github.com/gopherd/gonum@v0.0.4/mathext/internal/amos/amoslib/zbiry.f (about) 1 SUBROUTINE ZBIRY(ZR, ZI, ID, KODE, BIR, BII, IERR) 2 C***BEGIN PROLOGUE ZBIRY 3 C***DATE WRITTEN 830501 (YYMMDD) 4 C***REVISION DATE 890801 (YYMMDD) 5 C***CATEGORY NO. B5K 6 C***KEYWORDS AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD 7 C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES 8 C***PURPOSE TO COMPUTE AIRY FUNCTIONS BI(Z) AND DBI(Z) FOR COMPLEX Z 9 C***DESCRIPTION 10 C 11 C ***A DOUBLE PRECISION ROUTINE*** 12 C ON KODE=1, CBIRY COMPUTES THE COMPLEX AIRY FUNCTION BI(Z) OR 13 C ITS DERIVATIVE DBI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON 14 C KODE=2, A SCALING OPTION CEXP(-AXZTA)*BI(Z) OR CEXP(-AXZTA)* 15 C DBI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL BEHAVIOR IN 16 C BOTH THE LEFT AND RIGHT HALF PLANES WHERE 17 C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) AND AXZTA=ABS(XZTA). 18 C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF 19 C MATHEMATICAL FUNCTIONS (REF. 1). 20 C 21 C INPUT ZR,ZI ARE DOUBLE PRECISION 22 C ZR,ZI - Z=CMPLX(ZR,ZI) 23 C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1 24 C KODE - A PARAMETER TO INDICATE THE SCALING OPTION 25 C KODE= 1 RETURNS 26 C BI=BI(Z) ON ID=0 OR 27 C BI=DBI(Z)/DZ ON ID=1 28 C = 2 RETURNS 29 C BI=CEXP(-AXZTA)*BI(Z) ON ID=0 OR 30 C BI=CEXP(-AXZTA)*DBI(Z)/DZ ON ID=1 WHERE 31 C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) 32 C AND AXZTA=ABS(XZTA) 33 C 34 C OUTPUT BIR,BII ARE DOUBLE PRECISION 35 C BIR,BII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND 36 C KODE 37 C IERR - ERROR FLAG 38 C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED 39 C IERR=1, INPUT ERROR - NO COMPUTATION 40 C IERR=2, OVERFLOW - NO COMPUTATION, REAL(Z) 41 C TOO LARGE ON KODE=1 42 C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED 43 C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION 44 C PRODUCE LESS THAN HALF OF MACHINE ACCURACY 45 C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION 46 C COMPLETE LOSS OF ACCURACY BY ARGUMENT 47 C REDUCTION 48 C IERR=5, ERROR - NO COMPUTATION, 49 C ALGORITHM TERMINATION CONDITION NOT MET 50 C 51 C***LONG DESCRIPTION 52 C 53 C BI AND DBI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE I BESSEL 54 C FUNCTIONS BY 55 C 56 C BI(Z)=C*SQRT(Z)*( I(-1/3,ZTA) + I(1/3,ZTA) ) 57 C DBI(Z)=C * Z * ( I(-2/3,ZTA) + I(2/3,ZTA) ) 58 C C=1.0/SQRT(3.0) 59 C ZTA=(2/3)*Z**(3/2) 60 C 61 C WITH THE POWER SERIES FOR CABS(Z).LE.1.0. 62 C 63 C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE- 64 C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES 65 C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF 66 C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR), 67 C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR 68 C FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS 69 C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. 70 C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN 71 C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT 72 C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE 73 C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA 74 C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, 75 C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE 76 C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE 77 C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT- 78 C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG- 79 C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN 80 C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN 81 C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, 82 C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE 83 C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER 84 C MACHINES. 