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