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SUBROUTINE ZAIRY(ZR, ZI, ID, KODE, AIR, AII, NZ, IERR)
C***BEGIN PROLOGUE ZAIRY
C***DATE WRITTEN 830501 (YYMMDD)
C***REVISION DATE 890801 (YYMMDD)
C***CATEGORY NO. B5K
C***KEYWORDS AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD
C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C***PURPOSE TO COMPUTE AIRY FUNCTIONS AI(Z) AND DAI(Z) FOR COMPLEX Z
C***DESCRIPTION
C
C ***A DOUBLE PRECISION ROUTINE***
C ON KODE=1, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR
C ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON
C KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)*
C DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN
C -PI/3.LT.ARG(Z).LT.PI/3 AND THE EXPONENTIAL GROWTH IN
C PI/3.LT.ABS(ARG(Z)).LT.PI WHERE ZTA=(2/3)*Z*CSQRT(Z).
C
C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN
C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED
C FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS.
C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
C MATHEMATICAL FUNCTIONS (REF. 1).
C
C INPUT ZR,ZI ARE DOUBLE PRECISION
C ZR,ZI - Z=CMPLX(ZR,ZI)
C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1
C KODE - A PARAMETER TO INDICATE THE SCALING OPTION
C KODE= 1 RETURNS
C AI=AI(Z) ON ID=0 OR
C AI=DAI(Z)/DZ ON ID=1
C = 2 RETURNS
C AI=CEXP(ZTA)*AI(Z) ON ID=0 OR
C AI=CEXP(ZTA)*DAI(Z)/DZ ON ID=1 WHERE
C ZTA=(2/3)*Z*CSQRT(Z)
C
C OUTPUT AIR,AII ARE DOUBLE PRECISION
C AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND
C KODE
C NZ - UNDERFLOW INDICATOR
C NZ= 0 , NORMAL RETURN
C NZ= 1 , AI=CMPLX(0.0D0,0.0D0) DUE TO UNDERFLOW IN
C -PI/3.LT.ARG(Z).LT.PI/3 ON KODE=1
C IERR - ERROR FLAG
C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
C IERR=1, INPUT ERROR - NO COMPUTATION
C IERR=2, OVERFLOW - NO COMPUTATION, REAL(ZTA)
C TOO LARGE ON KODE=1
C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED
C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
C PRODUCE LESS THAN HALF OF MACHINE ACCURACY
C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION
C COMPLETE LOSS OF ACCURACY BY ARGUMENT
C REDUCTION
C IERR=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C***LONG DESCRIPTION
C
C AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL
C FUNCTIONS BY
C
C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA)
C C=1.0/(PI*SQRT(3.0))
C ZTA=(2/3)*Z**(3/2)
C
C WITH THE POWER SERIES FOR CABS(Z).LE.1.0.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES
C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
C FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS
C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT
C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA
C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER
C 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 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 ZACAI,ZBKNU,AZEXP,AZSQRT,I1MACH,D1MACH
C***END PROLOGUE ZAIRY
C COMPLEX AI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3
DOUBLE PRECISION AA, AD, AII, AIR, AK, ALIM, ATRM, AZ, AZ3, BK,
* CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, DIG,
* DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR,
* S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI,
* ZEROR, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, AZABS, ALAZ, BB
INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH
DIMENSION CYR(1), CYI(1)
DATA TTH, C1, C2, COEF /6.66666666666666667D-01,
* 3.55028053887817240D-01,2.58819403792806799D-01,
* 1.83776298473930683D-01/
DATA ZEROR, ZEROI, CONER, CONEI /0.0D0,0.0D0,1.0D0,0.0D0/
C***FIRST EXECUTABLE STATEMENT ZAIRY
IERR = 0
NZ=0
IF (ID.LT.0 .OR. ID.GT.1) IERR=1
IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
IF (IERR.NE.0) RETURN
AZ = AZABS(ZR,ZI)
TOL = DMAX1(D1MACH(4),1.0D-18)
FID = DBLE(FLOAT(ID))
IF (AZ.GT.1.0D0) GO TO 70
C-----------------------------------------------------------------------
C POWER SERIES FOR CABS(Z).LE.1.
