2019-03-19 16:42:47 +02:00
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SUBROUTINE ZAIRY(ZR, ZI, ID, KODE, AIR, AII, NZ, IERR)
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C***BEGIN PROLOGUE ZAIRY
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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 AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD
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C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
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C***PURPOSE TO COMPUTE AIRY FUNCTIONS AI(Z) AND DAI(Z) FOR COMPLEX Z
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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 ON KODE=1, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR
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C ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON
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C KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)*
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C DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN
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C -PI/3.LT.ARG(Z).LT.PI/3 AND THE EXPONENTIAL GROWTH IN
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C PI/3.LT.ABS(ARG(Z)).LT.PI WHERE ZTA=(2/3)*Z*CSQRT(Z).
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C
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C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN
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C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED
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C FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS.
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C DEFINITIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
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C MATHEMATICAL FUNCTIONS (REF. 1).
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C
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C INPUT ZR,ZI ARE DOUBLE PRECISION
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C ZR,ZI - Z=CMPLX(ZR,ZI)
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C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1
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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 AI=AI(Z) ON ID=0 OR
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C AI=DAI(Z)/DZ ON ID=1
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C = 2 RETURNS
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C AI=CEXP(ZTA)*AI(Z) ON ID=0 OR
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C AI=CEXP(ZTA)*DAI(Z)/DZ ON ID=1 WHERE
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C ZTA=(2/3)*Z*CSQRT(Z)
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C
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C OUTPUT AIR,AII ARE DOUBLE PRECISION
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C AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND
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C KODE
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C NZ - UNDERFLOW INDICATOR
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C NZ= 0 , NORMAL RETURN
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C NZ= 1 , AI=CMPLX(0.0D0,0.0D0) DUE TO UNDERFLOW IN
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C -PI/3.LT.ARG(Z).LT.PI/3 ON KODE=1
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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, REAL(ZTA)
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C TOO LARGE ON KODE=1
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C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED
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C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
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C PRODUCE LESS THAN HALF OF MACHINE ACCURACY
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C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION
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C COMPLETE LOSS OF ACCURACY BY ARGUMENT
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C 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 AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL
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C FUNCTIONS BY
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C
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C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA)
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C C=1.0/(PI*SQRT(3.0))
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C ZTA=(2/3)*Z**(3/2)
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C
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C WITH THE POWER SERIES FOR CABS(Z).LE.1.0.
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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 IS LARGE, LOSSES
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C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
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C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
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C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
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C FLAG 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 ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
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C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT
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C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
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C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA
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C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
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C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
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C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
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C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
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C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
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C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
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C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
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C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
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C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
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C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER
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C 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 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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2019-03-20 12:25:42 +02:00
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C***ROUTINES CALLED ZACAI,ZBKNU,AZEXP,AZSQRT,I1MACH,D1MACH
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C***END PROLOGUE ZAIRY
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C COMPLEX AI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3
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DOUBLE PRECISION AA, AD, AII, AIR, AK, ALIM, ATRM, AZ, AZ3, BK,
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* CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, DIG,
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* DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR,
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* S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI,
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* ZEROR, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, AZABS, ALAZ, BB
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INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH
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DIMENSION CYR(1), CYI(1)
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DATA TTH, C1, C2, COEF /6.66666666666666667D-01,
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* 3.55028053887817240D-01,2.58819403792806799D-01,
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* 1.83776298473930683D-01/
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DATA ZEROR, ZEROI, CONER, CONEI /0.0D0,0.0D0,1.0D0,0.0D0/
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C***FIRST EXECUTABLE STATEMENT ZAIRY
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IERR = 0
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NZ=0
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IF (ID.LT.0 .OR. ID.GT.1) IERR=1
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IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
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IF (IERR.NE.0) RETURN
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AZ = AZABS(ZR,ZI)
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TOL = DMAX1(D1MACH(4),1.0D-18)
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FID = DBLE(FLOAT(ID))
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IF (AZ.GT.1.0D0) GO TO 70
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C-----------------------------------------------------------------------
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C POWER SERIES FOR CABS(Z).LE.1.
