relibc_openlibm/slatec/cbesy.f

*DECK CBESY
      SUBROUTINE CBESY (Z, FNU, KODE, N, CY, NZ, CWRK, IERR)
C***BEGIN PROLOGUE  CBESY
C***PURPOSE  Compute a sequence of the Bessel functions Y(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  C10A4
C***TYPE      COMPLEX (CBESY-C, ZBESY-C)
C***KEYWORDS  BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
C             BESSEL FUNCTIONS OF SECOND KIND, WEBER'S FUNCTION,
C             Y BESSEL FUNCTIONS
C***AUTHOR  Amos, D. E., (SNL)
C***DESCRIPTION
C
C         On KODE=1, CBESY computes an N member sequence of complex
C         Bessel functions CY(L)=Y(FNU+L-1,Z) for real nonnegative
C         orders FNU+L-1, L=1,...,N and complex Z in the cut plane
C         -pi<arg(Z)<=pi.  On KODE=2, CBESY returns the scaled
C         functions
C
C            CY(L) = exp(-abs(Y))*Y(FNU+L-1,Z),  L=1,...,N, Y=Im(Z)
C
C         which remove the exponential growth in both the upper and
C         lower 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)=Y(FNU+L-1,Z), L=1,...,N
C                        =2  returns
C                            CY(L)=Y(FNU+L-1,Z)*exp(-abs(Y)), L=1,...,N
C                            where Y=Im(Z)
C           N      - Number of terms in the sequence, N>=1
C           CWRK   - A work vector of type COMPLEX and dimension N
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, usually on
C                            KODE=2 (the underflows may not be in an
C                            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         The computation is carried out by the formula
C
C            Y(a,z) = (H(1,a,z) - H(2,a,z))/(2*i)
C
C         where the Hankel functions are computed as described in CBESH.
C
C         For negative orders, the formula
C
C            Y(-a,z) = Y(a,z)*cos(a*pi) + J(a,z)*sin(a*pi)
C
C         can be used.  However, for large orders close to half odd
C         integers the function changes radically.  When a is a large
C         positive half odd integer, the magnitude of Y(-a,z)=J(a,z)*
C         sin(a*pi) is a large negative power of ten.  But when a is
C         not a half odd integer, Y(a,z) dominates in magnitude with a
C         large positive power of ten and the most that the second term
C         can be reduced is by unit roundoff from the coefficient.
C         Thus,  wide changes can occur within unit roundoff of a large
C         half odd integer.  Here, large means a>abs(z).
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  CBESH, 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  CBESY
C
      COMPLEX CWRK, CY, C1, C2, EX, HCI, Z, ZU, ZV
      REAL ELIM, EY, FNU, R1, R2, TAY, XX, YY, R1MACH, R1M5, ASCLE,
     *  RTOL, ATOL, TOL, AA, BB
      INTEGER I, IERR, K, KODE, K1, K2, N, NZ, NZ1, NZ2, I1MACH
      DIMENSION CY(N), CWRK(N)
C***FIRST EXECUTABLE STATEMENT  CBESY
      XX = REAL(Z)
      YY = AIMAG(Z)
      IERR = 0
      NZ=0
      IF (XX.EQ.0.0E0 .AND. YY.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
      HCI = CMPLX(0.0E0,0.5E0)
      CALL CBESH(Z, FNU, KODE, 1, N, CY, NZ1, IERR)
      IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
      CALL CBESH(Z, FNU, KODE, 2, N, CWRK, NZ2, IERR)
      IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
      NZ = MIN(NZ1,NZ2)
      IF (KODE.EQ.2) GO TO 60
      DO 50 I=1,N
        CY(I) = HCI*(CWRK(I)-CY(I))
   50 CONTINUE
      RETURN
   60 CONTINUE
      TOL = MAX(R1MACH(4),1.0E-18)
      K1 = I1MACH(12)
      K2 = I1MACH(13)
      K = MIN(ABS(K1),ABS(K2))
      R1M5 = R1MACH(5)
C-----------------------------------------------------------------------
C     ELIM IS THE APPROXIMATE EXPONENTIAL UNDER- AND OVERFLOW LIMIT
C-----------------------------------------------------------------------
      ELIM = 2.303E0*(K*R1M5-3.0E0)
      R1 = COS(XX)
      R2 = SIN(XX)
      EX = CMPLX(R1,R2)
      EY = 0.0E0
      TAY = ABS(YY+YY)
      IF (TAY.LT.ELIM) EY = EXP(-TAY)
      IF (YY.LT.0.0E0) GO TO 90
      C1 = EX*CMPLX(EY,0.0E0)
      C2 = CONJG(EX)
   70 CONTINUE
      NZ = 0
      RTOL = 1.0E0/TOL
      ASCLE = R1MACH(1)*RTOL*1.0E+3
      DO 80 I=1,N
C       CY(I) = HCI*(C2*CWRK(I)-C1*CY(I))
        ZV = CWRK(I)
        AA=REAL(ZV)
        BB=AIMAG(ZV)
        ATOL=1.0E0
        IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 75
          ZV = ZV*CMPLX(RTOL,0.0E0)
          ATOL = TOL
   75   CONTINUE
        ZV = ZV*C2*HCI
        ZV = ZV*CMPLX(ATOL,0.0E0)
        ZU=CY(I)
        AA=REAL(ZU)
        BB=AIMAG(ZU)
        ATOL=1.0E0
        IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 85
          ZU = ZU*CMPLX(RTOL,0.0E0)
          ATOL = TOL
   85   CONTINUE
        ZU = ZU*C1*HCI
        ZU = ZU*CMPLX(ATOL,0.0E0)
        CY(I) = ZV - ZU
        IF (CY(I).EQ.CMPLX(0.0E0,0.0E0) .AND. EY.EQ.0.0E0) NZ = NZ + 1
   80 CONTINUE
      RETURN
   90 CONTINUE
      C1 = EX
      C2 = CONJG(EX)*CMPLX(EY,0.0E0)
      GO TO 70
  170 CONTINUE
      NZ = 0
      RETURN
      END