relibc_openlibm/amos/zbesk.f

      SUBROUTINE ZBESK(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE  ZBESK
C***DATE WRITTEN   830501   (YYMMDD)
C***REVISION DATE  890801   (YYMMDD)
C***CATEGORY NO.  B5K
C***KEYWORDS  K-BESSEL FUNCTION,COMPLEX BESSEL FUNCTION,
C             MODIFIED BESSEL FUNCTION OF THE SECOND KIND,
C             BESSEL FUNCTION OF THE THIRD KIND
C***AUTHOR  AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C***PURPOSE  TO COMPUTE K-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C***DESCRIPTION
C
C                      ***A DOUBLE PRECISION ROUTINE***
C
C         ON KODE=1, CBESK COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C         BESSEL FUNCTIONS CY(J)=K(FNU+J-1,Z) FOR REAL, NONNEGATIVE
C         ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z.NE.CMPLX(0.0,0.0)
C         IN THE CUT PLANE -PI.LT.ARG(Z).LE.PI. ON KODE=2, CBESK
C         RETURNS THE SCALED K FUNCTIONS,
C
C         CY(J)=EXP(Z)*K(FNU+J-1,Z) , J=1,...,N,
C
C         WHICH REMOVE THE EXPONENTIAL BEHAVIOR IN BOTH THE LEFT AND
C         RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND
C         NOTATION ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL
C         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                    -PI.LT.ARG(Z).LE.PI
C           FNU    - ORDER OF INITIAL K FUNCTION, FNU.GE.0.0D0
C           N      - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C           KODE   - A PARAMETER TO INDICATE THE SCALING OPTION
C                    KODE= 1  RETURNS
C                             CY(I)=K(FNU+I-1,Z), I=1,...,N
C                        = 2  RETURNS
C                             CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
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(I)=K(FNU+I-1,Z), I=1,...,N OR
C                    CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
C                    DEPENDING ON KODE
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(I)=CMPLX(0.0D0,0.0D0),
C                              I=1,...,N WHEN X.GE.0.0. WHEN X.LT.0.0
C                              NZ STATES ONLY THE NUMBER OF UNDERFLOWS
C                              IN THE SEQUENCE.
C
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 IS
C                            TOO LARGE OR CABS(Z) IS 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         EQUATIONS OF THE REFERENCE ARE IMPLEMENTED FOR SMALL ORDERS
C         DNU AND DNU+1.0 IN THE RIGHT HALF PLANE X.GE.0.0. FORWARD
C         RECURRENCE GENERATES HIGHER ORDERS. K IS CONTINUED TO THE LEFT
C         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         FOR LARGE ORDERS, FNU.GT.FNUL, THE K FUNCTION IS COMPUTED
C         BY MEANS OF ITS UNIFORM ASYMPTOTIC EXPANSIONS.
C
C         FOR NEGATIVE ORDERS, THE FORMULA
C
C                       K(-FNU,Z) = K(FNU,Z)
C
C         CAN BE USED.
C
C         CBESK ASSUMES THAT A SIGNIFICANT DIGIT SINH(X) FUNCTION IS
C         AVAILABLE.
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,1.0E-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(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,ZABS,I1MACH,D1MACH
C***END PROLOGUE  ZBESK
C
C     COMPLEX CY,Z
      DOUBLE PRECISION AA, ALIM, ALN, ARG, AZ, CYI, CYR, DIG, ELIM, FN,
     * FNU, FNUL, RL, R1M5, TOL, UFL, ZI, ZR, D1MACH, ZABS, BB
      INTEGER IERR, K, KODE, K1, K2, MR, N, NN, NUF, NW, NZ, I1MACH
      DIMENSION CYR(N), CYI(N)
C***FIRST EXECUTABLE STATEMENT  ZBESK
      IERR = 0
      NZ=0
      IF (ZI.EQ.0.0E0 .AND. ZR.EQ.0.0E0) IERR=1
      IF (FNU.LT.0.0D0) 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
C-----------------------------------------------------------------------------
C     TEST FOR PROPER RANGE
C-----------------------------------------------------------------------
      AZ = ZABS(COMPLEX(ZR,ZI))
      FN = FNU + DBLE(FLOAT(NN-1))
      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-----------------------------------------------------------------------
C     UFL = DEXP(-ELIM)
      UFL = D1MACH(1)*1.0D+3
      IF (AZ.LT.UFL) GO TO 180
      IF (FNU.GT.FNUL) GO TO 80
      IF (FN.LE.1.0D0) GO TO 60
      IF (FN.GT.2.0D0) GO TO 50
      IF (AZ.GT.TOL) GO TO 60
      ARG = 0.5D0*AZ
      ALN = -FN*DLOG(ARG)
      IF (ALN.GT.ELIM) GO TO 180
      GO TO 60
   50 CONTINUE
      CALL ZUOIK(ZR, ZI, FNU, KODE, 2, NN, CYR, CYI, NUF, TOL, ELIM,
     * ALIM)
      IF (NUF.LT.0) GO TO 180
      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 100
   60 CONTINUE
      IF (ZR.LT.0.0D0) GO TO 70
C-----------------------------------------------------------------------
C     RIGHT HALF PLANE COMPUTATION, REAL(Z).GE.0.
C-----------------------------------------------------------------------
      CALL ZBKNU(ZR, ZI, FNU, KODE, NN, CYR, CYI, NW, TOL, ELIM, ALIM)
      IF (NW.LT.0) GO TO 200
      NZ=NW
      RETURN
C-----------------------------------------------------------------------
C     LEFT HALF PLANE COMPUTATION
C     PI/2.LT.ARG(Z).LE.PI AND -PI.LT.ARG(Z).LT.-PI/2.
C-----------------------------------------------------------------------
   70 CONTINUE
      IF (NZ.NE.0) GO TO 180
      MR = 1
      IF (ZI.LT.0.0D0) MR = -1
      CALL ZACON(ZR, ZI, FNU, KODE, MR, NN, CYR, CYI, NW, RL, FNUL,
     * TOL, ELIM, ALIM)
      IF (NW.LT.0) GO TO 200
      NZ=NW
      RETURN
C-----------------------------------------------------------------------
C     UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL
C-----------------------------------------------------------------------
   80 CONTINUE
      MR = 0
      IF (ZR.GE.0.0D0) GO TO 90
      MR = 1
      IF (ZI.LT.0.0D0) MR = -1
   90 CONTINUE
      CALL ZBUNK(ZR, ZI, FNU, KODE, MR, NN, CYR, CYI, NW, TOL, ELIM,
     * ALIM)
      IF (NW.LT.0) GO TO 200
      NZ = NZ + NW
      RETURN
  100 CONTINUE
      IF (ZR.LT.0.0D0) GO TO 180
      RETURN
  180 CONTINUE
      NZ = 0
      IERR=2
      RETURN
  200 CONTINUE
      IF(NW.EQ.(-1)) GO TO 180
      NZ=0
      IERR=5
      RETURN
  260 CONTINUE
      NZ=0
      IERR=4
      RETURN
      END