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C-----------------------------------------------------------------------
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S1R = CONER
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S1I = CONEI
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S2R = CONER
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S2I = CONEI
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IF (AZ.LT.TOL) GO TO 170
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AA = AZ*AZ
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IF (AA.LT.TOL/AZ) GO TO 40
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TRM1R = CONER
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TRM1I = CONEI
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TRM2R = CONER
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TRM2I = CONEI
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ATRM = 1.0D0
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STR = ZR*ZR - ZI*ZI
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STI = ZR*ZI + ZI*ZR
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Z3R = STR*ZR - STI*ZI
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Z3I = STR*ZI + STI*ZR
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AZ3 = AZ*AA
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AK = 2.0D0 + FID
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BK = 3.0D0 - FID - FID
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CK = 4.0D0 - FID
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DK = 3.0D0 + FID + FID
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D1 = AK*DK
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D2 = BK*CK
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AD = DMIN1(D1,D2)
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AK = 24.0D0 + 9.0D0*FID
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BK = 30.0D0 - 9.0D0*FID
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DO 30 K=1,25
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STR = (TRM1R*Z3R-TRM1I*Z3I)/D1
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TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1
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TRM1R = STR
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S1R = S1R + TRM1R
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S1I = S1I + TRM1I
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STR = (TRM2R*Z3R-TRM2I*Z3I)/D2
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TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2
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TRM2R = STR
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S2R = S2R + TRM2R
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S2I = S2I + TRM2I
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ATRM = ATRM*AZ3/AD
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D1 = D1 + AK
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D2 = D2 + BK
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AD = DMIN1(D1,D2)
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IF (ATRM.LT.TOL*AD) GO TO 40
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AK = AK + 18.0D0
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BK = BK + 18.0D0
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30 CONTINUE
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40 CONTINUE
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IF (ID.EQ.1) GO TO 50
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AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I)
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AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R)
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IF (KODE.EQ.1) RETURN
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CALL AZSQRT(ZR, ZI, STR, STI)
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ZTAR = TTH*(ZR*STR-ZI*STI)
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ZTAI = TTH*(ZR*STI+ZI*STR)
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CALL AZEXP(ZTAR, ZTAI, STR, STI)
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PTR = AIR*STR - AII*STI
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AII = AIR*STI + AII*STR
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AIR = PTR
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RETURN
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50 CONTINUE
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AIR = -S2R*C2
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AII = -S2I*C2
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IF (AZ.LE.TOL) GO TO 60
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STR = ZR*S1R - ZI*S1I
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STI = ZR*S1I + ZI*S1R
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CC = C1/(1.0D0+FID)
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AIR = AIR + CC*(STR*ZR-STI*ZI)
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AII = AII + CC*(STR*ZI+STI*ZR)
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60 CONTINUE
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IF (KODE.EQ.1) RETURN
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CALL AZSQRT(ZR, ZI, STR, STI)
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ZTAR = TTH*(ZR*STR-ZI*STI)
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ZTAI = TTH*(ZR*STI+ZI*STR)
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CALL AZEXP(ZTAR, ZTAI, STR, STI)
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PTR = STR*AIR - STI*AII
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AII = STR*AII + STI*AIR
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AIR = PTR
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RETURN
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C-----------------------------------------------------------------------
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C CASE FOR CABS(Z).GT.1.0
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C-----------------------------------------------------------------------
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70 CONTINUE
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FNU = (1.0D0+FID)/3.0D0
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C-----------------------------------------------------------------------
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C SET PARAMETERS RELATED TO MACHINE CONSTANTS.
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C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-18.
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C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
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C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND
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C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
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C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
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C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
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C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
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|
C-----------------------------------------------------------------------
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K1 = I1MACH(15)
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K2 = I1MACH(16)
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R1M5 = D1MACH(5)
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K = MIN0(IABS(K1),IABS(K2))
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ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
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|
K1 = I1MACH(14) - 1
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|
AA = R1M5*DBLE(FLOAT(K1))
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DIG = DMIN1(AA,18.0D0)
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AA = AA*2.303D0
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|
ALIM = ELIM + DMAX1(-AA,-41.45D0)
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RL = 1.2D0*DIG + 3.0D0
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ALAZ = DLOG(AZ)
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C--------------------------------------------------------------------------
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C TEST FOR PROPER RANGE
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|
C-----------------------------------------------------------------------
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AA=0.5D0/TOL
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BB=DBLE(FLOAT(I1MACH(9)))*0.5D0
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AA=DMIN1(AA,BB)
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AA=AA**TTH
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IF (AZ.GT.AA) GO TO 260
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AA=DSQRT(AA)
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IF (AZ.GT.AA) IERR=3
|
2019-03-20 12:25:42 +02:00
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|
CALL AZSQRT(ZR, ZI, CSQR, CSQI)
|
2019-03-19 16:42:47 +02:00
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|
ZTAR = TTH*(ZR*CSQR-ZI*CSQI)
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|
ZTAI = TTH*(ZR*CSQI+ZI*CSQR)
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|
C-----------------------------------------------------------------------
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|
C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL
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|
C-----------------------------------------------------------------------
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IFLAG = 0
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SFAC = 1.0D0
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AK = ZTAI
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|
IF (ZR.GE.0.0D0) GO TO 80
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|
BK = ZTAR
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|
CK = -DABS(BK)
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|
ZTAR = CK
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ZTAI = AK
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80 CONTINUE
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IF (ZI.NE.0.0D0) GO TO 90
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|
IF (ZR.GT.0.0D0) GO TO 90
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|
ZTAR = 0.0D0
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|
ZTAI = AK
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|
90 CONTINUE
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|
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|
AA = ZTAR
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|
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|
|
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
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|
|
|
|
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
|
