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author | John W. Eaton <jwe@octave.org> |
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date | Fri, 18 Jul 2008 17:42:48 -0400 |
parents | 82be108cc558 |
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SUBROUTINE CBESJ(Z, FNU, KODE, N, CY, NZ, IERR) C***BEGIN PROLOGUE CBESJ C***DATE WRITTEN 830501 (YYMMDD) C***REVISION DATE 890801 (YYMMDD) C***CATEGORY NO. B5K C***KEYWORDS J-BESSEL FUNCTION,BESSEL FUNCTION OF COMPLEX ARGUMENT, C BESSEL FUNCTION OF FIRST KIND C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES C***PURPOSE TO COMPUTE THE J-BESSEL FUNCTION OF A COMPLEX ARGUMENT C***DESCRIPTION C C ON KODE=1, CBESJ COMPUTES AN N MEMBER SEQUENCE OF COMPLEX C BESSEL FUNCTIONS CY(I)=J(FNU+I-1,Z) FOR REAL, NONNEGATIVE C ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE C -PI.LT.ARG(Z).LE.PI. ON KODE=2, CBESJ RETURNS THE SCALED C FUNCTIONS C C CY(I)=EXP(-ABS(Y))*J(FNU+I-1,Z) I = 1,...,N , Y=AIMAG(Z) C C WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND C LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION C ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS C (REF. 1). C C INPUT C Z - Z=CMPLX(X,Y), -PI.LT.ARG(Z).LE.PI C FNU - ORDER OF INITIAL J FUNCTION, FNU.GE.0.0E0 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION C KODE= 1 RETURNS C CY(I)=J(FNU+I-1,Z), I=1,...,N C = 2 RETURNS C CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,... C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1 C C OUTPUT C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN C VALUES FOR THE SEQUENCE C CY(I)=J(FNU+I-1,Z) OR C CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N C DEPENDING ON KODE, Y=AIMAG(Z). C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW, C NZ= 0 , NORMAL RETURN C NZ.GT.0 , LAST NZ COMPONENTS OF CY SET TO ZERO C DUE TO UNDERFLOW, CY(I)=CMPLX(0.0,0.0), C I = 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, AIMAG(Z) C TOO LARGE ON KODE=1 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 THE COMPUTATION IS CARRIED OUT BY THE FORMULA C C J(FNU,Z)=EXP( FNU*PI*I/2)*I(FNU,-I*Z) AIMAG(Z).GE.0.0 C C J(FNU,Z)=EXP(-FNU*PI*I/2)*I(FNU, I*Z) AIMAG(Z).LT.0.0 C C WHERE I**2 = -1 AND I(FNU,Z) IS THE I BESSEL FUNCTION. C C FOR NEGATIVE ORDERS,THE FORMULA C C J(-FNU,Z) = J(FNU,Z)*COS(PI*FNU) - Y(FNU,Z)*SIN(PI*FNU) C C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE C THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE C INTEGER,THE MAGNITUDE OF J(-FNU,Z)=J(FNU,Z)*COS(PI*FNU) IS A C LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT AN INTEGER, C Y(FNU,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 FNU. HERE, C LARGE MEANS FNU.GT.CABS(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=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 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 CBINU,I1MACH,R1MACH C***END PROLOGUE CBESJ C COMPLEX CI, CSGN, CY, Z, ZN REAL AA, ALIM, ARG, DIG, ELIM, FNU, FNUL, HPI, RL, R1, R1M5, R2, * TOL, YY, R1MACH, AZ, FN, BB, ASCLE, RTOL, ATOL INTEGER I, IERR, INU, INUH, IR, KODE, K1, K2, N, NL, NZ, I1MACH, K DIMENSION CY(N) DATA HPI /1.57079632679489662E0/ C C***FIRST EXECUTABLE STATEMENT CBESJ IERR = 0 NZ=0 IF (FNU.LT.0.0E0) 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 = AMAX1(R1MACH(4),1.0E-18) K1 = I1MACH(12) K2 = I1MACH(13) R1M5 = R1MACH(5) K = MIN0(IABS(K1),IABS(K2)) ELIM = 2.303E0*(FLOAT(K)*R1M5-3.0E0) K1 = I1MACH(11) - 1 AA = R1M5*FLOAT(K1) DIG = AMIN1(AA,18.0E0) AA = AA*2.303E0 ALIM = ELIM + AMAX1(-AA,-41.45E0) RL = 1.2E0*DIG + 3.0E0 FNUL = 10.0E0 + 6.0E0*(DIG-3.0E0) CI = CMPLX(0.0E0,1.0E0) YY = AIMAG(Z) AZ = CABS(Z) C----------------------------------------------------------------------- C TEST FOR RANGE C----------------------------------------------------------------------- AA = 0.5E0/TOL BB=FLOAT(I1MACH(9))*0.5E0 AA=AMIN1(AA,BB) FN=FNU+FLOAT(N-1) IF(AZ.GT.AA) GO TO 140 IF(FN.GT.AA) GO TO 140 AA=SQRT(AA) IF(AZ.GT.AA) IERR=3 IF(FN.GT.AA) IERR=3 C----------------------------------------------------------------------- C CALCULATE CSGN=EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE C WHEN FNU IS LARGE C----------------------------------------------------------------------- INU = INT(FNU) INUH = INU/2 IR = INU - 2*INUH ARG = (FNU-FLOAT(INU-IR))*HPI R1 = COS(ARG) R2 = SIN(ARG) CSGN = CMPLX(R1,R2) IF (MOD(INUH,2).EQ.1) CSGN = -CSGN C----------------------------------------------------------------------- C ZN IS IN THE RIGHT HALF PLANE C----------------------------------------------------------------------- ZN = -Z*CI IF (YY.GE.0.0E0) GO TO 40 ZN = -ZN CSGN = CONJG(CSGN) CI = CONJG(CI) 40 CONTINUE CALL CBINU(ZN, FNU, KODE, N, CY, NZ, RL, FNUL, TOL, ELIM, ALIM) IF (NZ.LT.0) GO TO 120 NL = N - NZ IF (NL.EQ.0) RETURN RTOL = 1.0E0/TOL ASCLE = R1MACH(1)*RTOL*1.0E+3 DO 50 I=1,NL C CY(I)=CY(I)*CSGN ZN=CY(I) AA=REAL(ZN) BB=AIMAG(ZN) ATOL=1.0E0 IF (AMAX1(ABS(AA),ABS(BB)).GT.ASCLE) GO TO 55 ZN = ZN*CMPLX(RTOL,0.0E0) ATOL = TOL 55 CONTINUE ZN = ZN*CSGN CY(I) = ZN*CMPLX(ATOL,0.0E0) CSGN = CSGN*CI 50 CONTINUE RETURN 120 CONTINUE IF(NZ.EQ.(-2)) GO TO 130 NZ = 0 IERR = 2 RETURN 130 CONTINUE NZ=0 IERR=5 RETURN 140 CONTINUE NZ=0 IERR=4 RETURN END