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OpenLibm/amos/zbesj.f
Viral B. Shah f7c5d2787a Add the amos library
2012-10-29 15:14:32 +05:30

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SUBROUTINE ZBESJ(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR)
C***BEGIN PROLOGUE ZBESJ
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 ***A DOUBLE PRECISION ROUTINE***
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 ZR,ZI,FNU ARE DOUBLE PRECISION
C ZR,ZI - Z=CMPLX(ZR,ZI), -PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL J FUNCTION, FNU.GE.0.0D0
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,...,N
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
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)=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 ZERO DUE
C TO UNDERFLOW, CY(I)=CMPLX(0.0D0,0.0D0),
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=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 ZBINU,I1MACH,D1MACH
C***END PROLOGUE ZBESJ
C
C COMPLEX CI,CSGN,CY,Z,ZN
DOUBLE PRECISION AA, ALIM, ARG, CII, CSGNI, CSGNR, CYI, CYR, DIG,
* ELIM, FNU, FNUL, HPI, RL, R1M5, STR, TOL, ZI, ZNI, ZNR, ZR,
* D1MACH, BB, FN, AZ, ZABS, ASCLE, RTOL, ATOL, STI
INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, N, NL, NZ, I1MACH
DIMENSION CYR(N), CYI(N)
DATA HPI /1.57079632679489662D0/
C
C***FIRST EXECUTABLE STATEMENT ZBESJ
IERR = 0
NZ=0
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
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)
RL = 1.2D0*DIG + 3.0D0
FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0)
C-----------------------------------------------------------------------
C TEST FOR PROPER RANGE
C-----------------------------------------------------------------------
AZ = ZABS(COMPLEX(ZR,ZI))
FN = FNU+DBLE(FLOAT(N-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 CALCULATE CSGN=EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
C WHEN FNU IS LARGE
C-----------------------------------------------------------------------
CII = 1.0D0
INU = INT(SNGL(FNU))
INUH = INU/2
IR = INU - 2*INUH
ARG = (FNU-DBLE(FLOAT(INU-IR)))*HPI
CSGNR = DCOS(ARG)
CSGNI = DSIN(ARG)
IF (MOD(INUH,2).EQ.0) GO TO 40
CSGNR = -CSGNR
CSGNI = -CSGNI
40 CONTINUE
C-----------------------------------------------------------------------
C ZN IS IN THE RIGHT HALF PLANE
C-----------------------------------------------------------------------
ZNR = ZI
ZNI = -ZR
IF (ZI.GE.0.0D0) GO TO 50
ZNR = -ZNR
ZNI = -ZNI
CSGNI = -CSGNI
CII = -CII
50 CONTINUE
CALL ZBINU(ZNR, ZNI, FNU, KODE, N, CYR, CYI, NZ, RL, FNUL, TOL,
* ELIM, ALIM)
IF (NZ.LT.0) GO TO 130
NL = N - NZ
IF (NL.EQ.0) RETURN
RTOL = 1.0D0/TOL
ASCLE = D1MACH(1)*RTOL*1.0D+3
DO 60 I=1,NL
C STR = CYR(I)*CSGNR - CYI(I)*CSGNI
C CYI(I) = CYR(I)*CSGNI + CYI(I)*CSGNR
C CYR(I) = STR
AA = CYR(I)
BB = CYI(I)
ATOL = 1.0D0
IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 55
AA = AA*RTOL
BB = BB*RTOL
ATOL = TOL
55 CONTINUE
STR = AA*CSGNR - BB*CSGNI
STI = AA*CSGNI + BB*CSGNR
CYR(I) = STR*ATOL
CYI(I) = STI*ATOL
STR = -CSGNI*CII
CSGNI = CSGNR*CII
CSGNR = STR
60 CONTINUE
RETURN
130 CONTINUE
IF(NZ.EQ.(-2)) GO TO 140
NZ = 0
IERR = 2
RETURN
140 CONTINUE
NZ=0
IERR=5
RETURN
260 CONTINUE
NZ=0
IERR=4
RETURN
END
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