zbesh.f

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      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,XZABS,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, xzabs, 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 = xzabs(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