amos/zairy.f

   1       SUBROUTINE ZAIRY(ZR, ZI, ID, KODE, AIR, AII, NZ, IERR)
   2 C***BEGIN PROLOGUE  ZAIRY
   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 AI(Z) AND DAI(Z) FOR COMPLEX Z
   9 C***DESCRIPTION
  10 C
  11 C                      ***A DOUBLE PRECISION ROUTINE***
  12 C         ON KODE=1, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR
  13 C         ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON
  14 C         KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)*
  15 C         DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN
  16 C         -PI/3.LT.ARG(Z).LT.PI/3 AND THE EXPONENTIAL GROWTH IN
  17 C         PI/3.LT.ABS(ARG(Z)).LT.PI WHERE ZTA=(2/3)*Z*CSQRT(Z).
  18 C
  19 C         WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN
  20 C         THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED
  21 C         FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS.
  22 C         DEFINITIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
  23 C         MATHEMATICAL FUNCTIONS (REF. 1).
  24 C
  25 C         INPUT      ZR,ZI ARE DOUBLE PRECISION
  26 C           ZR,ZI  - Z=CMPLX(ZR,ZI)
  27 C           ID     - ORDER OF DERIVATIVE, ID=0 OR ID=1
  28 C           KODE   - A PARAMETER TO INDICATE THE SCALING OPTION
  29 C                    KODE= 1  RETURNS
  30 C                             AI=AI(Z)                ON ID=0 OR
  31 C                             AI=DAI(Z)/DZ            ON ID=1
  32 C                        = 2  RETURNS
  33 C                             AI=CEXP(ZTA)*AI(Z)       ON ID=0 OR
  34 C                             AI=CEXP(ZTA)*DAI(Z)/DZ   ON ID=1 WHERE
  35 C                             ZTA=(2/3)*Z*CSQRT(Z)
  36 C
  37 C         OUTPUT     AIR,AII ARE DOUBLE PRECISION
  38 C           AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND
  39 C                    KODE
  40 C           NZ     - UNDERFLOW INDICATOR
  41 C                    NZ= 0   , NORMAL RETURN
  42 C                    NZ= 1   , AI=CMPLX(0.0D0,0.0D0) DUE TO UNDERFLOW IN
  43 C                              -PI/3.LT.ARG(Z).LT.PI/3 ON KODE=1
  44 C           IERR   - ERROR FLAG
  45 C                    IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
  46 C                    IERR=1, INPUT ERROR   - NO COMPUTATION
  47 C                    IERR=2, OVERFLOW      - NO COMPUTATION, REAL(ZTA)
  48 C                            TOO LARGE ON KODE=1
  49 C                    IERR=3, CABS(Z) LARGE      - COMPUTATION COMPLETED
  50 C                            LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
  51 C                            PRODUCE LESS THAN HALF OF MACHINE ACCURACY
  52 C                    IERR=4, CABS(Z) TOO LARGE  - NO COMPUTATION
  53 C                            COMPLETE LOSS OF ACCURACY BY ARGUMENT
  54 C                            REDUCTION
  55 C                    IERR=5, ERROR              - NO COMPUTATION,
  56 C                            ALGORITHM TERMINATION CONDITION NOT MET
  57 C
  58 C***LONG DESCRIPTION
  59 C
  60 C         AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL
  61 C         FUNCTIONS BY
  62 C
  63 C            AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA)
  64 C                           C=1.0/(PI*SQRT(3.0))
  65 C                            ZTA=(2/3)*Z**(3/2)
  66 C
  67 C         WITH THE POWER SERIES FOR CABS(Z).LE.1.0.
  68 C
  69 C         IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
  70 C         MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES
  71 C         OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
  72 C         THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
  73 C         THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
  74 C         FLAG IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS
  75 C         DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
  76 C         ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
  77 C         ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT
  78 C         FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
  79 C         LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA
  80 C         MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
  81 C         AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
  82 C         PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
  83 C         PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
  84 C         ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
  85 C         NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
  86 C         DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
  87 C         EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
  88 C         NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
  89 C         PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER
  90 C         MACHINES.
  91 C
  92 C         THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
  93 C         BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
  94 C         ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
  95 C         SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
  96 C         ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
  97 C         ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
  98 C         CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
  99 C         HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
 100 C         ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
 101 C         SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
 102 C         THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
 103 C         0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
 104 C         THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
 105 C         COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
 106 C         BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
 107 C         COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
 108 C         MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
 109 C         THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
 110 C         OR -PI/2+P.
 111 C
 112 C***REFERENCES  HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
 113 C                 AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
 114 C                 COMMERCE, 1955.
 115 C
 116 C               COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
 117 C                 AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
 118 C
 119 C               A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
 120 C                 ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
 121 C                 1018, MAY, 1985
 122 C
 123 C               A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
 124 C                 ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
 125 C                 MATH. SOFTWARE, 1986
 126 C
 127 C***ROUTINES CALLED  ZACAI,ZBKNU,AZEXP,AZSQRT,I1MACH,D1MACH
 128 C***END PROLOGUE  ZAIRY
 129 C     COMPLEX AI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3
 130       DOUBLE PRECISION AA, AD, AII, AIR, AK, ALIM, ATRM, AZ, AZ3, BK,
 131      * CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2, DIG,
 132      * DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR,
 133      * S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI,
 134      * ZEROR, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, AZABS, ALAZ, BB
