*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=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