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zbknu.f

      SUBROUTINE ZBKNU(ZR, ZI, FNU, KODE, N, YR, YI, NZ, TOL, ELIM,
     * ALIM)
C***BEGIN PROLOGUE  ZBKNU
C***REFER TO  ZBESI,ZBESK,ZAIRY,ZBESH
C
C     ZBKNU COMPUTES THE K BESSEL FUNCTION IN THE RIGHT HALF Z PLANE.
C
C***ROUTINES CALLED  DGAMLN,I1MACH,D1MACH,ZKSCL,ZSHCH,ZUCHK,XZABS,ZDIV,
C                    XZEXP,XZLOG,ZMLT,XZSQRT
C***END PROLOGUE  ZBKNU
C
      DOUBLE PRECISION AA, AK, ALIM, ASCLE, A1, A2, BB, BK, BRY, CAZ,
     * CBI, CBR, CC, CCHI, CCHR, CKI, CKR, COEFI, COEFR, CONEI, CONER,
     * CRSCR, CSCLR, CSHI, CSHR, CSI, CSR, CSRR, CSSR, CTWOR,
     * CZEROI, CZEROR, CZI, CZR, DNU, DNU2, DPI, ELIM, ETEST, FC, FHS,
     * FI, FK, FKS, FMUI, FMUR, FNU, FPI, FR, G1, G2, HPI, PI, PR, PTI,
     * PTR, P1I, P1R, P2I, P2M, P2R, QI, QR, RAK, RCAZ, RTHPI, RZI,
     * RZR, R1, S, SMUI, SMUR, SPI, STI, STR, S1I, S1R, S2I, S2R, TM,
     * TOL, TTH, T1, T2, YI, YR, ZI, ZR, DGAMLN, D1MACH, XZABS, ELM,
     * CELMR, ZDR, ZDI, AS, ALAS, HELIM, CYR, CYI
      INTEGER I, IFLAG, INU, K, KFLAG, KK, KMAX, KODE, KODED, N, NZ,
     * IDUM, I1MACH, J, IC, INUB, NW
      DIMENSION YR(N), YI(N), CC(8), CSSR(3), CSRR(3), BRY(3), CYR(2),
     * CYI(2)
C     COMPLEX Z,Y,A,B,RZ,SMU,FU,FMU,F,FLRZ,CZ,S1,S2,CSH,CCH
C     COMPLEX CK,P,Q,COEF,P1,P2,CBK,PT,CZERO,CONE,CTWO,ST,EZ,CS,DK
C
      DATA KMAX / 30 /
      DATA CZEROR,CZEROI,CONER,CONEI,CTWOR,R1/
     1  0.0D0 , 0.0D0 , 1.0D0 , 0.0D0 , 2.0D0 , 2.0D0 /
      DATA DPI, RTHPI, SPI ,HPI, FPI, TTH /
     1     3.14159265358979324D0,       1.25331413731550025D0,
     2     1.90985931710274403D0,       1.57079632679489662D0,
     3     1.89769999331517738D0,       6.66666666666666666D-01/
      DATA CC(1), CC(2), CC(3), CC(4), CC(5), CC(6), CC(7), CC(8)/
     1     5.77215664901532861D-01,    -4.20026350340952355D-02,
     2    -4.21977345555443367D-02,     7.21894324666309954D-03,
     3    -2.15241674114950973D-04,    -2.01348547807882387D-05,
     4     1.13302723198169588D-06,     6.11609510448141582D-09/
C
      CAZ = XZABS(ZR,ZI)
      CSCLR = 1.0D0/TOL
      CRSCR = TOL
      CSSR(1) = CSCLR
      CSSR(2) = 1.0D0
      CSSR(3) = CRSCR
      CSRR(1) = CRSCR
      CSRR(2) = 1.0D0
      CSRR(3) = CSCLR
      BRY(1) = 1.0D+3*D1MACH(1)/TOL
      BRY(2) = 1.0D0/BRY(1)
      BRY(3) = D1MACH(2)
      NZ = 0
      IFLAG = 0
      KODED = KODE
      RCAZ = 1.0D0/CAZ
      STR = ZR*RCAZ
      STI = -ZI*RCAZ
      RZR = (STR+STR)*RCAZ
      RZI = (STI+STI)*RCAZ
      INU = INT(SNGL(FNU+0.5D0))
      DNU = FNU - DBLE(FLOAT(INU))
      IF (DABS(DNU).EQ.0.5D0) GO TO 110
      DNU2 = 0.0D0
      IF (DABS(DNU).GT.TOL) DNU2 = DNU*DNU
      IF (CAZ.GT.R1) GO TO 110
C-----------------------------------------------------------------------
