245 lines
10 KiB
FortranFixed
245 lines
10 KiB
FortranFixed
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SUBROUTINE ZBESY(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, CWRKR, CWRKI,
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* IERR)
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C***BEGIN PROLOGUE ZBESY
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C***DATE WRITTEN 830501 (YYMMDD)
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C***REVISION DATE 890801 (YYMMDD)
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C***CATEGORY NO. B5K
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C***KEYWORDS Y-BESSEL FUNCTION,BESSEL FUNCTION OF COMPLEX ARGUMENT,
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C BESSEL FUNCTION OF SECOND KIND
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C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
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C***PURPOSE TO COMPUTE THE Y-BESSEL FUNCTION OF A COMPLEX ARGUMENT
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C***DESCRIPTION
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C
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C ***A DOUBLE PRECISION ROUTINE***
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C
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C ON KODE=1, CBESY COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
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C BESSEL FUNCTIONS CY(I)=Y(FNU+I-1,Z) FOR REAL, NONNEGATIVE
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C ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE
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C -PI.LT.ARG(Z).LE.PI. ON KODE=2, CBESY RETURNS THE SCALED
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C FUNCTIONS
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C
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C CY(I)=EXP(-ABS(Y))*Y(FNU+I-1,Z) I = 1,...,N , Y=AIMAG(Z)
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C
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C WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND
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C LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION
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C ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS
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C (REF. 1).
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C
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C INPUT ZR,ZI,FNU ARE DOUBLE PRECISION
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C ZR,ZI - Z=CMPLX(ZR,ZI), Z.NE.CMPLX(0.0D0,0.0D0),
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C -PI.LT.ARG(Z).LE.PI
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C FNU - ORDER OF INITIAL Y FUNCTION, FNU.GE.0.0D0
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C KODE - A PARAMETER TO INDICATE THE SCALING OPTION
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C KODE= 1 RETURNS
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C CY(I)=Y(FNU+I-1,Z), I=1,...,N
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C = 2 RETURNS
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C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,...,N
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C WHERE Y=AIMAG(Z)
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C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
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C CWRKR, - DOUBLE PRECISION WORK VECTORS OF DIMENSION AT
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C CWRKI AT LEAST N
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C
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C OUTPUT CYR,CYI ARE DOUBLE PRECISION
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C CYR,CYI- DOUBLE PRECISION VECTORS WHOSE FIRST N COMPONENTS
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C CONTAIN REAL AND IMAGINARY PARTS FOR THE SEQUENCE
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C CY(I)=Y(FNU+I-1,Z) OR
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C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N
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C DEPENDING ON KODE.
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C NZ - NZ=0 , A NORMAL RETURN
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C NZ.GT.0 , NZ COMPONENTS OF CY SET TO ZERO DUE TO
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C UNDERFLOW (GENERALLY ON KODE=2)
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C IERR - ERROR FLAG
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C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
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C IERR=1, INPUT ERROR - NO COMPUTATION
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C IERR=2, OVERFLOW - NO COMPUTATION, FNU IS
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C TOO LARGE OR CABS(Z) IS TOO SMALL OR BOTH
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C IERR=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
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C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
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C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
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C ACCURACY
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C IERR=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
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C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
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C CANCE BY ARGUMENT REDUCTION
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C IERR=5, ERROR - NO COMPUTATION,
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C ALGORITHM TERMINATION CONDITION NOT MET
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C
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C***LONG DESCRIPTION
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C
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C THE COMPUTATION IS CARRIED OUT BY THE FORMULA
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C
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C Y(FNU,Z)=0.5*(H(1,FNU,Z)-H(2,FNU,Z))/I
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C
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C WHERE I**2 = -1 AND THE HANKEL BESSEL FUNCTIONS H(1,FNU,Z)
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C AND H(2,FNU,Z) ARE CALCULATED IN CBESH.
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C
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C FOR NEGATIVE ORDERS,THE FORMULA
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C
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C Y(-FNU,Z) = Y(FNU,Z)*COS(PI*FNU) + J(FNU,Z)*SIN(PI*FNU)
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C
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C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO HALF ODD
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C INTEGERS THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE
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C POSITIVE HALF ODD INTEGER,THE MAGNITUDE OF Y(-FNU,Z)=J(FNU,Z)*
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C SIN(PI*FNU) IS A LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS
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C NOT A HALF ODD INTEGER, Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A
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C LARGE POSITIVE POWER OF TEN AND THE MOST THAT THE SECOND TERM
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C CAN BE REDUCED IS BY UNIT ROUNDOFF FROM THE COEFFICIENT. THUS,
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C WIDE CHANGES CAN OCCUR WITHIN UNIT ROUNDOFF OF A LARGE HALF
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C ODD INTEGER. HERE, LARGE MEANS FNU.GT.CABS(Z).
