00001 SUBROUTINE ZAIRY(ZR, ZI, ID, KODE, AIR, AII, NZ, IERR) 00002 C***BEGIN PROLOGUE ZAIRY 00003 C***DATE WRITTEN 830501 (YYMMDD) 00004 C***REVISION DATE 890801 (YYMMDD) 00005 C***CATEGORY NO. B5K 00006 C***KEYWORDS AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD 00007 C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES 00008 C***PURPOSE TO COMPUTE AIRY FUNCTIONS AI(Z) AND DAI(Z) FOR COMPLEX Z 00009 C***DESCRIPTION 00010 C 00011 C ***A DOUBLE PRECISION ROUTINE*** 00012 C ON KODE=1, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR 00013 C ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON 00014 C KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)* 00015 C DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN 00016 C -PI/3.LT.ARG(Z).LT.PI/3 AND THE EXPONENTIAL GROWTH IN 00017 C PI/3.LT.ABS(ARG(Z)).LT.PI WHERE ZTA=(2/3)*Z*CSQRT(Z). 00018 C 00019 C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN 00020 C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED 00021 C FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS. 00022 C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF 00023 C MATHEMATICAL FUNCTIONS (REF. 1). 00024 C 00025 C INPUT ZR,ZI ARE DOUBLE PRECISION 00026 C ZR,ZI - Z=CMPLX(ZR,ZI) 00027 C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1 00028 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION 00029 C KODE= 1 RETURNS 00030 C AI=AI(Z) ON ID=0 OR 00031 C AI=DAI(Z)/DZ ON ID=1 00032 C = 2 RETURNS 00033 C AI=CEXP(ZTA)*AI(Z) ON ID=0 OR 00034 C AI=CEXP(ZTA)*DAI(Z)/DZ ON ID=1 WHERE 00035 C ZTA=(2/3)*Z*CSQRT(Z) 00036 C 00037 C OUTPUT AIR,AII ARE DOUBLE PRECISION 00038 C AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND 00039 C KODE 00040 C NZ - UNDERFLOW INDICATOR 00041 C NZ= 0 , NORMAL RETURN 00042 C NZ= 1 , AI=CMPLX(0.0D0,0.0D0) DUE TO UNDERFLOW IN 00043 C -PI/3.LT.ARG(Z).LT.PI/3 ON KODE=1 00044 C IERR - ERROR FLAG 00045 C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED 00046 C IERR=1, INPUT ERROR - NO COMPUTATION 00047 C IERR=2, OVERFLOW - NO COMPUTATION, REAL(ZTA) 00048 C TOO LARGE ON KODE=1 00049 C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED 00050 C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION 00051 C PRODUCE LESS THAN HALF OF MACHINE ACCURACY 00052 C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION 00053 C COMPLETE LOSS OF ACCURACY BY ARGUMENT 00054 C REDUCTION 00055 C IERR=5, ERROR - NO COMPUTATION, 00056 C ALGORITHM TERMINATION CONDITION NOT MET 00057 C 00058 C***LONG DESCRIPTION 00059 C 00060 C AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL 00061 C FUNCTIONS BY 00062 C 00063 C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA) 00064 C C=1.0/(PI*SQRT(3.0)) 00065 C ZTA=(2/3)*Z**(3/2) 00066 C 00067 C WITH THE POWER SERIES FOR CABS(Z).LE.1.0. 00068 C 00069 C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE- 00070 C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES 00071 C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF 00072 C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR), 00073 C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR 00074 C FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS 00075 C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. 