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- SUBROUTINE ZBESY(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, CWRKR, CWRKI,
- * IERR)
- C***BEGIN PROLOGUE ZBESY
- C***DATE WRITTEN 830501 (YYMMDD)
- C***REVISION DATE 890801 (YYMMDD)
- C***CATEGORY NO. B5K
- C***KEYWORDS Y-BESSEL FUNCTION,BESSEL FUNCTION OF COMPLEX ARGUMENT,
- C BESSEL FUNCTION OF SECOND KIND
- C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
- C***PURPOSE TO COMPUTE THE Y-BESSEL FUNCTION OF A COMPLEX ARGUMENT
- C***DESCRIPTION
- C
- C ***A DOUBLE PRECISION ROUTINE***
- C
- C ON KODE=1, CBESY COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
- C BESSEL FUNCTIONS CY(I)=Y(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, CBESY RETURNS THE SCALED
- C FUNCTIONS
- C
- C CY(I)=EXP(-ABS(Y))*Y(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 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 Y FUNCTION, FNU.GE.0.0D0
- C KODE - A PARAMETER TO INDICATE THE SCALING OPTION
- C KODE= 1 RETURNS
- C CY(I)=Y(FNU+I-1,Z), I=1,...,N
- C = 2 RETURNS
- C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,...,N
- C WHERE Y=AIMAG(Z)
- C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
- C CWRKR, - DOUBLE PRECISION WORK VECTORS OF DIMENSION AT
- C CWRKI AT LEAST 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)=Y(FNU+I-1,Z) OR
- C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N
- C DEPENDING ON KODE.
- C NZ - NZ=0 , A NORMAL RETURN
- C NZ.GT.0 , NZ COMPONENTS OF CY SET TO ZERO DUE TO
- C UNDERFLOW (GENERALLY ON KODE=2)
- 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 THE COMPUTATION IS CARRIED OUT BY THE FORMULA
- C
- C Y(FNU,Z)=0.5*(H(1,FNU,Z)-H(2,FNU,Z))/I
- C
- C WHERE I**2 = -1 AND THE HANKEL BESSEL FUNCTIONS H(1,FNU,Z)
- C AND H(2,FNU,Z) ARE CALCULATED IN CBESH.
- C
- C FOR NEGATIVE ORDERS,THE FORMULA
- C
- C Y(-FNU,Z) = Y(FNU,Z)*COS(PI*FNU) + J(FNU,Z)*SIN(PI*FNU)
- C
- C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO HALF ODD
- C INTEGERS THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE
- C POSITIVE HALF ODD INTEGER,THE MAGNITUDE OF Y(-FNU,Z)=J(FNU,Z)*
- C SIN(PI*FNU) IS A LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS
- C NOT A HALF ODD INTEGER, Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A
- C LARGE POSITIVE POWER OF TEN AND THE MOST THAT THE SECOND TERM
- C CAN BE REDUCED IS BY UNIT ROUNDOFF FROM THE COEFFICIENT. THUS,
- C WIDE CHANGES CAN OCCUR WITHIN UNIT ROUNDOFF OF A LARGE HALF
- C ODD INTEGER. HERE, 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=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 ZBESH,I1MACH,D1MACH
- C***END PROLOGUE ZBESY
- C
- C COMPLEX CWRK,CY,C1,C2,EX,HCI,Z,ZU,ZV
- DOUBLE PRECISION CWRKI, CWRKR, CYI, CYR, C1I, C1R, C2I, C2R,
- * ELIM, EXI, EXR, EY, FNU, HCII, STI, STR, TAY, ZI, ZR, DEXP,
- * D1MACH, ASCLE, RTOL, ATOL, AA, BB, TOL
- INTEGER I, IERR, K, KODE, K1, K2, N, NZ, NZ1, NZ2, I1MACH
- DIMENSION CYR(N), CYI(N), CWRKR(N), CWRKI(N)
- C***FIRST EXECUTABLE STATEMENT ZBESY
- IERR = 0
- NZ=0
- IF (ZR.EQ.0.0D0 .AND. ZI.EQ.0.0D0) 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
- HCII = 0.5D0
- CALL ZBESH(ZR, ZI, FNU, KODE, 1, N, CYR, CYI, NZ1, IERR)
- IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
- CALL ZBESH(ZR, ZI, FNU, KODE, 2, N, CWRKR, CWRKI, NZ2, IERR)
- IF (IERR.NE.0.AND.IERR.NE.3) GO TO 170
- NZ = MIN0(NZ1,NZ2)
- IF (KODE.EQ.2) GO TO 60
- DO 50 I=1,N
- STR = CWRKR(I) - CYR(I)
- STI = CWRKI(I) - CYI(I)
- CYR(I) = -STI*HCII
- CYI(I) = STR*HCII
- 50 CONTINUE
- RETURN
- 60 CONTINUE
- TOL = DMAX1(D1MACH(4),1.0D-18)
- K1 = I1MACH(15)
- K2 = I1MACH(16)
- K = MIN0(IABS(K1),IABS(K2))
- R1M5 = D1MACH(5)
- C-----------------------------------------------------------------------
- C ELIM IS THE APPROXIMATE EXPONENTIAL UNDER- AND OVERFLOW LIMIT
- C-----------------------------------------------------------------------
- ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
- EXR = DCOS(ZR)
- EXI = DSIN(ZR)
- EY = 0.0D0
- TAY = DABS(ZI+ZI)
- IF (TAY.LT.ELIM) EY = DEXP(-TAY)
- IF (ZI.LT.0.0D0) GO TO 90
- C1R = EXR*EY
- C1I = EXI*EY
- C2R = EXR
- C2I = -EXI
- 70 CONTINUE
- NZ = 0
- RTOL = 1.0D0/TOL
- ASCLE = D1MACH(1)*RTOL*1.0D+3
- DO 80 I=1,N
- C STR = C1R*CYR(I) - C1I*CYI(I)
- C STI = C1R*CYI(I) + C1I*CYR(I)
- C STR = -STR + C2R*CWRKR(I) - C2I*CWRKI(I)
- C STI = -STI + C2R*CWRKI(I) + C2I*CWRKR(I)
- C CYR(I) = -STI*HCII
- C CYI(I) = STR*HCII
- AA = CWRKR(I)
- BB = CWRKI(I)
- ATOL = 1.0D0
- IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 75
- AA = AA*RTOL
- BB = BB*RTOL
- ATOL = TOL
- 75 CONTINUE
- STR = (AA*C2R - BB*C2I)*ATOL
- STI = (AA*C2I + BB*C2R)*ATOL
- AA = CYR(I)
- BB = CYI(I)
- ATOL = 1.0D0
- IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 85
- AA = AA*RTOL
- BB = BB*RTOL
- ATOL = TOL
- 85 CONTINUE
- STR = STR - (AA*C1R - BB*C1I)*ATOL
- STI = STI - (AA*C1I + BB*C1R)*ATOL
- CYR(I) = -STI*HCII
- CYI(I) = STR*HCII
- IF (STR.EQ.0.0D0 .AND. STI.EQ.0.0D0 .AND. EY.EQ.0.0D0) NZ = NZ
- * + 1
- 80 CONTINUE
- RETURN
- 90 CONTINUE
- C1R = EXR
- C1I = EXI
- C2R = EXR*EY
- C2I = -EXI*EY
- GO TO 70
- 170 CONTINUE
- NZ = 0
- RETURN
- END
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