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SUBROUTINE ZBESK(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE ZBESK
C***DATE WRITTEN 830501 (YYMMDD)
C***REVISION DATE 890801 (YYMMDD)
C***CATEGORY NO. B5K
C***KEYWORDS K-BESSEL FUNCTION,COMPLEX BESSEL FUNCTION,
C MODIFIED BESSEL FUNCTION OF THE SECOND KIND,
C BESSEL FUNCTION OF THE THIRD KIND
C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C***PURPOSE TO COMPUTE K-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C***DESCRIPTION
C
C ***A DOUBLE PRECISION ROUTINE***
C
C ON KODE=1, CBESK COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C BESSEL FUNCTIONS CY(J)=K(FNU+J-1,Z) FOR REAL, NONNEGATIVE
C ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z.NE.CMPLX(0.0,0.0)
C IN THE CUT PLANE -PI.LT.ARG(Z).LE.PI. ON KODE=2, CBESK
C RETURNS THE SCALED K FUNCTIONS,
C
C CY(J)=EXP(Z)*K(FNU+J-1,Z) , J=1,...,N,
C
C WHICH REMOVE THE EXPONENTIAL BEHAVIOR IN BOTH THE LEFT AND
C RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND
C NOTATION ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL
C FUNCTIONS (REF. 1).
C
C INPUT ZR,ZI,FNU ARE DOUBLE PRECISION
C ZR,ZI - Z=CMPLX(ZR,ZI), Z.NE.CMPLX(0.0D0,0.0D0),
C -PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL K FUNCTION, FNU.GE.0.0D0
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C KODE - A PARAMETER TO INDICATE THE SCALING OPTION
C KODE= 1 RETURNS
C CY(I)=K(FNU+I-1,Z), I=1,...,N
C = 2 RETURNS
C CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
C
C OUTPUT CYR,CYI ARE DOUBLE PRECISION
C CYR,CYI- DOUBLE PRECISION VECTORS WHOSE FIRST N COMPONENTS
C CONTAIN REAL AND IMAGINARY PARTS FOR THE SEQUENCE
C CY(I)=K(FNU+I-1,Z), I=1,...,N OR
C CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
C DEPENDING ON KODE
C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW.
C NZ= 0 , NORMAL RETURN
C NZ.GT.0 , FIRST NZ COMPONENTS OF CY SET TO ZERO DUE
C TO UNDERFLOW, CY(I)=CMPLX(0.0D0,0.0D0),
C I=1,...,N WHEN X.GE.0.0. WHEN X.LT.0.0
C NZ STATES ONLY THE NUMBER OF UNDERFLOWS
C IN THE SEQUENCE.
C
C IERR - ERROR FLAG
C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
C IERR=1, INPUT ERROR - NO COMPUTATION
C IERR=2, OVERFLOW - NO COMPUTATION, FNU IS
C TOO LARGE OR CABS(Z) IS TOO SMALL OR BOTH
C IERR=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
C ACCURACY
C IERR=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
C CANCE BY ARGUMENT REDUCTION
C IERR=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C***LONG DESCRIPTION
C
C EQUATIONS OF THE REFERENCE ARE IMPLEMENTED FOR SMALL ORDERS
C DNU AND DNU+1.0 IN THE RIGHT HALF PLANE X.GE.0.0. FORWARD
C RECURRENCE GENERATES HIGHER ORDERS. K IS CONTINUED TO THE LEFT
C HALF PLANE BY THE RELATION
C
C K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z)
C MP=MR*PI*I, MR=+1 OR -1, RE(Z).GT.0, I**2=-1
C
C WHERE I(FNU,Z) IS THE I BESSEL FUNCTION.
C
C FOR LARGE ORDERS, FNU.GT.FNUL, THE K FUNCTION IS COMPUTED
C BY MEANS OF ITS UNIFORM ASYMPTOTIC EXPANSIONS.
C
C FOR NEGATIVE ORDERS, THE FORMULA
C
C K(-FNU,Z) = K(FNU,Z)
C
C CAN BE USED.
C
C CBESK ASSUMES THAT A SIGNIFICANT DIGIT SINH(X) FUNCTION IS
C AVAILABLE.
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=DMAX1(D1MACH(4),1.0D-18) IS
C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
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 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY 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(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) 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 HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983.