00076 C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN 00077 C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT 00078 C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE 00079 C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA 00080 C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, 00081 C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE 00082 C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE 00083 C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT- 00084 C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG- 00085 C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN 00086 C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN 00087 C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, 00088 C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE 00089 C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER 00090 C MACHINES. 00091 C 00092 C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX 00093 C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT 00094 C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE- 00095 C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE 00096 C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))), 00097 C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF 00098 C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY 00099 C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN 00100 C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY 00101 C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER 00102 C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, 00103 C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS 00104 C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER 00105 C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY 00106 C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER 00107 C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE 00108 C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES, 00109 C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, 00110 C OR -PI/2+P. 00111 C 00112 C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ 00113 C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF 00114 C COMMERCE, 1955. 00115 C 00116 C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT 00117 C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 00118 C 00119 C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX 00120 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- 00121 C 1018, MAY, 1985 00122 C 00123 C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX 00124 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. 00125 C MATH. SOFTWARE, 1986 00126 C 00127 C***ROUTINES CALLED ZACAI,ZBKNU,XZEXP,XZSQRT,I1MACH,D1MACH 00128 C***END PROLOGUE ZAIRY 00129 C COMPLEX AI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3 00130 DOUBLE PRECISION AA, AD, AII, AIR, AK, ALIM, ATRM, AZ, AZ3, BK, 00131 * CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, DIG, 00132 * DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR, 00133 * S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI, 00134 * ZEROR, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, XZABS, ALAZ, BB 00135 INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH 00136 DIMENSION CYR(1), CYI(1) 00137 DATA TTH, C1, C2, COEF /6.66666666666666667D-01, 00138 * 3.55028053887817240D-01,2.58819403792806799D-01, 00139 * 1.83776298473930683D-01/ 00140 DATA ZEROR, ZEROI, CONER, CONEI /0.0D0,0.0D0,1.0D0,0.0D0/ 00141 C***FIRST EXECUTABLE STATEMENT ZAIRY 00142 IERR = 0 00143 NZ=0 00144 IF (ID.LT.0 .OR. ID.GT.1) IERR=1 00145 IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1 00146 IF (IERR.NE.0) RETURN 00147 AZ = XZABS(ZR,ZI) 00148 TOL = DMAX1(D1MACH(4),1.0D-18) 00149 FID = DBLE(FLOAT(ID)) 00150 IF (AZ.GT.1.0D0) GO TO 70 00151 C----------------------------------------------------------------------- 00152 C POWER SERIES FOR CABS(Z).LE.1. 