 135       INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH
 136       DIMENSION CYR(1), CYI(1)
 137       DATA TTH, C1, C2, COEF /6.66666666666666667D-01,
 138      * 3.55028053887817240D-01,2.58819403792806799D-01,
 139      * 1.83776298473930683D-01/
 140       DATA ZEROR, ZEROI, CONER, CONEI /0.0D0,0.0D0,1.0D0,0.0D0/
 141 C***FIRST EXECUTABLE STATEMENT  ZAIRY
 142       IERR = 0
 143       NZ=0
 144       IF (ID.LT.0 .OR. ID.GT.1) IERR=1
 145       IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
 146       IF (IERR.NE.0) RETURN
 147       AZ = AZABS(ZR,ZI)
 148       TOL = DMAX1(D1MACH(4),1.0D-18)
 149       FID = DBLE(FLOAT(ID))
 150       IF (AZ.GT.1.0D0) GO TO 70
 151 C-----------------------------------------------------------------------
 152 C     POWER SERIES FOR CABS(Z).LE.1.
 153 C-----------------------------------------------------------------------
 154       S1R = CONER
 155       S1I = CONEI
 156       S2R = CONER
 157       S2I = CONEI
 158       IF (AZ.LT.TOL) GO TO 170
 159       AA = AZ*AZ
 160       IF (AA.LT.TOL/AZ) GO TO 40
 161       TRM1R = CONER
 162       TRM1I = CONEI
 163       TRM2R = CONER
 164       TRM2I = CONEI
 165       ATRM = 1.0D0
 166       STR = ZR*ZR - ZI*ZI
 167       STI = ZR*ZI + ZI*ZR
 168       Z3R = STR*ZR - STI*ZI
 169       Z3I = STR*ZI + STI*ZR
 170       AZ3 = AZ*AA
 171       AK = 2.0D0 + FID
 172       BK = 3.0D0 - FID - FID
 173       CK = 4.0D0 - FID
 174       DK = 3.0D0 + FID + FID
 175       D1 = AK*DK
 176       D2 = BK*CK
 177       AD = DMIN1(D1,D2)
 178       AK = 24.0D0 + 9.0D0*FID
 179       BK = 30.0D0 - 9.0D0*FID
 180       DO 30 K=1,25
 181         STR = (TRM1R*Z3R-TRM1I*Z3I)/D1
 182         TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1
 183         TRM1R = STR
 184         S1R = S1R + TRM1R
 185         S1I = S1I + TRM1I
 186         STR = (TRM2R*Z3R-TRM2I*Z3I)/D2
 187         TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2
 188         TRM2R = STR
 189         S2R = S2R + TRM2R
 190         S2I = S2I + TRM2I
 191         ATRM = ATRM*AZ3/AD
 192         D1 = D1 + AK
 193         D2 = D2 + BK
 194         AD = DMIN1(D1,D2)
 195         IF (ATRM.LT.TOL*AD) GO TO 40
 196         AK = AK + 18.0D0
 197         BK = BK + 18.0D0
 198    30 CONTINUE
 199    40 CONTINUE
 200       IF (ID.EQ.1) GO TO 50
 201       AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I)
 202       AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R)
 203       IF (KODE.EQ.1) RETURN
 204       CALL AZSQRT(ZR, ZI, STR, STI)
 205       ZTAR = TTH*(ZR*STR-ZI*STI)
 206       ZTAI = TTH*(ZR*STI+ZI*STR)
 207       CALL AZEXP(ZTAR, ZTAI, STR, STI)
 208       PTR = AIR*STR - AII*STI
 209       AII = AIR*STI + AII*STR
 210       AIR = PTR
 211       RETURN
 212    50 CONTINUE
 213       AIR = -S2R*C2
 214       AII = -S2I*C2
 215       IF (AZ.LE.TOL) GO TO 60
 216       STR = ZR*S1R - ZI*S1I
 217       STI = ZR*S1I + ZI*S1R
 218       CC = C1/(1.0D0+FID)
 219       AIR = AIR + CC*(STR*ZR-STI*ZI)
 220       AII = AII + CC*(STR*ZI+STI*ZR)
 221    60 CONTINUE
 222       IF (KODE.EQ.1) RETURN
 223       CALL AZSQRT(ZR, ZI, STR, STI)
 224       ZTAR = TTH*(ZR*STR-ZI*STI)
 225       ZTAI = TTH*(ZR*STI+ZI*STR)
 226       CALL AZEXP(ZTAR, ZTAI, STR, STI)
 227       PTR = STR*AIR - STI*AII
 228       AII = STR*AII + STI*AIR
 229       AIR = PTR
 230       RETURN
 231 C-----------------------------------------------------------------------
 232 C     CASE FOR CABS(Z).GT.1.0
 233 C-----------------------------------------------------------------------
 234    70 CONTINUE
 235       FNU = (1.0D0+FID)/3.0D0
 236 C-----------------------------------------------------------------------
 237 C     SET PARAMETERS RELATED TO MACHINE CONSTANTS.
 238 C     TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-18.
 239 C     ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
