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OpenLibm/slatec/cbesh.f
Viral B. Shah c977aa998f Add Makefile.extras to build libopenlibm-extras. 
Replace amos with slatec
2012-12-31 16:37:05 -05:00

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*DECK CBESH
SUBROUTINE CBESH (Z, FNU, KODE, M, N, CY, NZ, IERR)
C***BEGIN PROLOGUE CBESH
C***PURPOSE Compute a sequence of the Hankel functions H(m,a,z)
C for superscript m=1 or 2, real nonnegative orders a=b,
C b+1,... where b>0, and nonzero complex argument z. A
C scaling option is available to help avoid overflow.
C***LIBRARY SLATEC
C***CATEGORY C10A4
C***TYPE COMPLEX (CBESH-C, ZBESH-C)
C***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT,
C BESSEL FUNCTIONS OF THE THIRD KIND, H BESSEL FUNCTIONS,
C HANKEL FUNCTIONS
C***AUTHOR Amos, D. E., (SNL)
C***DESCRIPTION
C
C On KODE=1, CBESH computes an N member sequence of complex
C Hankel (Bessel) functions CY(L)=H(M,FNU+L-1,Z) for super-
C script M=1 or 2, real nonnegative orders FNU+L-1, L=1,...,
C N, and complex nonzero Z in the cut plane -pi<arg(Z)<=pi.
C On KODE=2, CBESH returns the scaled functions
C
C CY(L) = H(M,FNU+L-1,Z)*exp(-(3-2*M)*Z*i), i**2=-1
C
C which removes the exponential behavior in both the upper
C and lower half planes. Definitions and notation are found
C in the NBS Handbook of Mathematical 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)=H(M,FNU+L-1,Z), L=1,...,N
C =2 returns
C CY(L)=H(M,FNU+L-1,Z)*exp(-(3-2M)*Z*i),
C L=1,...,N
C M - Superscript of Hankel function, M=1 or 2
C N - Number of terms in the sequence, N>=1
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 (if M=1 and
C Im(Z)>0 or if M=2 and Im(Z)<0, then
C CY(L)=0 for L=1,...,NZ; in the com-
C plementary half planes, the underflows
C may not be in an 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 H(m,a,z) = (1/t)*exp(-a*t)*K(a,z*exp(-t))
C t = (3-2*m)*i*pi/2
C
C where the K Bessel function is computed as described in the
C prologue to CBESK.
C
C Exponential decay of H(m,a,z) occurs in the upper half z
C plane for m=1 and the lower half z plane for m=2. Exponential
C growth occurs in the complementary half planes. Scaling
C by exp(-(3-2*m)*z*i) removes the exponential behavior in the
C whole z plane as z goes to infinity.
C
C For negative orders, the formula
C
C H(m,-a,z) = H(m,a,z)*exp((3-2*m)*a*pi*i)
C
C can be used.
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 CACON, CBKNU, CBUNK, CUOIK, 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 CBESH
C
COMPLEX CY, Z, ZN, ZT, CSGN
REAL AA, ALIM, ALN, ARG, AZ, CPN, DIG, ELIM, FMM, FN, FNU, FNUL,
* HPI, RHPI, RL, R1M5, SGN, SPN, TOL, UFL, XN, XX, YN, YY, R1MACH,
* BB, ASCLE, RTOL, ATOL
INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, M,
* MM, MR, N, NN, NUF, NW, NZ, I1MACH
DIMENSION CY(N)
C
DATA HPI /1.57079632679489662E0/
C
C***FIRST EXECUTABLE STATEMENT CBESH
NZ=0
XX = REAL(Z)
YY = AIMAG(Z)
IERR = 0
IF (XX.EQ.0.0E0 .AND. YY.EQ.0.0E0) IERR=1
IF (FNU.LT.0.0E0) IERR=1
IF (M.LT.1 .OR. M.GT.2) IERR=1
IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
IF (N.LT.1) IERR=1
IF (IERR.NE.0) RETURN
NN = N
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(R1MACH(4),1.0E-18)
K1 = I1MACH(12)
K2 = I1MACH(13)
R1M5 = R1MACH(5)
K = MIN(ABS(K1),ABS(K2))
ELIM = 2.303E0*(K*R1M5-3.0E0)
K1 = I1MACH(11) - 1
AA = R1M5*K1
DIG = MIN(AA,18.0E0)
AA = AA*2.303E0
ALIM = ELIM + MAX(-AA,-41.45E0)
FNUL = 10.0E0 + 6.0E0*(DIG-3.0E0)
RL = 1.2E0*DIG + 3.0E0
FN = FNU + (NN-1)
MM = 3 - M - M
FMM = MM
ZN = Z*CMPLX(0.0E0,-FMM)
XN = REAL(ZN)
YN = AIMAG(ZN)
AZ = ABS(Z)