85 C 86 C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX 87 C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT 88 C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE- 89 C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE 90 C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))), 91 C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF 92 C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY 93 C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN 94 C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY 95 C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER 96 C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, 97 C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS 98 C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER 99 C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY 100 C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER 101 C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE 102 C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES, 103 C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, 104 C OR -PI/2+P. 105 C 106 C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ 107 C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF 108 C COMMERCE, 1955. 109 C 110 C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT 111 C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 112 C 113 C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX 114 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- 115 C 1018, MAY, 1985 116 C 117 C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX 118 C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. 119 C MATH. SOFTWARE, 1986 120 C 121 C***ROUTINES CALLED ZBINU,ZABS,ZDIV,ZSQRT,D1MACH,I1MACH 122 C***END PROLOGUE ZBIRY 123 C COMPLEX BI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3 124 DOUBLE PRECISION AA, AD, AK, ALIM, ATRM, AZ, AZ3, BB, BII, BIR, 125 * BK, CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, 126 * DIG, DK, D1, D2, EAA, ELIM, FID, FMR, FNU, FNUL, PI, RL, R1M5, 127 * SFAC, STI, STR, S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, 128 * TRM2R, TTH, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, ZABS 129 INTEGER ID, IERR, K, KODE, K1, K2, NZ, I1MACH 130 DIMENSION CYR(2), CYI(2) 131 DATA TTH, C1, C2, COEF, PI /6.66666666666666667D-01, 132 * 6.14926627446000736D-01,4.48288357353826359D-01, 133 * 5.77350269189625765D-01,3.14159265358979324D+00/ 134 DATA CONER, CONEI /1.0D0,0.0D0/ 135 C***FIRST EXECUTABLE STATEMENT ZBIRY 136 IERR = 0 137 NZ=0 138 IF (ID.LT.0 .OR. ID.GT.1) IERR=1 139 IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1 140 IF (IERR.NE.0) RETURN 141 AZ = ZABS(CMPLX(ZR,ZI,kind=KIND(1.0D0))) 142 TOL = DMAX1(D1MACH(4),1.0D-18) 143 FID = DBLE(FLOAT(ID)) 144 IF (AZ.GT.1.0E0) GO TO 70 145 C----------------------------------------------------------------------- 146 C POWER SERIES FOR CABS(Z).LE.1. 147 C----------------------------------------------------------------------- 148 S1R = CONER 149 S1I = CONEI 150 S2R = CONER 151 S2I = CONEI 152 IF (AZ.LT.TOL) GO TO 130 153 AA = AZ*AZ 154 IF (AA.LT.TOL/AZ) GO TO 40 155 TRM1R = CONER 156 TRM1I = CONEI 157 TRM2R = CONER 158 TRM2I = CONEI 159 ATRM = 1.0D0 160 STR = ZR*ZR - ZI*ZI 161 STI = ZR*ZI + ZI*ZR 162 Z3R = STR*ZR - STI*ZI 163 Z3I = STR*ZI + STI*ZR 164 AZ3 = AZ*AA 165 AK = 2.0D0 + FID 166 BK = 3.0D0 - FID - FID 167 CK = 4.0D0 - FID 168 DK = 3.0D0 + FID + FID 169 D1 = AK*DK 170 D2 = BK*CK 171 AD = DMIN1(D1,D2) 172 AK = 24.0D0 + 9.0D0*FID 173 BK = 30.0D0 - 9.0D0*FID 174 DO 30 K=1,25 175 STR = (TRM1R*Z3R-TRM1I*Z3I)/D1 176 TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1 177 TRM1R = STR 178 S1R = S1R + TRM1R 179 S1I = S1I + TRM1I 180 STR = (TRM2R*Z3R-TRM2I*Z3I)/D2 181 TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2 182 TRM2R = STR 183 S2R = S2R + TRM2R 184 S2I = S2I + TRM2I 185 ATRM = ATRM*AZ3/AD 186 D1 = D1 + AK 187 D2 = D2 + BK 188 AD = DMIN1(D1,D2) 189 IF (ATRM.LT.TOL*AD) GO TO 40 190 AK = AK + 18.0D0 191 BK = BK + 18.0D0 192 30 CONTINUE 193 40 CONTINUE 194 IF (ID.EQ.1) GO TO 50 195 BIR = C1*S1R + C2*(ZR*S2R-ZI*S2I) 196 BII = C1*S1I + C2*(ZR*S2I+ZI*S2R) 197 IF (KODE.EQ.1) RETURN 198 CALL ZSQRT(ZR, ZI, STR, STI) 199 ZTAR = TTH*(ZR*STR-ZI*STI) 200 ZTAI = TTH*(ZR*STI+ZI*STR) 201 AA = ZTAR 202 AA = -DABS(AA) 203 EAA = DEXP(AA) 204 BIR = BIR*EAA 205 BII = BII*EAA 206 RETURN 207 50 CONTINUE 208 BIR = S2R*C2 209 BII = S2I*C2 210 IF (AZ.LE.TOL) GO TO 60 211 CC = C1/(1.0D0+FID) 212 STR = S1R*ZR - S1I*ZI 213 STI = S1R*ZI + S1I*ZR 214 BIR = BIR + CC*(STR*ZR-STI*ZI) 215 BII = BII + CC*(STR*ZI+STI*ZR) 216 60 CONTINUE 217 IF (KODE.EQ.1) RETURN 218 CALL ZSQRT(ZR, ZI, STR, STI) 219 ZTAR = TTH*(ZR*STR-ZI*STI) 220 ZTAI = TTH*(ZR*STI+ZI*STR) 221 AA = ZTAR 222 AA = -DABS(AA) 223 EAA = DEXP(AA) 224 BIR = BIR*EAA 225 BII = BII*EAA 226 RETURN 227 C----------------------------------------------------------------------- 228 C CASE FOR CABS(Z).GT.1.0 229 C----------------------------------------------------------------------- 230 70 CONTINUE 231 FNU = (1.0D0+FID)/3.0D0 232 C----------------------------------------------------------------------- 233 C SET PARAMETERS RELATED TO MACHINE CONSTANTS. 