C     SERIES FOR CABS(Z).LE.R1
C-----------------------------------------------------------------------
      FC = 1.0D0
      CALL XZLOG(RZR, RZI, SMUR, SMUI, IDUM)
      FMUR = SMUR*DNU
      FMUI = SMUI*DNU
      CALL ZSHCH(FMUR, FMUI, CSHR, CSHI, CCHR, CCHI)
      IF (DNU.EQ.0.0D0) GO TO 10
      FC = DNU*DPI
      FC = FC/DSIN(FC)
      SMUR = CSHR/DNU
      SMUI = CSHI/DNU
   10 CONTINUE
      A2 = 1.0D0 + DNU
C-----------------------------------------------------------------------
C     GAM(1-Z)*GAM(1+Z)=PI*Z/SIN(PI*Z), T1=1/GAM(1-DNU), T2=1/GAM(1+DNU)
C-----------------------------------------------------------------------
      T2 = DEXP(-DGAMLN(A2,IDUM))
      T1 = 1.0D0/(T2*FC)
      IF (DABS(DNU).GT.0.1D0) GO TO 40
C-----------------------------------------------------------------------
C     SERIES FOR F0 TO RESOLVE INDETERMINACY FOR SMALL ABS(DNU)
C-----------------------------------------------------------------------
      AK = 1.0D0
      S = CC(1)
      DO 20 K=2,8
        AK = AK*DNU2
        TM = CC(K)*AK
        S = S + TM
        IF (DABS(TM).LT.TOL) GO TO 30
   20 CONTINUE
   30 G1 = -S
      GO TO 50
   40 CONTINUE
      G1 = (T1-T2)/(DNU+DNU)
   50 CONTINUE
      G2 = (T1+T2)*0.5D0
      FR = FC*(CCHR*G1+SMUR*G2)
      FI = FC*(CCHI*G1+SMUI*G2)
      CALL XZEXP(FMUR, FMUI, STR, STI)
      PR = 0.5D0*STR/T2
      PI = 0.5D0*STI/T2
      CALL ZDIV(0.5D0, 0.0D0, STR, STI, PTR, PTI)
      QR = PTR/T1
      QI = PTI/T1
      S1R = FR
      S1I = FI
      S2R = PR
      S2I = PI
      AK = 1.0D0
      A1 = 1.0D0
      CKR = CONER
      CKI = CONEI
      BK = 1.0D0 - DNU2
      IF (INU.GT.0 .OR. N.GT.1) GO TO 80
C-----------------------------------------------------------------------
C     GENERATE K(FNU,Z), 0.0D0 .LE. FNU .LT. 0.5D0 AND N=1
C-----------------------------------------------------------------------
      IF (CAZ.LT.TOL) GO TO 70
      CALL ZMLT(ZR, ZI, ZR, ZI, CZR, CZI)
      CZR = 0.25D0*CZR
      CZI = 0.25D0*CZI
      T1 = 0.25D0*CAZ*CAZ
   60 CONTINUE
      FR = (FR*AK+PR+QR)/BK
      FI = (FI*AK+PI+QI)/BK
      STR = 1.0D0/(AK-DNU)
      PR = PR*STR
      PI = PI*STR
      STR = 1.0D0/(AK+DNU)
      QR = QR*STR
      QI = QI*STR
      STR = CKR*CZR - CKI*CZI
      RAK = 1.0D0/AK
      CKI = (CKR*CZI+CKI*CZR)*RAK
      CKR = STR*RAK
      S1R = CKR*FR - CKI*FI + S1R
      S1I = CKR*FI + CKI*FR + S1I
      A1 = A1*T1*RAK
      BK = BK + AK + AK + 1.0D0
      AK = AK + 1.0D0
      IF (A1.GT.TOL) GO TO 60
   70 CONTINUE
      YR(1) = S1R
      YI(1) = S1I
      IF (KODED.EQ.1) RETURN
      CALL XZEXP(ZR, ZI, STR, STI)
      CALL ZMLT(S1R, S1I, STR, STI, YR(1), YI(1))
      RETURN
C-----------------------------------------------------------------------
