*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=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