, N, NN, NUF, NW, NZ, I1MACH DIMENSION CY(N) C DATA HPI /1.57079632679489662E0/ C C***FIRST EXECUTABLE STATEMENT CBESH NZ=0 XX = REAL(Z) YY = AIMAG(Z) IERR = 0 IF (XX.EQ.0.0E0 .AND. YY.EQ.0.0E0) IERR=1 IF (FNU.LT.0.0E0) IERR=1 IF (M.LT.1 .OR. M.GT.2) IERR=1 IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1 IF (N.LT.1) IERR=1 IF (IERR.NE.0) RETURN NN = N C----------------------------------------------------------------------- C SET PARAMETERS RELATED TO MACHINE CONSTANTS. C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18. C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG). C FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU C----------------------------------------------------------------------- TOL = MAX(R1MACH(4),1.0E-18) K1 = I1MACH(12) K2 = I1MACH(13) R1M5 = R1MACH(5) K = MIN(ABS(K1),ABS(K2)) ELIM = 2.303E0*(K*R1M5-3.0E0) K1 = I1MACH(11) - 1 AA = R1M5*K1 DIG = MIN(AA,18.0E0) AA = AA*2.303E0 ALIM = ELIM + MAX(-AA,-41.45E0) FNUL = 10.0E0 + 6.0E0*(DIG-3.0E0) RL = 1.2E0*DIG + 3.0E0 FN = FNU + (NN-1) MM = 3 - M - M FMM = MM ZN = Z*CMPLX(0.0E0,-FMM) XN = REAL(ZN) YN = AIMAG(ZN) AZ = ABS(Z) C----------------------------------------------------------------------- C TEST FOR RANGE C----------------------------------------------------------------------- AA = 0.5E0/TOL BB=I1MACH(9)*0.5E0 AA=MIN(AA,BB) IF(AZ.GT.AA) GO TO 240 IF(FN.GT.AA) GO TO 240 AA=SQRT(AA) IF(AZ.GT.AA) IERR=3 IF(FN.GT.AA) IERR=3 C----------------------------------------------------------------------- C OVERFLOW TEST ON THE LAST MEMBER OF THE SEQUENCE C----------------------------------------------------------------------- UFL = R1MACH(1)*1.0E+3 IF (AZ.LT.UFL) GO TO 220 IF (FNU.GT.FNUL) GO TO 90 IF (FN.LE.1.0E0) GO TO 70 IF (FN.GT.2.0E0) GO TO 60 IF (AZ.GT.TOL) GO TO 70 ARG = 0.5E0*AZ ALN = -FN*ALOG(ARG) IF (ALN.GT.ELIM) GO TO 220 GO TO 70 60 CONTINUE CALL CUOIK(ZN, FNU, KODE, 2, NN, CY, NUF, TOL, ELIM, ALIM) IF (NUF.LT.0) GO TO 220 NZ = NZ + NUF NN = NN - NUF C----------------------------------------------------------------------- C HERE NN=N OR NN=0 SINCE NUF=0,NN, OR -1 ON RETURN FROM CUOIK C IF NUF=NN, THEN CY(I)=CZERO FOR ALL I C----------------------------------------------------------------------- IF (NN.EQ.0) GO TO 130 70 CONTINUE IF ((XN.LT.0.0E0) .OR. (XN.EQ.0.0E0 .AND. YN.LT.0.0E0 .AND. * M.EQ.2)) GO TO 80 C----------------------------------------------------------------------- C RIGHT HALF PLANE COMPUTATION, XN.GE.0. .AND. (XN.NE.0. .OR. C YN.GE.0. .OR. M=1) C----------------------------------------------------------------------- CALL CBKNU(ZN, FNU, KODE, NN, CY, NZ, TOL, ELIM, ALIM) GO TO 110 C----------------------------------------------------------------------- C LEFT HALF PLANE COMPUTATION C----------------------------------------------------------------------- 80 CONTINUE MR = -MM CALL CACON(ZN, FNU, KODE, MR, NN, CY, NW, RL, FNUL, TOL, ELIM, * ALIM) IF (NW.LT.0) GO TO 230 NZ=NW GO TO 110 90 CONTINUE C----------------------------------------------------------------------- C UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL C----------------------------------------------------------------------- MR = 0 IF ((XN.GE.0.0E0) .AND. (XN.NE.0.0E0 .OR. YN.GE.0.0E0 .OR. * M.NE.2)) GO TO 100 MR = -MM IF (XN.EQ.0.0E0 .AND. YN.LT.0.0E0) ZN = -ZN 100 CONTINUE CALL CBUNK(ZN, FNU, KODE, MR, NN, CY, NW, TOL, ELIM, ALIM) IF (NW.LT.0) GO TO 230 NZ = NZ + NW 110 CONTINUE C----------------------------------------------------------------------- C H(M,FNU,Z) = -FMM*(I/HPI)*(ZT**FNU)*K(FNU,-Z*ZT) C C ZT=EXP(-FMM*HPI*I) = CMPLX(0.0,-FMM), FMM=3-2*M, M=1,2 C----------------------------------------------------------------------- SGN = SIGN(HPI,-FMM) C----------------------------------------------------------------------- C CALCULATE EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE C WHEN FNU IS LARGE C----------------------------------------------------------------------- INU = FNU INUH = INU/2 IR = INU - 2*INUH ARG = (FNU-(INU-IR))*SGN RHPI = 1.0E0/SGN CPN = RHPI*COS(ARG) SPN = RHPI*SIN(ARG) C ZN = CMPLX(-SPN,CPN) CSGN = CMPLX(-SPN,CPN) C IF (MOD(INUH,2).EQ.1) ZN = -ZN IF (MOD(INUH,2).EQ.1) CSGN = -CSGN ZT = CMPLX(0.0E0,-FMM) RTOL = 1.0E0/TOL ASCLE = UFL*RTOL DO 120 I=1,NN C CY(I) = CY(I)*ZN C ZN = ZN*ZT ZN=CY(I) AA=REAL(ZN) BB=AIMAG(ZN) ATOL=1.0E0 IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 125 ZN = ZN*CMPLX(RTOL,0.0E0) ATOL = TOL 125 CONTINUE ZN = ZN*CSGN CY(I) = ZN*CMPLX(ATOL,0.0E0) CSGN = CSGN*ZT 120 CONTINUE RETURN 130 CONTINUE IF (XN.LT.0.0E0) GO TO 220 RETURN 220 CONTINUE IERR=2 NZ=0 RETURN 230 CONTINUE IF(NW.EQ.(-1)) GO TO 220 NZ=0 IERR=5 RETURN 240 CONTINUE NZ=0 IERR=4 RETURN END