*DECK ZBESI SUBROUTINE ZBESI (ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR) C***BEGIN PROLOGUE ZBESI C***PURPOSE Compute a sequence of the Bessel functions I(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 (CBESI-C, ZBESI-C) C***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT, I BESSEL FUNCTIONS, C MODIFIED BESSEL FUNCTIONS C***AUTHOR Amos, D. E., (SNL) C***DESCRIPTION C C ***A DOUBLE PRECISION ROUTINE*** C On KODE=1, ZBESI computes an N-member sequence of complex C Bessel functions CY(L)=I(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 where Z=ZR+i*ZI. On KODE=2, CBESI returns C the scaled functions C C CY(L) = exp(-abs(X))*I(FNU+L-1,Z), L=1,...,N and X=Re(Z) C C which removes the exponential growth in both the left and C right half-planes as Z goes to infinity. C C Input C ZR - DOUBLE PRECISION real part of argument Z C ZI - DOUBLE PRECISION imag part of argument Z C FNU - DOUBLE PRECISION initial order, FNU>=0 C KODE - A parameter to indicate the scaling option C KODE=1 returns C CY(L)=I(FNU+L-1,Z), L=1,...,N C =2 returns C CY(L)=exp(-abs(X))*I(FNU+L-1,Z), L=1,...,N C where X=Re(Z) C N - Number of terms in the sequence, N>=1 C C Output C CYR - DOUBLE PRECISION real part of result vector C CYI - DOUBLE PRECISION imag part of result vector C NZ - Number of underflows set to zero C NZ=0 Normal return C NZ>0 CY(L)=0, L=N-NZ+1,...,N 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 (Re(Z) too large on KODE=1) 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 of I(a,z) is carried out by the power series C for small abs(z), the asymptotic expansion for large abs(z), C the Miller algorithm normalized by the Wronskian and a C Neumann series for intermediate magnitudes of z, and the C uniform asymptotic expansions for I(a,z) and J(a,z) for C large orders a. Backward recurrence is used to generate C sequences or reduce orders when necessary. C C The calculations above are done in the right half plane and C continued into the left half plane by the formula C C I(a,z*exp(t)) = exp(t*a)*I(a,z), Re(z)>0 C t = i*pi or -i*pi C C For negative orders, the formula C C I(-a,z) = I(a,z) + (2/pi)*sin(pi*a)*K(a,z) C C can be used. However, for large orders close to integers the C the function changes radically. When a is a large positive C integer, the magnitude of I(-a,z)=I(a,z) is a large C negative power of ten. But when a is not an integer, C K(a,z) dominates in magnitude with a large positive power of C ten and the most that the second term can be reduced is by C unit roundoff from the coefficient. Thus, wide changes can C occur within unit roundoff of a large integer for a. Here, C 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=MAX(D1MACH(4),1.0D-18) is double C precision unit roundoff limited to 18 digits precision. 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 D1MACH, I1MACH, ZABS, ZBINU 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 ZBESI C COMPLEX CONE,CSGN,CW,CY,CZERO,Z,ZN DOUBLE PRECISION AA, ALIM, ARG, CONEI, CONER, CSGNI, CSGNR, CYI, * CYR, DIG, ELIM, FNU, FNUL, PI, RL, R1M5, STR, TOL, ZI, ZNI, ZNR, * ZR, D1MACH, AZ, BB, FN, ZABS, ASCLE, RTOL, ATOL, STI INTEGER I, IERR, INU, K, KODE, K1,K2,N,NZ,NN, I1MACH DIMENSION CYR(N), CYI(N) EXTERNAL ZABS DATA PI /3.14159265358979324D0/ DATA CONER, CONEI /1.0D0,0.0D0/ C C***FIRST EXECUTABLE STATEMENT ZBESI IERR = 0 NZ=0 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 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(D1MACH(4),1.0D-18) K1 = I1MACH(15) K2 = I1MACH(16) R1M5 = D1MACH(5) K = MIN(ABS(K1),ABS(K2)) ELIM = 2.303D0*(K*R1M5-3.0D0) K1 = I1MACH(14) - 1 AA = R1M5*K1 DIG = MIN(AA,18.0D0) AA = AA*2.303D0 ALIM = ELIM + MAX(-AA,-41.45D0) RL = 1.2D0*DIG + 3.0D0 FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0) C----------------------------------------------------------------------- C TEST FOR PROPER RANGE C----------------------------------------------------------------------- AZ = ZABS(ZR,ZI) FN = FNU+(N-1) AA = 0.5D0/TOL BB=I1MACH(9)*0.5D0 AA = MIN(AA,BB) IF (AZ.GT.AA) GO TO 260 IF (FN.GT.AA) GO TO 260 AA = SQRT(AA) IF (AZ.GT.AA) IERR=3 IF (FN.GT.AA) IERR=3 ZNR = ZR ZNI = ZI CSGNR = CONER CSGNI = CONEI IF (ZR.GE.0.0D0) GO TO 40 ZNR = -ZR ZNI = -ZI C----------------------------------------------------------------------- C CALCULATE CSGN=EXP(FNU*PI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE C WHEN FNU IS LARGE C----------------------------------------------------------------------- INU = FNU ARG = (FNU-INU)*PI IF (ZI.LT.0.0D0) ARG = -ARG CSGNR = COS(ARG) CSGNI = SIN(ARG) IF (MOD(INU,2).EQ.0) GO TO 40 CSGNR = -CSGNR CSGNI = -CSGNI 40 CONTINUE CALL ZBINU(ZNR, ZNI, FNU, KODE, N, CYR, CYI, NZ, RL, FNUL, TOL, * ELIM, ALIM) IF (NZ.LT.0) GO TO 120 IF (ZR.GE.0.0D0) RETURN C----------------------------------------------------------------------- C ANALYTIC CONTINUATION TO THE LEFT HALF PLANE C----------------------------------------------------------------------- NN = N - NZ IF (NN.EQ.0) RETURN RTOL = 1.0D0/TOL ASCLE = D1MACH(1)*RTOL*1.0D+3 DO 50 I=1,NN C STR = CYR(I)*CSGNR - CYI(I)*CSGNI C CYI(I) = CYR(I)*CSGNI + CYI(I)*CSGNR C CYR(I) = STR AA = CYR(I) BB = CYI(I) ATOL = 1.0D0 IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 55 AA = AA*RTOL BB = BB*RTOL ATOL = TOL 55 CONTINUE STR = AA*CSGNR - BB*CSGNI STI = AA*CSGNI + BB*CSGNR CYR(I) = STR*ATOL CYI(I) = STI*ATOL CSGNR = -CSGNR CSGNI = -CSGNI 50 CONTINUE RETURN 120 CONTINUE IF(NZ.EQ.(-2)) GO TO 130 NZ = 0 IERR=2 RETURN 130 CONTINUE NZ=0 IERR=5 RETURN 260 CONTINUE NZ=0 IERR=4 RETURN END

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