SUBROUTINE ZBESH(ZR, ZI, FNU, KODE, M, N, CYR, CYI, NZ, IERR) C***BEGIN PROLOGUE ZBESH C***DATE WRITTEN 830501 (YYMMDD) C***REVISION DATE 890801 (YYMMDD) C***CATEGORY NO. B5K C***KEYWORDS H-BESSEL FUNCTIONS,BESSEL FUNCTIONS OF COMPLEX ARGUMENT, C BESSEL FUNCTIONS OF THIRD KIND,HANKEL FUNCTIONS C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES C***PURPOSE TO COMPUTE THE H-BESSEL FUNCTIONS OF A COMPLEX ARGUMENT 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(J)=H(M,FNU+J-1,Z) FOR KINDS M=1 C OR 2, REAL, NONNEGATIVE ORDERS FNU+J-1, J=1,...,N, AND COMPLEX C Z.NE.CMPLX(0.0,0.0) IN THE CUT PLANE -PI.LT.ARG(Z).LE.PI. C ON KODE=2, ZBESH RETURNS THE SCALED HANKEL FUNCTIONS C C CY(I)=EXP(-MM*Z*I)*H(M,FNU+J-1,Z) MM=3-2*M, I**2=-1. C C WHICH REMOVES THE EXPONENTIAL BEHAVIOR IN BOTH THE UPPER AND C LOWER HALF PLANES. DEFINITIONS AND NOTATION ARE FOUND IN THE C NBS HANDBOOK OF MATHEMATICAL FUNCTIONS (REF. 1). C C INPUT ZR,ZI,FNU ARE DOUBLE PRECISION C ZR,ZI - Z=CMPLX(ZR,ZI), Z.NE.CMPLX(0.0D0,0.0D0), C -PT.LT.ARG(Z).LE.PI C FNU - ORDER OF INITIAL H FUNCTION, FNU.GE.0.0D0 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION C KODE= 1 RETURNS C CY(J)=H(M,FNU+J-1,Z), J=1,...,N C = 2 RETURNS C CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M)) C J=1,...,N , I**2=-1 C M - KIND OF HANKEL FUNCTION, M=1 OR 2 C N - NUMBER OF MEMBERS IN THE SEQUENCE, N.GE.1 C C OUTPUT CYR,CYI ARE DOUBLE PRECISION C CYR,CYI- DOUBLE PRECISION VECTORS WHOSE FIRST N COMPONENTS C CONTAIN REAL AND IMAGINARY PARTS FOR THE SEQUENCE C CY(J)=H(M,FNU+J-1,Z) OR C CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M)) J=1,...,N C DEPENDING ON KODE, I**2=-1. C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW, C NZ= 0 , NORMAL RETURN C NZ.GT.0 , FIRST NZ COMPONENTS OF CY SET TO ZERO DUE C TO UNDERFLOW, CY(J)=CMPLX(0.0D0,0.0D0) C J=1,...,NZ WHEN Y.GT.0.0 AND M=1 OR C Y.LT.0.0 AND M=2. FOR THE COMPLMENTARY C HALF PLANES, NZ STATES ONLY THE NUMBER C OF UNDERFLOWS. C IERR - ERROR FLAG C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED C IERR=1, INPUT ERROR - NO COMPUTATION C IERR=2, OVERFLOW - NO COMPUTATION, FNU TOO C LARGE OR CABS(Z) TOO SMALL OR BOTH C IERR=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT C REDUCTION PRODUCE LESS THAN HALF OF MACHINE C ACCURACY C IERR=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA- C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI- C CANCE BY ARGUMENT REDUCTION C IERR=5, ERROR - NO COMPUTATION, C ALGORITHM TERMINATION CONDITION NOT MET C C***LONG DESCRIPTION C C THE COMPUTATION IS CARRIED OUT BY THE RELATION C C H(M,FNU,Z)=(1/MP)*EXP(-MP*FNU)*K(FNU,Z*EXP(-MP)) C MP=MM*HPI*I, MM=3-2*M, HPI=PI/2, I**2=-1 C C FOR M=1 OR 2 WHERE THE K BESSEL FUNCTION IS COMPUTED FOR THE C RIGHT HALF PLANE RE(Z).GE.0.0. THE K FUNCTION IS CONTINUED C TO THE LEFT