234 C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18. 235 C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. 236 C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND 237 C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR 238 C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. 239 C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. 240 C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG). 241 C FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU. 242 C----------------------------------------------------------------------- 243 K1 = I1MACH(15) 244 K2 = I1MACH(16) 245 R1M5 = D1MACH(5) 246 K = MIN0(IABS(K1),IABS(K2)) 247 ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0) 248 K1 = I1MACH(14) - 1 249 AA = R1M5*DBLE(FLOAT(K1)) 250 DIG = DMIN1(AA,18.0D0) 251 AA = AA*2.303D0 252 ALIM = ELIM + DMAX1(-AA,-41.45D0) 253 RL = 1.2D0*DIG + 3.0D0 254 FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0) 255 C----------------------------------------------------------------------- 256 C TEST FOR RANGE 257 C----------------------------------------------------------------------- 258 AA=0.5D0/TOL 259 BB=DBLE(FLOAT(I1MACH(9)))*0.5D0 260 AA=DMIN1(AA,BB) 261 AA=AA**TTH 262 IF (AZ.GT.AA) GO TO 260 263 AA=DSQRT(AA) 264 IF (AZ.GT.AA) IERR=3 265 CALL ZSQRT(ZR, ZI, CSQR, CSQI) 266 ZTAR = TTH*(ZR*CSQR-ZI*CSQI) 267 ZTAI = TTH*(ZR*CSQI+ZI*CSQR) 268 C----------------------------------------------------------------------- 269 C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL 270 C----------------------------------------------------------------------- 271 SFAC = 1.0D0 272 AK = ZTAI 273 IF (ZR.GE.0.0D0) GO TO 80 274 BK = ZTAR 275 CK = -DABS(BK) 276 ZTAR = CK 277 ZTAI = AK 278 80 CONTINUE 279 IF (ZI.NE.0.0D0 .OR. ZR.GT.0.0D0) GO TO 90 280 ZTAR = 0.0D0 281 ZTAI = AK 282 90 CONTINUE 283 AA = ZTAR 284 IF (KODE.EQ.2) GO TO 100 285 C----------------------------------------------------------------------- 286 C OVERFLOW TEST 287 C----------------------------------------------------------------------- 288 BB = DABS(AA) 289 IF (BB.LT.ALIM) GO TO 100 290 BB = BB + 0.25D0*DLOG(AZ) 291 SFAC = TOL 292 IF (BB.GT.ELIM) GO TO 190 293 100 CONTINUE 294 FMR = 0.0D0 295 IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110 296 FMR = PI 297 IF (ZI.LT.0.0D0) FMR = -PI 298 ZTAR = -ZTAR 299 ZTAI = -ZTAI 300 110 CONTINUE 301 C----------------------------------------------------------------------- 302 C AA=FACTOR FOR ANALYTIC CONTINUATION OF I(FNU,ZTA) 303 C KODE=2 RETURNS EXP(-ABS(XZTA))*I(FNU,ZTA) FROM CBESI 304 C----------------------------------------------------------------------- 305 CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, RL, FNUL, TOL, 306 * ELIM, ALIM) 307 IF (NZ.LT.0) GO TO 200 308 AA = FMR*FNU 309 Z3R = SFAC 310 STR = DCOS(AA) 311 STI = DSIN(AA) 312 S1R = (STR*CYR(1)-STI*CYI(1))*Z3R 313 S1I = (STR*CYI(1)+STI*CYR(1))*Z3R 314 FNU = (2.0D0-FID)/3.0D0 315 CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 2, CYR, CYI, NZ, RL, FNUL, TOL, 316 * ELIM, ALIM) 317 CYR(1) = CYR(1)*Z3R 318 CYI(1) = CYI(1)*Z3R 319 CYR(2) = CYR(2)*Z3R 320 CYI(2) = CYI(2)*Z3R 321 C----------------------------------------------------------------------- 322 C BACKWARD RECUR ONE STEP FOR ORDERS -1/3 OR -2/3 323 C----------------------------------------------------------------------- 324 CALL ZDIV(CYR(1), CYI(1), ZTAR, ZTAI, STR, STI) 325 S2R = (FNU+FNU)*STR + CYR(2) 326 S2I = (FNU+FNU)*STI + CYI(2) 327 AA = FMR*(FNU-1.0D0) 328 STR = DCOS(AA) 329 STI = DSIN(AA) 330 S1R = COEF*(S1R+S2R*STR-S2I*STI) 331 S1I = COEF*(S1I+S2R*STI+S2I*STR) 332 IF (ID.EQ.1) GO TO 120 333 STR = CSQR*S1R - CSQI*S1I 334 S1I = CSQR*S1I + CSQI*S1R 335 S1R = STR 336 BIR = S1R/SFAC 337 BII = S1I/SFAC 338 RETURN 339 120 CONTINUE 340 STR = ZR*S1R - ZI*S1I 341 S1I = ZR*S1I + ZI*S1R 342 S1R = STR 343 BIR = S1R/SFAC 344 BII = S1I/SFAC 345 RETURN 346 130 CONTINUE 347 AA = C1*(1.0D0-FID) + FID*C2 348 BIR = AA 349 BII = 0.0D0 350 RETURN 351 190 CONTINUE 352 IERR=2 353 NZ=0 354 RETURN 355 200 CONTINUE 356 IF(NZ.EQ.(-1)) GO TO 190 357 NZ=0 358 IERR=5 359 RETURN 360 260 CONTINUE 361 IERR=4 362 NZ=0 363 RETURN 364 END