C-----------------------------------------------------------------------
S1R = CONER
S1I = CONEI
S2R = CONER
S2I = CONEI
IF (AZ.LT.TOL) GO TO 170
AA = AZ*AZ
IF (AA.LT.TOL/AZ) GO TO 40
TRM1R = CONER
TRM1I = CONEI
TRM2R = CONER
TRM2I = CONEI
ATRM = 1.0D0
STR = ZR*ZR - ZI*ZI
STI = ZR*ZI + ZI*ZR
Z3R = STR*ZR - STI*ZI
Z3I = STR*ZI + STI*ZR
AZ3 = AZ*AA
AK = 2.0D0 + FID
BK = 3.0D0 - FID - FID
CK = 4.0D0 - FID
DK = 3.0D0 + FID + FID
D1 = AK*DK
D2 = BK*CK
AD = DMIN1(D1,D2)
AK = 24.0D0 + 9.0D0*FID
BK = 30.0D0 - 9.0D0*FID
DO 30 K=1,25
STR = (TRM1R*Z3R-TRM1I*Z3I)/D1
TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1
TRM1R = STR
S1R = S1R + TRM1R
S1I = S1I + TRM1I
STR = (TRM2R*Z3R-TRM2I*Z3I)/D2
TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2
TRM2R = STR
S2R = S2R + TRM2R
S2I = S2I + TRM2I
ATRM = ATRM*AZ3/AD
D1 = D1 + AK
D2 = D2 + BK
AD = DMIN1(D1,D2)
IF (ATRM.LT.TOL*AD) GO TO 40
AK = AK + 18.0D0
BK = BK + 18.0D0
30 CONTINUE
40 CONTINUE
IF (ID.EQ.1) GO TO 50
AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I)
AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R)
IF (KODE.EQ.1) RETURN
CALL AZSQRT(ZR, ZI, STR, STI)
ZTAR = TTH*(ZR*STR-ZI*STI)
ZTAI = TTH*(ZR*STI+ZI*STR)
CALL AZEXP(ZTAR, ZTAI, STR, STI)
PTR = AIR*STR - AII*STI
AII = AIR*STI + AII*STR
AIR = PTR
RETURN
50 CONTINUE
AIR = -S2R*C2
AII = -S2I*C2
IF (AZ.LE.TOL) GO TO 60
STR = ZR*S1R - ZI*S1I
STI = ZR*S1I + ZI*S1R
CC = C1/(1.0D0+FID)
AIR = AIR + CC*(STR*ZR-STI*ZI)
AII = AII + CC*(STR*ZI+STI*ZR)
60 CONTINUE
IF (KODE.EQ.1) RETURN
CALL AZSQRT(ZR, ZI, STR, STI)
ZTAR = TTH*(ZR*STR-ZI*STI)
ZTAI = TTH*(ZR*STI+ZI*STR)
CALL AZEXP(ZTAR, ZTAI, STR, STI)
PTR = STR*AIR - STI*AII
AII = STR*AII + STI*AIR
AIR = PTR
RETURN
C-----------------------------------------------------------------------
C CASE FOR CABS(Z).GT.1.0
C-----------------------------------------------------------------------
70 CONTINUE
FNU = (1.0D0+FID)/3.0D0
C-----------------------------------------------------------------------
C SET PARAMETERS RELATED TO MACHINE CONSTANTS.