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C***ROUTINES CALLED ZACON,ZBKNU,ZBUNK,ZUOIK,AZABS,I1MACH,D1MACH
C***END PROLOGUE ZBESK
C
C COMPLEX CY,Z
DOUBLE PRECISION AA, ALIM, ALN, ARG, AZ, CYI, CYR, DIG, ELIM, FN,
* FNU, FNUL, RL, R1M5, TOL, UFL, ZI, ZR, D1MACH, AZABS, BB
INTEGER IERR, K, KODE, K1, K2, MR, N, NN, NUF, NW, NZ, I1MACH
DIMENSION CYR(N), CYI(N)
C***FIRST EXECUTABLE STATEMENT ZBESK
IERR = 0
NZ=0
IF (ZI.EQ.0.0E0 .AND. ZR.EQ.0.0E0) IERR=1
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
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 = DMAX1(D1MACH(4),1.0D-18)
K1 = I1MACH(15)
K2 = I1MACH(16)
R1M5 = D1MACH(5)
K = MIN0(IABS(K1),IABS(K2))
ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
K1 = I1MACH(14) - 1
AA = R1M5*DBLE(FLOAT(K1))
DIG = DMIN1(AA,18.0D0)
AA = AA*2.303D0
ALIM = ELIM + DMAX1(-AA,-41.45D0)
FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0)
RL = 1.2D0*DIG + 3.0D0
C-----------------------------------------------------------------------------
C TEST FOR PROPER RANGE
C-----------------------------------------------------------------------
AZ = AZABS(ZR,ZI)
FN = FNU + DBLE(FLOAT(NN-1))
AA = 0.5D0/TOL
BB=DBLE(FLOAT(I1MACH(9)))*0.5D0
AA = DMIN1(AA,BB)
IF (AZ.GT.AA) GO TO 260
IF (FN.GT.AA) GO TO 260
AA = DSQRT(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-----------------------------------------------------------------------
C UFL = DEXP(-ELIM)
UFL = D1MACH(1)*1.0D+3
IF (AZ.LT.UFL) GO TO 180
IF (FNU.GT.FNUL) GO TO 80
IF (FN.LE.1.0D0) GO TO 60
IF (FN.GT.2.0D0) GO TO 50
IF (AZ.GT.TOL) GO TO 60
ARG = 0.5D0*AZ
ALN = -FN*DLOG(ARG)
IF (ALN.GT.ELIM) GO TO 180
GO TO 60
50 CONTINUE
CALL ZUOIK(ZR, ZI, FNU, KODE, 2, NN, CYR, CYI, NUF, TOL, ELIM,
* ALIM)
IF (NUF.LT.0) GO TO 180
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 100
60 CONTINUE
IF (ZR.LT.0.0D0) GO TO 70
C-----------------------------------------------------------------------
C RIGHT HALF PLANE COMPUTATION, REAL(Z).GE.0.
C-----------------------------------------------------------------------
CALL ZBKNU(ZR, ZI, FNU, KODE, NN, CYR, CYI, NW, TOL, ELIM, ALIM)
IF (NW.LT.0) GO TO 200
NZ=NW
RETURN
C-----------------------------------------------------------------------
C LEFT HALF PLANE COMPUTATION
C PI/2.LT.ARG(Z).LE.PI AND -PI.LT.ARG(Z).LT.-PI/2.
C-----------------------------------------------------------------------
70 CONTINUE
IF (NZ.NE.0) GO TO 180
MR = 1
IF (ZI.LT.0.0D0) MR = -1
CALL ZACON(ZR, ZI, FNU, KODE, MR, NN, CYR, CYI, NW, RL, FNUL,
* TOL, ELIM, ALIM)
IF (NW.LT.0) GO TO 200
NZ=NW
RETURN
C-----------------------------------------------------------------------
C UNIFORM ASYMPTOTIC EXPANSIONS FOR FNU.GT.FNUL
C-----------------------------------------------------------------------
80 CONTINUE
MR = 0
IF (ZR.GE.0.0D0) GO TO 90
MR = 1
IF (ZI.LT.0.0D0) MR = -1
90 CONTINUE
CALL ZBUNK(ZR, ZI, FNU, KODE, MR, NN, CYR, CYI, NW, TOL, ELIM,
* ALIM)
IF (NW.LT.0) GO TO 200
NZ = NZ + NW
RETURN
100 CONTINUE
IF (ZR.LT.0.0D0) GO TO 180
RETURN
180 CONTINUE
NZ = 0
IERR=2
RETURN
200 CONTINUE
IF(NW.EQ.(-1)) GO TO 180
NZ=0
IERR=5
RETURN
260 CONTINUE
NZ=0
IERR=4
RETURN
END
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