C     GENERATE K(DNU,Z) AND K(DNU+1,Z) FOR FORWARD RECURRENCE
C-----------------------------------------------------------------------
   80 CONTINUE
      IF (CAZ.LT.TOL) GO TO 100
      CALL ZMLT(ZR, ZI, ZR, ZI, CZR, CZI)
      CZR = 0.25D0*CZR
      CZI = 0.25D0*CZI
      T1 = 0.25D0*CAZ*CAZ
   90 CONTINUE
      FR = (FR*AK+PR+QR)/BK
      FI = (FI*AK+PI+QI)/BK
      STR = 1.0D0/(AK-DNU)
      PR = PR*STR
      PI = PI*STR
      STR = 1.0D0/(AK+DNU)
      QR = QR*STR
      QI = QI*STR
      STR = CKR*CZR - CKI*CZI
      RAK = 1.0D0/AK
      CKI = (CKR*CZI+CKI*CZR)*RAK
      CKR = STR*RAK
      S1R = CKR*FR - CKI*FI + S1R
      S1I = CKR*FI + CKI*FR + S1I
      STR = PR - FR*AK
      STI = PI - FI*AK
      S2R = CKR*STR - CKI*STI + S2R
      S2I = CKR*STI + CKI*STR + S2I
      A1 = A1*T1*RAK
      BK = BK + AK + AK + 1.0D0
      AK = AK + 1.0D0
      IF (A1.GT.TOL) GO TO 90
  100 CONTINUE
      KFLAG = 2
      A1 = FNU + 1.0D0
      AK = A1*DABS(SMUR)
      IF (AK.GT.ALIM) KFLAG = 3
      STR = CSSR(KFLAG)
      P2R = S2R*STR
      P2I = S2I*STR
      CALL ZMLT(P2R, P2I, RZR, RZI, S2R, S2I)
      S1R = S1R*STR
      S1I = S1I*STR
      IF (KODED.EQ.1) GO TO 210
      CALL XZEXP(ZR, ZI, FR, FI)
      CALL ZMLT(S1R, S1I, FR, FI, S1R, S1I)
      CALL ZMLT(S2R, S2I, FR, FI, S2R, S2I)
      GO TO 210
C-----------------------------------------------------------------------
C     IFLAG=0 MEANS NO UNDERFLOW OCCURRED
C     IFLAG=1 MEANS AN UNDERFLOW OCCURRED- COMPUTATION PROCEEDS WITH
C     KODED=2 AND A TEST FOR ON SCALE VALUES IS MADE DURING FORWARD
C     RECURSION
C-----------------------------------------------------------------------
  110 CONTINUE
      CALL XZSQRT(ZR, ZI, STR, STI)
      CALL ZDIV(RTHPI, CZEROI, STR, STI, COEFR, COEFI)
      KFLAG = 2
      IF (KODED.EQ.2) GO TO 120
      IF (ZR.GT.ALIM) GO TO 290
C     BLANK LINE
      STR = DEXP(-ZR)*CSSR(KFLAG)
      STI = -STR*DSIN(ZI)
      STR = STR*DCOS(ZI)
      CALL ZMLT(COEFR, COEFI, STR, STI, COEFR, COEFI)
  120 CONTINUE
      IF (DABS(DNU).EQ.0.5D0) GO TO 300
C-----------------------------------------------------------------------
C     MILLER ALGORITHM FOR CABS(Z).GT.R1
C-----------------------------------------------------------------------
      AK = DCOS(DPI*DNU)
      AK = DABS(AK)
      IF (AK.EQ.CZEROR) GO TO 300
      FHS = DABS(0.25D0-DNU2)
      IF (FHS.EQ.CZEROR) GO TO 300
C-----------------------------------------------------------------------
C     COMPUTE R2=F(E). IF CABS(Z).GE.R2, USE FORWARD RECURRENCE TO
C     DETERMINE THE BACKWARD INDEX K. R2=F(E) IS A STRAIGHT LINE ON
C     12.LE.E.LE.60. E IS COMPUTED FROM 2**(-E)=B**(1-I1MACH(14))=
C     TOL WHERE B IS THE BASE OF THE ARITHMETIC.