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C
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C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
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C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
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C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
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C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
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C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
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C IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS
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C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
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C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
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C LOST AND IERR=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
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C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
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C INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
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C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
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C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
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C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
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C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
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C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
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C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
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C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
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C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
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C
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C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
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C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
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C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
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C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
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C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
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C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
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C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
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C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
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C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
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C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
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C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
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C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
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C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
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C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
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C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
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C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
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C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
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C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
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C OR -PI/2+P.
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C
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C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
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C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
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C COMMERCE, 1955.
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C
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C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
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C BY D. E. AMOS, SAND83-0083, MAY, 1983.
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C
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C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
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C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
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C
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C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
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C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
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C 1018, MAY, 1985
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C
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C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
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C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
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C MATH. SOFTWARE, 1986
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C
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C***ROUTINES CALLED ZBESH,I1MACH,D1MACH
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C***END PROLOGUE ZBESY
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C
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C COMPLEX CWRK,CY,C1,C2,EX,HCI,Z,ZU,ZV
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DOUBLE PRECISION CWRKI, CWRKR, CYI, CYR, C1I, C1R, C2I, C2R,
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* ELIM, EXI, EXR, EY, FNU, HCII, STI, STR, TAY, ZI, ZR, DEXP,
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* D1MACH, ASCLE, RTOL, ATOL, AA, BB, TOL
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INTEGER I, IERR, K, KODE, K1, K2, N, NZ, NZ1, NZ2, I1MACH
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DIMENSION CYR(N), CYI(N), CWRKR(N), CWRKI(N)
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C***FIRST EXECUTABLE STATEMENT ZBESY
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IERR = 0
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NZ=0
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IF (ZR.EQ.0.0D0 .AND. ZI.EQ.0.0D0) IERR=1
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IF (FNU.LT.0.0D0) IERR=1
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IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
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IF (N.LT.1) IERR=1
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IF (IERR.NE.0) RETURN
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HCII = 0.5D0
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CALL ZBESH(ZR, ZI, FNU, KODE, 1, N, CYR, CYI, NZ1, IERR)
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IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
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CALL ZBESH(ZR, ZI, FNU, KODE, 2, N, CWRKR, CWRKI, NZ2, IERR)
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IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
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NZ = MIN0(NZ1,NZ2)
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IF (KODE.EQ.2) GO TO 60
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DO 50 I=1,N
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STR = CWRKR(I) - CYR(I)
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STI = CWRKI(I) - CYI(I)
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CYR(I) = -STI*HCII
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CYI(I) = STR*HCII
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50 CONTINUE
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RETURN
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60 CONTINUE
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TOL = DMAX1(D1MACH(4),1.0D-18)
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K1 = I1MACH(15)
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K2 = I1MACH(16)
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K = MIN0(IABS(K1),IABS(K2))
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R1M5 = D1MACH(5)
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C-----------------------------------------------------------------------
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C ELIM IS THE APPROXIMATE EXPONENTIAL UNDER- AND OVERFLOW LIMIT
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C-----------------------------------------------------------------------
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ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
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EXR = DCOS(ZR)
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EXI = DSIN(ZR)
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EY = 0.0D0
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TAY = DABS(ZI+ZI)
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IF (TAY.LT.ELIM) EY = DEXP(-TAY)
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IF (ZI.LT.0.0D0) GO TO 90
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C1R = EXR*EY
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C1I = EXI*EY
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C2R = EXR
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C2I = -EXI
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70 CONTINUE
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NZ = 0
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RTOL = 1.0D0/TOL
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ASCLE = D1MACH(1)*RTOL*1.0D+3
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DO 80 I=1,N
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C STR = C1R*CYR(I) - C1I*CYI(I)
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C STI = C1R*CYI(I) + C1I*CYR(I)
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C STR = -STR + C2R*CWRKR(I) - C2I*CWRKI(I)
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C STI = -STI + C2R*CWRKI(I) + C2I*CWRKR(I)
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C CYR(I) = -STI*HCII
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C CYI(I) = STR*HCII
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AA = CWRKR(I)
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BB = CWRKI(I)
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ATOL = 1.0D0
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IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 75
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AA = AA*RTOL
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BB = BB*RTOL
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ATOL = TOL
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75 CONTINUE
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STR = (AA*C2R - BB*C2I)*ATOL
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STI = (AA*C2I + BB*C2R)*ATOL
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AA = CYR(I)
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BB = CYI(I)
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ATOL = 1.0D0
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IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 85
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AA = AA*RTOL
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BB = BB*RTOL
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ATOL = TOL
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85 CONTINUE
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STR = STR - (AA*C1R - BB*C1I)*ATOL
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STI = STI - (AA*C1I + BB*C1R)*ATOL
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CYR(I) = -STI*HCII
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CYI(I) = STR*HCII
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IF (STR.EQ.0.0D0 .AND. STI.EQ.0.0D0 .AND. EY.EQ.0.0D0) NZ = NZ
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* + 1
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80 CONTINUE
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RETURN
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90 CONTINUE
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C1R = EXR
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C1I = EXI
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C2R = EXR*EY
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C2I = -EXI*EY
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GO TO 70
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170 CONTINUE
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NZ = 0
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RETURN
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END
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