relibc_openlibm/slatec/cbesk.f

*DECK CBESK
      SUBROUTINE CBESK (Z, FNU, KODE, N, CY, NZ, IERR)
C***BEGIN PROLOGUE  CBESK
C***PURPOSE  Compute a sequence of the Bessel functions K(a,z) for
C            complex argument z and real nonnegative orders a=b,b+1,
C            b+2,... where b>0.  A scaling option is available to
C            help avoid overflow.
C***LIBRARY   SLATEC
C***CATEGORY  C10B4
C***TYPE      COMPLEX (CBESK-C, ZBESK-C)
C***KEYWORDS  BESSEL FUNCTIONS OF COMPLEX ARGUMENT, K BESSEL FUNCTIONS,
C             MODIFIED BESSEL FUNCTIONS
C***AUTHOR  Amos, D. E., (SNL)
C***DESCRIPTION
C
C         On KODE=1, CBESK computes an N member sequence of complex
C         Bessel functions CY(L)=K(FNU+L-1,Z) for real nonnegative
C         orders FNU+L-1, L=1,...,N and complex Z.NE.0 in the cut
C         plane -pi<arg(Z)<=pi.  On KODE=2, CBESJ returns the scaled
C         functions
C
C            CY(L) = exp(Z)*K(FNU+L-1,Z),  L=1,...,N
C
C         which remove the exponential growth in both the left and
C         right half planes as Z goes to infinity.  Definitions and
C         notation are found in the NBS Handbook of Mathematical
C         Functions (Ref. 1).
C
C         Input
C           Z      - Nonzero argument of type COMPLEX
C           FNU    - Initial order of type REAL, FNU>=0
C           KODE   - A parameter to indicate the scaling option
C                    KODE=1  returns
C                            CY(L)=K(FNU+L-1,Z), L=1,...,N
C                        =2  returns
C                            CY(L)=K(FNU+L-1,Z)*EXP(Z), L=1,...,N
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 Re(Z)>0
C                            then CY(L)=0 for L=1,...,NZ; in the
C                            complementary half plane 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         Equations of the reference are implemented to compute K(a,z)
C         for small orders a and a+1 in the right half plane Re(z)>=0.
C         Forward recurrence generates higher orders.  The formula
C
C            K(a,z*exp((t)) = exp(-t)*K(a,z) - t*I(a,z),  Re(z)>0
C                         t = i*pi or -i*pi
C
C         continues K to the left half plane.
C
C         For large orders, K(a,z) is computed by means of its uniform
C         asymptotic expansion.
C
C         For negative orders, the formula
C
C            K(-a,z) = K(a,z)
C
C         can be used.
C
C         CBESK assumes that a significant digit sinh 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=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  CBESK
C
      COMPLEX CY, Z
      REAL AA, ALIM, ALN, ARG, AZ, DIG, ELIM, FN, FNU, FNUL, RL, R1M5,
     * TOL, UFL, XX, YY, R1MACH, BB
      INTEGER IERR, K, KODE, K1, K2, MR, N, NN, NUF, NW, NZ, I1MACH
      DIMENSION CY(N)
C***FIRST EXECUTABLE STATEMENT  CBESK
      IERR = 0
      NZ=0
      XX = REAL(Z)
      YY = AIMAG(Z)
      IF (YY.EQ.0.0E0 .AND. XX.EQ.0.0E0) IERR=1
      IF (FNU.LT.0.0E0) 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
      AZ = ABS(Z)
      FN = FNU + (NN-1)
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 210
      IF(FN.GT.AA) GO TO 210
      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-----------------------------------------------------------------------
C     UFL = EXP(-ELIM)
      UFL = R1MACH(1)*1.0E+3
      IF (AZ.LT.UFL) GO TO 180
      IF (FNU.GT.FNUL) GO TO 80
      IF (FN.LE.1.0E0) GO TO 60
      IF (FN.GT.2.0E0) GO TO 50
      IF (AZ.GT.TOL) GO TO 60
      ARG = 0.5E0*AZ
      ALN = -FN*ALOG(ARG)
      IF (ALN.GT.ELIM) GO TO 180
      GO TO 60
   50 CONTINUE
      CALL CUOIK(Z, FNU, KODE, 2, NN, CY, 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 (XX.LT.0.0E0) GO TO 70
C-----------------------------------------------------------------------
C     RIGHT HALF PLANE COMPUTATION, REAL(Z).GE.0.
C-----------------------------------------------------------------------
      CALL CBKNU(Z, FNU, KODE, NN, CY, 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 (YY.LT.0.0E0) MR = -1
      CALL CACON(Z, FNU, KODE, MR, NN, CY, 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 (XX.GE.0.0E0) GO TO 90
      MR = 1
      IF (YY.LT.0.0E0) MR = -1
   90 CONTINUE
      CALL CBUNK(Z, FNU, KODE, MR, NN, CY, NW, TOL, ELIM, ALIM)
      IF (NW.LT.0) GO TO 200
      NZ = NZ + NW
      RETURN
  100 CONTINUE
      IF (XX.LT.0.0E0) 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
  210 CONTINUE
      NZ=0
      IERR=4
      RETURN
      END