HALF PLANE BY THE RELATION C C K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z) C MP=MR*PI*I, MR=+1 OR -1, RE(Z).GT.0, I**2=-1 C C WHERE I(FNU,Z) IS THE I BESSEL FUNCTION. C C EXPONENTIAL DECAY OF H(M,FNU,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(-MM*Z*I) REMOVES THE EXPONENTIAL BEHAVIOR IN THE C WHOLE Z PLANE FOR Z TO INFINITY. C C FOR NEGATIVE ORDERS,THE FORMULAE C C H(1,-FNU,Z) = H(1,FNU,Z)*CEXP( PI*FNU*I) C H(2,-FNU,Z) = H(2,FNU,Z)*CEXP(-PI*FNU*I) C I**2=-1 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=DMAX1(D1MACH(4),1.0D-18) IS C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. 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 IS C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3 C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC. C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES. C C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT C ROUNDOFF,1.0D-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(CABS(Z))), C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN C ONE COMPONENT (IN ABSOLUTE VALUE) 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 HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF C COMMERCE, 1955. C C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT C BY D. E. AMOS, SAND83-0083, MAY, 1983. C C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 C C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- C 1018, MAY, 1985 C C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. C MATH. SOFTWARE, 1986 C C***ROUTINES CALLED ZACON,ZBKNU,ZBUNK,ZUOIK,AZABS,I1MACH,D1MACH C***END PROLOGUE ZBESH C C COMPLEX CY,Z,ZN,ZT,CSGN DOUBLE PRECISION AA, ALIM, ALN, ARG, AZ, CYI, CYR, DIG, ELIM, * FMM, FN, FNU, FNUL, HPI, RHPI, RL, R1M5, SGN, STR, TOL, UFL, ZI, * ZNI, ZNR, ZR, ZTI, D1MACH, AZABS, BB, ASCLE, RTOL, ATOL, STI, * CSGNR, CSGNI INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, M, * MM, MR, N, NN, NUF, NW, NZ, I1MACH DIMENSION CYR(N), CYI(N) C DATA HPI /1.57079632679489662D0/ C C***FIRST EXECUTABLE STATEMENT ZBESH IERR = 0 NZ=0 IF (ZR.EQ.0.0D0 .AND. ZI.EQ.0.0D0) IERR=1 IF (FNU.LT.0.0D0) 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 = DMAX1(D1MACH(4),1.0D-18) K1 = I1MACH(15) K2 = I1MACH(16) R1M5 = D1MACH(5) K = MIN0(IABS(K1),IABS(K2)) ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0) K1 = I1MACH(14) - 1 AA = R1M5*DBLE(FLOAT(K1)) DIG = DMIN1(AA,18.0D0) AA = AA*2.303D0 ALIM = ELIM + DMAX1(-AA,-41.45D0) FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0) RL = 1.2D0*DIG + 3.0D0 FN = FNU + DBLE(FLOAT(NN-1)) MM = 3 - M - M FMM = DBLE(FLOAT(MM)) ZNR = FMM*ZI ZNI = -FMM*ZR C----------------------------------------------------------------------- C TEST FOR PROPER RANGE C----------------------------------------------------------------------- AZ = AZABS(ZR,ZI) AA = 0.5D0/TOL