C-----------------------------------------------------------------------
      T1 = DBLE(FLOAT(I1MACH(14)-1))
      T1 = T1*D1MACH(5)*3.321928094D0
      T1 = DMAX1(T1,12.0D0)
      T1 = DMIN1(T1,60.0D0)
      T2 = TTH*T1 - 6.0D0
      IF (ZR.NE.0.0D0) GO TO 130
      T1 = HPI
      GO TO 140
  130 CONTINUE
      T1 = DATAN(ZI/ZR)
      T1 = DABS(T1)
  140 CONTINUE
      IF (T2.GT.CAZ) GO TO 170
C-----------------------------------------------------------------------
C     FORWARD RECURRENCE LOOP WHEN CABS(Z).GE.R2
C-----------------------------------------------------------------------
      ETEST = AK/(DPI*CAZ*TOL)
      FK = CONER
      IF (ETEST.LT.CONER) GO TO 180
      FKS = CTWOR
      CKR = CAZ + CAZ + CTWOR
      P1R = CZEROR
      P2R = CONER
      DO 150 I=1,KMAX
        AK = FHS/FKS
        CBR = CKR/(FK+CONER)
        PTR = P2R
        P2R = CBR*P2R - P1R*AK
        P1R = PTR
        CKR = CKR + CTWOR
        FKS = FKS + FK + FK + CTWOR
        FHS = FHS + FK + FK
        FK = FK + CONER
        STR = DABS(P2R)*FK
        IF (ETEST.LT.STR) GO TO 160
  150 CONTINUE
      GO TO 310
  160 CONTINUE
      FK = FK + SPI*T1*DSQRT(T2/CAZ)
      FHS = DABS(0.25D0-DNU2)
      GO TO 180
  170 CONTINUE
C-----------------------------------------------------------------------
C     COMPUTE BACKWARD INDEX K FOR CABS(Z).LT.R2
C-----------------------------------------------------------------------
      A2 = DSQRT(CAZ)
      AK = FPI*AK/(TOL*DSQRT(A2))
      AA = 3.0D0*T1/(1.0D0+CAZ)
      BB = 14.7D0*T1/(28.0D0+CAZ)
      AK = (DLOG(AK)+CAZ*DCOS(AA)/(1.0D0+0.008D0*CAZ))/DCOS(BB)
      FK = 0.12125D0*AK*AK/CAZ + 1.5D0
  180 CONTINUE
C-----------------------------------------------------------------------
C     BACKWARD RECURRENCE LOOP FOR MILLER ALGORITHM
C-----------------------------------------------------------------------
      K = INT(SNGL(FK))
      FK = DBLE(FLOAT(K))
      FKS = FK*FK
      P1R = CZEROR
      P1I = CZEROI
      P2R = TOL
      P2I = CZEROI
      CSR = P2R
      CSI = P2I
      DO 190 I=1,K
        A1 = FKS - FK
        AK = (FKS+FK)/(A1+FHS)
        RAK = 2.0D0/(FK+CONER)
        CBR = (FK+ZR)*RAK
        CBI = ZI*RAK
        PTR = P2R
        PTI = P2I
        P2R = (PTR*CBR-PTI*CBI-P1R)*AK
        P2I = (PTI*CBR+PTR*CBI-P1I)*AK
        P1R = PTR
        P1I = PTI
        CSR = CSR + P2R
        CSI = CSI + P2I
        FKS = A1 - FK + CONER
        FK = FK - CONER
  190 CONTINUE
C-----------------------------------------------------------------------
C     COMPUTE (P2/CS)=(P2/CABS(CS))*(CONJG(CS)/CABS(CS)) FOR BETTER
C     SCALING
C-----------------------------------------------------------------------
      TM = XZABS(CSR,CSI)
      PTR = 1.0D0/TM
      S1R = P2R*PTR
      S1I = P2I*PTR
      CSR = CSR*PTR
      CSI = -CSI*PTR
      CALL ZMLT(COEFR, COEFI, S1R, S1I, STR, STI)
      CALL ZMLT(STR, STI, CSR, CSI, S1R, S1I)
      IF (INU.GT.0 .OR. N.GT.1) GO TO 200
      ZDR = ZR
      ZDI = ZI
