*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