BB=DBLE(FLOAT(I1MACH(9)))*0.5D0 AA = DMIN1(AA,BB) IF (AZ.GT.AA) GO TO 260 IF (FN.GT.AA) GO TO 260 AA = DSQRT(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 = D1MACH(1)*1.0D+3 IF (AZ.LT.UFL) GO TO 230 IF (FNU.GT.FNUL) GO TO 90 IF (FN.LE.1.0D0) GO TO 70 IF (FN.GT.2.0D0) GO TO 60 IF (AZ.GT.TOL) GO TO 70 ARG = 0.5D0*AZ ALN = -FN*DLOG(ARG) IF (ALN.GT.ELIM) GO TO 230 GO TO 70 60 CONTINUE CALL ZUOIK(ZNR, ZNI, FNU, KODE, 2, NN, CYR, CYI, NUF, TOL, ELIM, * ALIM) IF (NUF.LT.0) GO TO 230 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 140 70 CONTINUE IF ((ZNR.LT.0.0D0) .OR. (ZNR.EQ.0.0D0 .AND. ZNI.LT.0.0D0 .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 ZBKNU(ZNR, ZNI, FNU, KODE, NN, CYR, CYI, NZ, TOL, ELIM, ALIM) GO TO 110 C----------------------------------------------------------------------- C LEFT HALF PLANE COMPUTATION C----------------------------------------------------------------------- 80 CONTINUE MR = -MM CALL ZACON(ZNR, ZNI, FNU, KODE, MR, NN, CYR, CYI, NW, RL, FNUL, * TOL, ELIM, ALIM) IF (NW.LT.0) GO TO 240 NZ=NW GO TO 110 90 CONTINUE C----------------------------------------------------------------------- C UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL C----------------------------------------------------------------------- MR = 0 IF ((ZNR.GE.0.0D0) .AND. (ZNR.NE.0.0D0 .OR. ZNI.GE.0.0D0 .OR. * M.NE.2)) GO TO 100 MR = -MM IF (ZNR.NE.0.0D0 .OR. ZNI.GE.0.0D0) GO TO 100 ZNR = -ZNR ZNI = -ZNI 100 CONTINUE CALL ZBUNK(ZNR, ZNI, FNU, KODE, MR, NN, CYR, CYI, NW, TOL, ELIM, * ALIM) IF (NW.LT.0) GO TO 240 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 = DSIGN(HPI,-FMM) C----------------------------------------------------------------------- C CALCULATE EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE C WHEN FNU IS LARGE C----------------------------------------------------------------------- INU = INT(SNGL(FNU)) INUH = INU/2 IR = INU - 2*INUH ARG = (FNU-DBLE(FLOAT(INU-IR)))*SGN RHPI = 1.0D0/SGN C ZNI = RHPI*DCOS(ARG) C ZNR = -RHPI*DSIN(ARG) CSGNI = RHPI*DCOS(ARG) CSGNR = -RHPI*DSIN(ARG) IF (MOD(INUH,2).EQ.0) GO TO 120 C ZNR = -ZNR C ZNI = -ZNI CSGNR = -CSGNR CSGNI = -CSGNI 120 CONTINUE ZTI = -FMM RTOL = 1.0D0/TOL ASCLE = UFL*RTOL DO 130 I=1,NN C STR = CYR(I)*ZNR - CYI(I)*ZNI C CYI(I) = CYR(I)*ZNI + CYI(I)*ZNR C CYR(I) = STR C STR = -ZNI*ZTI C ZNI = ZNR*ZTI C ZNR = STR AA = CYR(I) BB = CYI(I) ATOL = 1.0D0 IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 135 AA = AA*RTOL BB = BB*RTOL ATOL = TOL 135 CONTINUE STR = AA*CSGNR - BB*CSGNI STI = AA*CSGNI + BB*CSGNR CYR(I) = STR*ATOL CYI(I) = STI*ATOL STR = -CSGNI*ZTI CSGNI = CSGNR*ZTI CSGNR = STR 130 CONTINUE RETURN 140 CONTINUE IF (ZNR.LT.0.0D0) GO TO 230 RETURN 230 CONTINUE NZ=0 IERR=2 RETURN 240 CONTINUE IF(NW.EQ.(-1)) GO TO 230 NZ=0 IERR=5 RETURN 260 CONTINUE NZ=0 IERR=4 RETURN END