      IF(IFLAG.EQ.1) GO TO 270
      GO TO 240
  200 CONTINUE
C-----------------------------------------------------------------------
C     COMPUTE P1/P2=(P1/CABS(P2)*CONJG(P2)/CABS(P2) FOR SCALING
C-----------------------------------------------------------------------
      TM = XZABS(P2R,P2I)
      PTR = 1.0D0/TM
      P1R = P1R*PTR
      P1I = P1I*PTR
      P2R = P2R*PTR
      P2I = -P2I*PTR
      CALL ZMLT(P1R, P1I, P2R, P2I, PTR, PTI)
      STR = DNU + 0.5D0 - PTR
      STI = -PTI
      CALL ZDIV(STR, STI, ZR, ZI, STR, STI)
      STR = STR + 1.0D0
      CALL ZMLT(STR, STI, S1R, S1I, S2R, S2I)
C-----------------------------------------------------------------------
C     FORWARD RECURSION ON THE THREE TERM RECURSION WITH RELATION WITH
C     SCALING NEAR EXPONENT EXTREMES ON KFLAG=1 OR KFLAG=3
C-----------------------------------------------------------------------
  210 CONTINUE
      STR = DNU + 1.0D0
      CKR = STR*RZR
      CKI = STR*RZI
      IF (N.EQ.1) INU = INU - 1
      IF (INU.GT.0) GO TO 220
      IF (N.GT.1) GO TO 215
      S1R = S2R
      S1I = S2I
  215 CONTINUE
      ZDR = ZR
      ZDI = ZI
      IF(IFLAG.EQ.1) GO TO 270
      GO TO 240
  220 CONTINUE
      INUB = 1
      IF(IFLAG.EQ.1) GO TO 261
  225 CONTINUE
      P1R = CSRR(KFLAG)
      ASCLE = BRY(KFLAG)
      DO 230 I=INUB,INU
        STR = S2R
        STI = S2I
        S2R = CKR*STR - CKI*STI + S1R
        S2I = CKR*STI + CKI*STR + S1I
        S1R = STR
        S1I = STI
        CKR = CKR + RZR
        CKI = CKI + RZI
        IF (KFLAG.GE.3) GO TO 230
        P2R = S2R*P1R
        P2I = S2I*P1R
        STR = DABS(P2R)
        STI = DABS(P2I)
        P2M = DMAX1(STR,STI)
        IF (P2M.LE.ASCLE) GO TO 230
        KFLAG = KFLAG + 1
        ASCLE = BRY(KFLAG)
        S1R = S1R*P1R
        S1I = S1I*P1R
        S2R = P2R
        S2I = P2I
        STR = CSSR(KFLAG)
        S1R = S1R*STR
        S1I = S1I*STR
        S2R = S2R*STR
        S2I = S2I*STR
        P1R = CSRR(KFLAG)
  230 CONTINUE
      IF (N.NE.1) GO TO 240
      S1R = S2R
      S1I = S2I
  240 CONTINUE
      STR = CSRR(KFLAG)
      YR(1) = S1R*STR
      YI(1) = S1I*STR
      IF (N.EQ.1) RETURN
      YR(2) = S2R*STR
      YI(2) = S2I*STR
      IF (N.EQ.2) RETURN
      KK = 2
  250 CONTINUE
      KK = KK + 1
      IF (KK.GT.N) RETURN
      P1R = CSRR(KFLAG)
      ASCLE = BRY(KFLAG)
      DO 260 I=KK,N
        P2R = S2R
        P2I = S2I
        S2R = CKR*P2R - CKI*P2I + S1R
        S2I = CKI*P2R + CKR*P2I + S1I
        S1R = P2R
        S1I = P2I
        CKR = CKR + RZR
        CKI = CKI + RZI
        P2R = S2R*P1R
        P2I = S2I*P1R
        YR(I) = P2R
        YI(I) = P2I
        IF (KFLAG.GE.3) GO TO 260
        STR = DABS(P2R)
        STI = DABS(P2I)
        P2M = DMAX1(STR,STI)
        IF (P2M.LE.ASCLE) GO TO 260
        KFLAG = KFLAG + 1
        ASCLE = BRY(KFLAG)
        S1R = S1R*P1R
        S1I = S1I*P1R
        S2R = P2R
        S2I = P2I
        STR = CSSR(KFLAG)
        S1R = S1R*STR