00153 C----------------------------------------------------------------------- 00154 S1R = CONER 00155 S1I = CONEI 00156 S2R = CONER 00157 S2I = CONEI 00158 IF (AZ.LT.TOL) GO TO 170 00159 AA = AZ*AZ 00160 IF (AA.LT.TOL/AZ) GO TO 40 00161 TRM1R = CONER 00162 TRM1I = CONEI 00163 TRM2R = CONER 00164 TRM2I = CONEI 00165 ATRM = 1.0D0 00166 STR = ZR*ZR - ZI*ZI 00167 STI = ZR*ZI + ZI*ZR 00168 Z3R = STR*ZR - STI*ZI 00169 Z3I = STR*ZI + STI*ZR 00170 AZ3 = AZ*AA 00171 AK = 2.0D0 + FID 00172 BK = 3.0D0 - FID - FID 00173 CK = 4.0D0 - FID 00174 DK = 3.0D0 + FID + FID 00175 D1 = AK*DK 00176 D2 = BK*CK 00177 AD = DMIN1(D1,D2) 00178 AK = 24.0D0 + 9.0D0*FID 00179 BK = 30.0D0 - 9.0D0*FID 00180 DO 30 K=1,25 00181 STR = (TRM1R*Z3R-TRM1I*Z3I)/D1 00182 TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1 00183 TRM1R = STR 00184 S1R = S1R + TRM1R 00185 S1I = S1I + TRM1I 00186 STR = (TRM2R*Z3R-TRM2I*Z3I)/D2 00187 TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2 00188 TRM2R = STR 00189 S2R = S2R + TRM2R 00190 S2I = S2I + TRM2I 00191 ATRM = ATRM*AZ3/AD 00192 D1 = D1 + AK 00193 D2 = D2 + BK 00194 AD = DMIN1(D1,D2) 00195 IF (ATRM.LT.TOL*AD) GO TO 40 00196 AK = AK + 18.0D0 00197 BK = BK + 18.0D0 00198 30 CONTINUE 00199 40 CONTINUE 00200 IF (ID.EQ.1) GO TO 50 00201 AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I) 00202 AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R) 00203 IF (KODE.EQ.1) RETURN 00204 CALL XZSQRT(ZR, ZI, STR, STI) 00205 ZTAR = TTH*(ZR*STR-ZI*STI) 00206 ZTAI = TTH*(ZR*STI+ZI*STR) 00207 CALL XZEXP(ZTAR, ZTAI, STR, STI) 00208 PTR = AIR*STR - AII*STI 00209 AII = AIR*STI + AII*STR 00210 AIR = PTR 00211 RETURN 00212 50 CONTINUE 00213 AIR = -S2R*C2 00214 AII = -S2I*C2 00215 IF (AZ.LE.TOL) GO TO 60 00216 STR = ZR*S1R - ZI*S1I 00217 STI = ZR*S1I + ZI*S1R 00218 CC = C1/(1.0D0+FID) 00219 AIR = AIR + CC*(STR*ZR-STI*ZI) 00220 AII = AII + CC*(STR*ZI+STI*ZR) 00221 60 CONTINUE 00222 IF (KODE.EQ.1) RETURN 00223 CALL XZSQRT(ZR, ZI, STR, STI) 00224 ZTAR = TTH*(ZR*STR-ZI*STI) 00225 ZTAI = TTH*(ZR*STI+ZI*STR) 00226 CALL XZEXP(ZTAR, ZTAI, STR, STI) 00227 PTR = STR*AIR - STI*AII 00228 AII = STR*AII + STI*AIR 00229 AIR = PTR 00230 RETURN 00231 C----------------------------------------------------------------------- 00232 C CASE FOR CABS(Z).GT.1.0 00233 C----------------------------------------------------------------------- 00234 70 CONTINUE 00235 FNU = (1.0D0+FID)/3.0D0 00236 C----------------------------------------------------------------------- 00237 C SET PARAMETERS RELATED TO MACHINE CONSTANTS. 00238 C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-18. 00239 C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. 00240 C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND 00241 C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR 00242 C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. 00243 C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. 00244 C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG). 00245 C----------------------------------------------------------------------- 00246 K1 = I1MACH(15) 00247 K2 = I1MACH(16) 00248 R1M5 = D1MACH(5) 00249 K = MIN0(IABS(K1),IABS(K2)) 00250 ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0) 00251 K1 = I1MACH(14) - 1 00252 AA = R1M5*DBLE(FLOAT(K1)) 00253 DIG = DMIN1(AA,18.0D0) 00254 AA = AA*2.303D0 00255 ALIM = ELIM + DMAX1(-AA,-41.45D0) 00256 RL = 1.2D0*DIG + 3.0D0 00257 ALAZ = DLOG(AZ) 00258 C-------------------------------------------------------------------------- 00259 C TEST FOR PROPER RANGE 00260 C----------------------------------------------------------------------- 