C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-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-----------------------------------------------------------------------
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
ALAZ = DLOG(AZ)
C--------------------------------------------------------------------------
C TEST FOR PROPER RANGE
C-----------------------------------------------------------------------
AA=0.5D0/TOL
BB=DBLE(FLOAT(I1MACH(9)))*0.5D0
AA=DMIN1(AA,BB)
AA=AA**TTH
IF (AZ.GT.AA) GO TO 260
AA=DSQRT(AA)
IF (AZ.GT.AA) IERR=3
CALL AZSQRT(ZR, ZI, CSQR, CSQI)
ZTAR = TTH*(ZR*CSQR-ZI*CSQI)
ZTAI = TTH*(ZR*CSQI+ZI*CSQR)
C-----------------------------------------------------------------------
C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL
C-----------------------------------------------------------------------
IFLAG = 0
SFAC = 1.0D0
AK = ZTAI
IF (ZR.GE.0.0D0) GO TO 80
BK = ZTAR
CK = -DABS(BK)
ZTAR = CK
ZTAI = AK
80 CONTINUE
IF (ZI.NE.0.0D0) GO TO 90
IF (ZR.GT.0.0D0) GO TO 90
ZTAR = 0.0D0
ZTAI = AK
90 CONTINUE
AA = ZTAR
IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110
IF (KODE.EQ.2) GO TO 100
C-----------------------------------------------------------------------
C OVERFLOW TEST
C-----------------------------------------------------------------------
IF (AA.GT.(-ALIM)) GO TO 100
AA = -AA + 0.25D0*ALAZ
IFLAG = 1
SFAC = TOL
IF (AA.GT.ELIM) GO TO 270
100 CONTINUE
C-----------------------------------------------------------------------
C CBKNU AND CACON RETURN EXP(ZTA)*K(FNU,ZTA) ON KODE=2
C-----------------------------------------------------------------------
MR = 1
IF (ZI.LT.0.0D0) MR = -1
CALL ZACAI(ZTAR, ZTAI, FNU, KODE, MR, 1, CYR, CYI, NN, RL, TOL,
* ELIM, ALIM)
IF (NN.LT.0) GO TO 280
NZ = NZ + NN
GO TO 130
110 CONTINUE
IF (KODE.EQ.2) GO TO 120
C-----------------------------------------------------------------------
C UNDERFLOW TEST
C-----------------------------------------------------------------------
IF (AA.LT.ALIM) GO TO 120
AA = -AA - 0.25D0*ALAZ
IFLAG = 2
SFAC = 1.0D0/TOL
IF (AA.LT.(-ELIM)) GO TO 210
120 CONTINUE
CALL ZBKNU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, TOL, ELIM,
* ALIM)
130 CONTINUE
S1R = CYR(1)*COEF
S1I = CYI(1)*COEF
IF (IFLAG.NE.0) GO TO 150
IF (ID.EQ.1) GO TO 140
AIR = CSQR*S1R - CSQI*S1I
AII = CSQR*S1I + CSQI*S1R
RETURN
140 CONTINUE
AIR = -(ZR*S1R-ZI*S1I)
AII = -(ZR*S1I+ZI*S1R)
RETURN
150 CONTINUE
S1R = S1R*SFAC
S1I = S1I*SFAC
IF (ID.EQ.1) GO TO 160
STR = S1R*CSQR - S1I*CSQI
S1I = S1R*CSQI + S1I*CSQR
S1R = STR
AIR = S1R/SFAC
AII = S1I/SFAC
RETURN
160 CONTINUE
STR = -(S1R*ZR-S1I*ZI)
S1I = -(S1R*ZI+S1I*ZR)
S1R = STR
AIR = S1R/SFAC
AII = S1I/SFAC
RETURN
170 CONTINUE
AA = 1.0D+3*D1MACH(1)
S1R = ZEROR
S1I = ZEROI
IF (ID.EQ.1) GO TO 190
IF (AZ.LE.AA) GO TO 180
S1R = C2*ZR
S1I = C2*ZI
180 CONTINUE
AIR = C1 - S1R
AII = -S1I
RETURN
190 CONTINUE
AIR = -C2
AII = 0.0D0
AA = DSQRT(AA)
IF (AZ.LE.AA) GO TO 200
S1R = 0.5D0*(ZR*ZR-ZI*ZI)
S1I = ZR*ZI
200 CONTINUE
AIR = AIR + C1*S1R
AII = AII + C1*S1I
RETURN
210 CONTINUE
NZ = 1
AIR = ZEROR
AII = ZEROI
RETURN
270 CONTINUE
NZ = 0
IERR=2
RETURN
280 CONTINUE
IF(NN.EQ.(-1)) GO TO 270
NZ=0
IERR=5
RETURN
260 CONTINUE
IERR=4
NZ=0
RETURN
END
|