        S1I = S1I*STR
        S2R = S2R*STR
        S2I = S2I*STR
        P1R = CSRR(KFLAG)
  260 CONTINUE
      RETURN
C-----------------------------------------------------------------------
C     IFLAG=1 CASES, FORWARD RECURRENCE ON SCALED VALUES ON UNDERFLOW
C-----------------------------------------------------------------------
  261 CONTINUE
      HELIM = 0.5D0*ELIM
      ELM = DEXP(-ELIM)
      CELMR = ELM
      ASCLE = BRY(1)
      ZDR = ZR
      ZDI = ZI
      IC = -1
      J = 2
      DO 262 I=1,INU
        STR = S2R
        STI = S2I
        S2R = STR*CKR-STI*CKI+S1R
        S2I = STI*CKR+STR*CKI+S1I
        S1R = STR
        S1I = STI
        CKR = CKR+RZR
        CKI = CKI+RZI
        AS = XZABS(S2R,S2I)
        ALAS = DLOG(AS)
        P2R = -ZDR+ALAS
        IF(P2R.LT.(-ELIM)) GO TO 263
        CALL XZLOG(S2R,S2I,STR,STI,IDUM)
        P2R = -ZDR+STR
        P2I = -ZDI+STI
        P2M = DEXP(P2R)/TOL
        P1R = P2M*DCOS(P2I)
        P1I = P2M*DSIN(P2I)
        CALL ZUCHK(P1R,P1I,NW,ASCLE,TOL)
        IF(NW.NE.0) GO TO 263
        J = 3 - J
        CYR(J) = P1R
        CYI(J) = P1I
        IF(IC.EQ.(I-1)) GO TO 264
        IC = I
        GO TO 262
  263   CONTINUE
        IF(ALAS.LT.HELIM) GO TO 262
        ZDR = ZDR-ELIM
        S1R = S1R*CELMR
        S1I = S1I*CELMR
        S2R = S2R*CELMR
        S2I = S2I*CELMR
  262 CONTINUE
      IF(N.NE.1) GO TO 270
      S1R = S2R
      S1I = S2I
      GO TO 270
  264 CONTINUE
      KFLAG = 1
      INUB = I+1
      S2R = CYR(J)
      S2I = CYI(J)
      J = 3 - J
      S1R = CYR(J)
      S1I = CYI(J)
      IF(INUB.LE.INU) GO TO 225
      IF(N.NE.1) GO TO 240
      S1R = S2R
      S1I = S2I
      GO TO 240
  270 CONTINUE
      YR(1) = S1R
      YI(1) = S1I
      IF(N.EQ.1) GO TO 280
      YR(2) = S2R
      YI(2) = S2I
  280 CONTINUE
      ASCLE = BRY(1)
      CALL ZKSCL(ZDR,ZDI,FNU,N,YR,YI,NZ,RZR,RZI,ASCLE,TOL,ELIM)
      INU = N - NZ
      IF (INU.LE.0) RETURN
      KK = NZ + 1
      S1R = YR(KK)
      S1I = YI(KK)
      YR(KK) = S1R*CSRR(1)
      YI(KK) = S1I*CSRR(1)
      IF (INU.EQ.1) RETURN
      KK = NZ + 2
      S2R = YR(KK)
      S2I = YI(KK)
      YR(KK) = S2R*CSRR(1)
      YI(KK) = S2I*CSRR(1)
      IF (INU.EQ.2) RETURN
      T2 = FNU + DBLE(FLOAT(KK-1))
      CKR = T2*RZR
      CKI = T2*RZI
      KFLAG = 1
      GO TO 250
  290 CONTINUE
C-----------------------------------------------------------------------
C     SCALE BY DEXP(Z), IFLAG = 1 CASES
C-----------------------------------------------------------------------
      KODED = 2
      IFLAG = 1
      KFLAG = 2
      GO TO 120
C-----------------------------------------------------------------------
C     FNU=HALF ODD INTEGER CASE, DNU=-0.5
C-----------------------------------------------------------------------
  300 CONTINUE
      S1R = COEFR
      S1I = COEFI
      S2R = COEFR
      S2I = COEFI
      GO TO 210
C
C
  310 CONTINUE
      NZ=-2
      RETURN
      END

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