00261 AA=0.5D0/TOL 00262 BB=DBLE(FLOAT(I1MACH(9)))*0.5D0 00263 AA=DMIN1(AA,BB) 00264 AA=AA**TTH 00265 IF (AZ.GT.AA) GO TO 260 00266 AA=DSQRT(AA) 00267 IF (AZ.GT.AA) IERR=3 00268 CALL XZSQRT(ZR, ZI, CSQR, CSQI) 00269 ZTAR = TTH*(ZR*CSQR-ZI*CSQI) 00270 ZTAI = TTH*(ZR*CSQI+ZI*CSQR) 00271 C----------------------------------------------------------------------- 00272 C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL 00273 C----------------------------------------------------------------------- 00274 IFLAG = 0 00275 SFAC = 1.0D0 00276 AK = ZTAI 00277 IF (ZR.GE.0.0D0) GO TO 80 00278 BK = ZTAR 00279 CK = -DABS(BK) 00280 ZTAR = CK 00281 ZTAI = AK 00282 80 CONTINUE 00283 IF (ZI.NE.0.0D0) GO TO 90 00284 IF (ZR.GT.0.0D0) GO TO 90 00285 ZTAR = 0.0D0 00286 ZTAI = AK 00287 90 CONTINUE 00288 AA = ZTAR 00289 IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110 00290 IF (KODE.EQ.2) GO TO 100 00291 C----------------------------------------------------------------------- 00292 C OVERFLOW TEST 00293 C----------------------------------------------------------------------- 00294 IF (AA.GT.(-ALIM)) GO TO 100 00295 AA = -AA + 0.25D0*ALAZ 00296 IFLAG = 1 00297 SFAC = TOL 00298 IF (AA.GT.ELIM) GO TO 270 00299 100 CONTINUE 00300 C----------------------------------------------------------------------- 00301 C CBKNU AND CACON RETURN EXP(ZTA)*K(FNU,ZTA) ON KODE=2 00302 C----------------------------------------------------------------------- 00303 MR = 1 00304 IF (ZI.LT.0.0D0) MR = -1 00305 CALL ZACAI(ZTAR, ZTAI, FNU, KODE, MR, 1, CYR, CYI, NN, RL, TOL, 00306 * ELIM, ALIM) 00307 IF (NN.LT.0) GO TO 280 00308 NZ = NZ + NN 00309 GO TO 130 00310 110 CONTINUE 00311 IF (KODE.EQ.2) GO TO 120 00312 C----------------------------------------------------------------------- 00313 C UNDERFLOW TEST 00314 C----------------------------------------------------------------------- 00315 IF (AA.LT.ALIM) GO TO 120 00316 AA = -AA - 0.25D0*ALAZ 00317 IFLAG = 2 00318 SFAC = 1.0D0/TOL 00319 IF (AA.LT.(-ELIM)) GO TO 210 00320 120 CONTINUE 00321 CALL ZBKNU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, TOL, ELIM, 00322 * ALIM) 00323 130 CONTINUE 00324 S1R = CYR(1)*COEF 00325 S1I = CYI(1)*COEF 00326 IF (IFLAG.NE.0) GO TO 150 00327 IF (ID.EQ.1) GO TO 140 00328 AIR = CSQR*S1R - CSQI*S1I 00329 AII = CSQR*S1I + CSQI*S1R 00330 RETURN 00331 140 CONTINUE 00332 AIR = -(ZR*S1R-ZI*S1I) 00333 AII = -(ZR*S1I+ZI*S1R) 00334 RETURN 00335 150 CONTINUE 00336 S1R = S1R*SFAC 00337 S1I = S1I*SFAC 00338 IF (ID.EQ.1) GO TO 160 00339 STR = S1R*CSQR - S1I*CSQI 00340 S1I = S1R*CSQI + S1I*CSQR 00341 S1R = STR 00342 AIR = S1R/SFAC 00343 AII = S1I/SFAC 00344 RETURN 00345 160 CONTINUE 00346 STR = -(S1R*ZR-S1I*ZI) 00347 S1I = -(S1R*ZI+S1I*ZR) 00348 S1R = STR 00349 AIR = S1R/SFAC 00350 AII = S1I/SFAC 00351 RETURN 00352 170 CONTINUE 00353 AA = 1.0D+3*D1MACH(1) 00354 S1R = ZEROR 00355 S1I = ZEROI 00356 IF (ID.EQ.1) GO TO 190 00357 IF (AZ.LE.AA) GO TO 180 00358 S1R = C2*ZR 00359 S1I = C2*ZI 00360 180 CONTINUE 00361 AIR = C1 - S1R 00362 AII = -S1I 00363 RETURN 00364 190 CONTINUE 00365 AIR = -C2 00366 AII = 0.0D0 00367 AA = DSQRT(AA) 00368 IF (AZ.LE.AA) GO TO 200 00369 S1R = 0.5D0*(ZR*ZR-ZI*ZI) 00370 S1I = ZR*ZI 00371 200 CONTINUE 00372 AIR = AIR + C1*S1R 00373 AII = AII + C1*S1I 00374 RETURN 00375 210 CONTINUE 00376 NZ = 1 00377 AIR = ZEROR 00378 AII = ZEROI 00379 RETURN 00380 270 CONTINUE 00381 NZ = 0 00382 IERR=2 00383 RETURN 00384 280 CONTINUE 00385 IF(NN.EQ.(-1)) GO TO 270 00386 NZ=0 00387 IERR=5 00388 RETURN 00389 260 CONTINUE 00390 IERR=4 00391 NZ=0 00392 RETURN 00393 END
1.7.1
