SUBROUTINE ZBIRY(ZR, ZI, ID, KODE, BIR, BII, IERR) C***BEGIN PROLOGUE ZBIRY 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 BI(Z) AND DBI(Z) FOR COMPLEX Z C***DESCRIPTION C C ***A DOUBLE PRECISION ROUTINE*** C ON KODE=1, CBIRY COMPUTES THE COMPLEX AIRY FUNCTION BI(Z) OR C ITS DERIVATIVE DBI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON C KODE=2, A SCALING OPTION CEXP(-AXZTA)*BI(Z) OR CEXP(-AXZTA)* C DBI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL BEHAVIOR IN C BOTH THE LEFT AND RIGHT HALF PLANES WHERE C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) AND AXZTA=ABS(XZTA). C DEFINITIONS 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 BI=BI(Z) ON ID=0 OR C BI=DBI(Z)/DZ ON ID=1 C = 2 RETURNS C BI=CEXP(-AXZTA)*BI(Z) ON ID=0 OR C BI=CEXP(-AXZTA)*DBI(Z)/DZ ON ID=1 WHERE C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) C AND AXZTA=ABS(XZTA) C C OUTPUT BIR,BII ARE DOUBLE PRECISION C BIR,BII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND C KODE 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(Z) 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 BI AND DBI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE I BESSEL C FUNCTIONS BY C C BI(Z)=C*SQRT(Z)*( I(-1/3,ZTA) + I(1/3,ZTA) ) C DBI(Z)=C * Z * ( I(-2/3,ZTA) + I(2/3,ZTA) ) C C=1.0/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 ZBINU,AZABS,ZDIV,AZSQRT,D1MACH,I1MACH C***END PROLOGUE ZBIRY C COMPLEX BI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3 DOUBLE PRECISION AA, AD, AK, ALIM, ATRM, AZ, AZ3, BB, BII, BIR, * BK, CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, * DIG, DK, D1, D2, EAA, ELIM, FID, FMR, FNU, FNUL, PI, RL, R1M5, * SFAC, STI, STR, S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, * TRM2R, TTH, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, AZABS INTEGER ID, IERR, K, KODE, K1, K2, NZ, I1MACH DIMENSION CYR(2), CYI(2) DATA TTH, C1, C2, COEF, PI /6.66666666666666667D-01, * 6.14926627446000736D-01,4.48288357353826359D-01, * 5.77350269189625765D-01,3.14159265358979324D+00/ DATA CONER, CONEI /1.0D0,0.0D0/ C***FIRST EXECUTABLE STATEMENT ZBIRY 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.1.0E0) 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 130 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 BIR = C1*S1R + C2*(ZR*S2R-ZI*S2I) BII = C1*S1I + 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) AA = ZTAR AA = -DABS(AA) EAA = DEXP(AA) BIR = BIR*EAA BII = BII*EAA RETURN 50 CONTINUE BIR = S2R*C2 BII = S2I*C2 IF (AZ.LE.TOL) GO TO 60 CC = C1/(1.0D0+FID) STR = S1R*ZR - S1I*ZI STI = S1R*ZI + S1I*ZR BIR = BIR + CC*(STR*ZR-STI*ZI) BII = BII + 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) AA = ZTAR AA = -DABS(AA) EAA = DEXP(AA) BIR = BIR*EAA BII = BII*EAA 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 1.0E-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----------------------------------------------------------------------- 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 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----------------------------------------------------------------------- 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 .OR. ZR.GT.0.0D0) GO TO 90 ZTAR = 0.0D0 ZTAI = AK 90 CONTINUE AA = ZTAR IF (KODE.EQ.2) GO TO 100 C----------------------------------------------------------------------- C OVERFLOW TEST C----------------------------------------------------------------------- BB = DABS(AA) IF (BB.LT.ALIM) GO TO 100 BB = BB + 0.25D0*DLOG(AZ) SFAC = TOL IF (BB.GT.ELIM) GO TO 190 100 CONTINUE FMR = 0.0D0 IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110 FMR = PI IF (ZI.LT.0.0D0) FMR = -PI ZTAR = -ZTAR ZTAI = -ZTAI 110 CONTINUE C----------------------------------------------------------------------- C AA=FACTOR FOR ANALYTIC CONTINUATION OF I(FNU,ZTA) C KODE=2 RETURNS EXP(-ABS(XZTA))*I(FNU,ZTA) FROM CBESI C----------------------------------------------------------------------- CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, RL, FNUL, TOL, * ELIM, ALIM) IF (NZ.LT.0) GO TO 200 AA = FMR*FNU Z3R = SFAC STR = DCOS(AA) STI = DSIN(AA) S1R = (STR*CYR(1)-STI*CYI(1))*Z3R S1I = (STR*CYI(1)+STI*CYR(1))*Z3R FNU = (2.0D0-FID)/3.0D0 CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 2, CYR, CYI, NZ, RL, FNUL, TOL, * ELIM, ALIM) CYR(1) = CYR(1)*Z3R CYI(1) = CYI(1)*Z3R CYR(2) = CYR(2)*Z3R CYI(2) = CYI(2)*Z3R C----------------------------------------------------------------------- C BACKWARD RECUR ONE STEP FOR ORDERS -1/3 OR -2/3 C----------------------------------------------------------------------- CALL ZDIV(CYR(1), CYI(1), ZTAR, ZTAI, STR, STI) S2R = (FNU+FNU)*STR + CYR(2) S2I = (FNU+FNU)*STI + CYI(2) AA = FMR*(FNU-1.0D0) STR = DCOS(AA) STI = DSIN(AA) S1R = COEF*(S1R+S2R*STR-S2I*STI) S1I = COEF*(S1I+S2R*STI+S2I*STR) IF (ID.EQ.1) GO TO 120 STR = CSQR*S1R - CSQI*S1I S1I = CSQR*S1I + CSQI*S1R S1R = STR BIR = S1R/SFAC BII = S1I/SFAC RETURN 120 CONTINUE STR = ZR*S1R - ZI*S1I S1I = ZR*S1I + ZI*S1R S1R = STR BIR = S1R/SFAC BII = S1I/SFAC RETURN 130 CONTINUE AA = C1*(1.0D0-FID) + FID*C2 BIR = AA BII = 0.0D0 RETURN 190 CONTINUE IERR=2 NZ=0 RETURN 200 CONTINUE IF(NZ.EQ.(-1)) GO TO 190 NZ=0 IERR=5 RETURN 260 CONTINUE IERR=4 NZ=0 RETURN END