C-----------------------------------------------------------------------
C TEST FOR RANGE
C-----------------------------------------------------------------------
AA = 0.5E0/TOL
BB=I1MACH(9)*0.5E0
AA=MIN(AA,BB)
IF(AZ.GT.AA) GO TO 240
IF(FN.GT.AA) GO TO 240
AA=SQRT(AA)
IF(AZ.GT.AA) IERR=3
IF(FN.GT.AA) IERR=3
C-----------------------------------------------------------------------
C OVERFLOW TEST ON THE LAST MEMBER OF THE SEQUENCE
C-----------------------------------------------------------------------
UFL = R1MACH(1)*1.0E+3
IF (AZ.LT.UFL) GO TO 220
IF (FNU.GT.FNUL) GO TO 90
IF (FN.LE.1.0E0) GO TO 70
IF (FN.GT.2.0E0) GO TO 60
IF (AZ.GT.TOL) GO TO 70
ARG = 0.5E0*AZ
ALN = -FN*ALOG(ARG)
IF (ALN.GT.ELIM) GO TO 220
GO TO 70
60 CONTINUE
CALL CUOIK(ZN, FNU, KODE, 2, NN, CY, NUF, TOL, ELIM, ALIM)
IF (NUF.LT.0) GO TO 220
NZ = NZ + NUF
NN = NN - NUF
C-----------------------------------------------------------------------
C HERE NN=N OR NN=0 SINCE NUF=0,NN, OR -1 ON RETURN FROM CUOIK
C IF NUF=NN, THEN CY(I)=CZERO FOR ALL I
C-----------------------------------------------------------------------
IF (NN.EQ.0) GO TO 130
70 CONTINUE
IF ((XN.LT.0.0E0) .OR. (XN.EQ.0.0E0 .AND. YN.LT.0.0E0 .AND.
* M.EQ.2)) GO TO 80
C-----------------------------------------------------------------------
C RIGHT HALF PLANE COMPUTATION, XN.GE.0. .AND. (XN.NE.0. .OR.
C YN.GE.0. .OR. M=1)
C-----------------------------------------------------------------------
CALL CBKNU(ZN, FNU, KODE, NN, CY, NZ, TOL, ELIM, ALIM)
GO TO 110
C-----------------------------------------------------------------------
C LEFT HALF PLANE COMPUTATION
C-----------------------------------------------------------------------
80 CONTINUE
MR = -MM
CALL CACON(ZN, FNU, KODE, MR, NN, CY, NW, RL, FNUL, TOL, ELIM,
* ALIM)
IF (NW.LT.0) GO TO 230
NZ=NW
GO TO 110
90 CONTINUE
C-----------------------------------------------------------------------
C UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL
C-----------------------------------------------------------------------
MR = 0
IF ((XN.GE.0.0E0) .AND. (XN.NE.0.0E0 .OR. YN.GE.0.0E0 .OR.
* M.NE.2)) GO TO 100
MR = -MM
IF (XN.EQ.0.0E0 .AND. YN.LT.0.0E0) ZN = -ZN
100 CONTINUE
CALL CBUNK(ZN, FNU, KODE, MR, NN, CY, NW, TOL, ELIM, ALIM)
IF (NW.LT.0) GO TO 230
NZ = NZ + NW
110 CONTINUE
C-----------------------------------------------------------------------
C H(M,FNU,Z) = -FMM*(I/HPI)*(ZT**FNU)*K(FNU,-Z*ZT)
C
C ZT=EXP(-FMM*HPI*I) = CMPLX(0.0,-FMM), FMM=3-2*M, M=1,2
C-----------------------------------------------------------------------
SGN = SIGN(HPI,-FMM)
C-----------------------------------------------------------------------
C CALCULATE EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
C WHEN FNU IS LARGE
C-----------------------------------------------------------------------
INU = FNU
INUH = INU/2
IR = INU - 2*INUH
ARG = (FNU-(INU-IR))*SGN
RHPI = 1.0E0/SGN
CPN = RHPI*COS(ARG)
SPN = RHPI*SIN(ARG)
C ZN = CMPLX(-SPN,CPN)
CSGN = CMPLX(-SPN,CPN)
C IF (MOD(INUH,2).EQ.1) ZN = -ZN
IF (MOD(INUH,2).EQ.1) CSGN = -CSGN
ZT = CMPLX(0.0E0,-FMM)
RTOL = 1.0E0/TOL
ASCLE = UFL*RTOL
DO 120 I=1,NN
C CY(I) = CY(I)*ZN
C ZN = ZN*ZT
ZN=CY(I)
AA=REAL(ZN)
BB=AIMAG(ZN)
ATOL=1.0E0
IF (MAX(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 125
ZN = ZN*CMPLX(RTOL,0.0E0)
ATOL = TOL
125 CONTINUE
ZN = ZN*CSGN
CY(I) = ZN*CMPLX(ATOL,0.0E0)
CSGN = CSGN*ZT
120 CONTINUE
RETURN
130 CONTINUE
IF (XN.LT.0.0E0) GO TO 220
RETURN
220 CONTINUE
IERR=2
NZ=0
RETURN
230 CONTINUE
IF(NW.EQ.(-1)) GO TO 220
NZ=0
IERR=5
RETURN
240 CONTINUE
NZ=0
IERR=4
RETURN
END
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