 240 C     EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL    AND
 241 C     EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL       ARE INTERVALS NEAR
 242 C     UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
 243 C     RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
 244 C     DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
 245 C-----------------------------------------------------------------------
 246       K1 = I1MACH(15)
 247       K2 = I1MACH(16)
 248       R1M5 = D1MACH(5)
 249       K = MIN0(IABS(K1),IABS(K2))
 250       ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
 251       K1 = I1MACH(14) - 1
 252       AA = R1M5*DBLE(FLOAT(K1))
 253       DIG = DMIN1(AA,18.0D0)
 254       AA = AA*2.303D0
 255       ALIM = ELIM + DMAX1(-AA,-41.45D0)
 256       RL = 1.2D0*DIG + 3.0D0
 257       ALAZ = DLOG(AZ)
 258 C--------------------------------------------------------------------------
 259 C     TEST FOR PROPER RANGE
 260 C-----------------------------------------------------------------------
 261       AA=0.5D0/TOL
 262       BB=DBLE(FLOAT(I1MACH(9)))*0.5D0
 263       AA=DMIN1(AA,BB)
 264       AA=AA**TTH
 265       IF (AZ.GT.AA) GO TO 260
 266       AA=DSQRT(AA)
 267       IF (AZ.GT.AA) IERR=3
 268       CALL AZSQRT(ZR, ZI, CSQR, CSQI)
 269       ZTAR = TTH*(ZR*CSQR-ZI*CSQI)
 270       ZTAI = TTH*(ZR*CSQI+ZI*CSQR)
 271 C-----------------------------------------------------------------------
 272 C     RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL
 273 C-----------------------------------------------------------------------
 274       IFLAG = 0
 275       SFAC = 1.0D0
 276       AK = ZTAI
 277       IF (ZR.GE.0.0D0) GO TO 80
 278       BK = ZTAR
 279       CK = -DABS(BK)
 280       ZTAR = CK
 281       ZTAI = AK
 282    80 CONTINUE
 283       IF (ZI.NE.0.0D0) GO TO 90
 284       IF (ZR.GT.0.0D0) GO TO 90
 285       ZTAR = 0.0D0
 286       ZTAI = AK
 287    90 CONTINUE
 288       AA = ZTAR
 289       IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110
 290       IF (KODE.EQ.2) GO TO 100
 291 C-----------------------------------------------------------------------
 292 C     OVERFLOW TEST
 293 C-----------------------------------------------------------------------
 294       IF (AA.GT.(-ALIM)) GO TO 100
 295       AA = -AA + 0.25D0*ALAZ
 296       IFLAG = 1
 297       SFAC = TOL
 298       IF (AA.GT.ELIM) GO TO 270
 299   100 CONTINUE
 300 C-----------------------------------------------------------------------
 301 C     CBKNU AND CACON RETURN EXP(ZTA)*K(FNU,ZTA) ON KODE=2
 302 C-----------------------------------------------------------------------
 303       MR = 1
 304       IF (ZI.LT.0.0D0) MR = -1
 305       CALL ZACAI(ZTAR, ZTAI, FNU, KODE, MR, 1, CYR, CYI, NN, RL, TOL,
 306      * ELIM, ALIM)
 307       IF (NN.LT.0) GO TO 280
 308       NZ = NZ + NN
 309       GO TO 130
 310   110 CONTINUE
 311       IF (KODE.EQ.2) GO TO 120
 312 C-----------------------------------------------------------------------
 313 C     UNDERFLOW TEST
 314 C-----------------------------------------------------------------------
 315       IF (AA.LT.ALIM) GO TO 120
 316       AA = -AA - 0.25D0*ALAZ
 317       IFLAG = 2
 318       SFAC = 1.0D0/TOL
 319       IF (AA.LT.(-ELIM)) GO TO 210
 320   120 CONTINUE
 321       CALL ZBKNU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, TOL, ELIM,
 322      * ALIM)
 323   130 CONTINUE
 324       S1R = CYR(1)*COEF
 325       S1I = CYI(1)*COEF
 326       IF (IFLAG.NE.0) GO TO 150
 327       IF (ID.EQ.1) GO TO 140
 328       AIR = CSQR*S1R - CSQI*S1I
 329       AII = CSQR*S1I + CSQI*S1R
 330       RETURN
 331   140 CONTINUE
 332       AIR = -(ZR*S1R-ZI*S1I)
 333       AII = -(ZR*S1I+ZI*S1R)
 334       RETURN
 335   150 CONTINUE
 336       S1R = S1R*SFAC
 337       S1I = S1I*SFAC
 338       IF (ID.EQ.1) GO TO 160
 339       STR = S1R*CSQR - S1I*CSQI
 340       S1I = S1R*CSQI + S1I*CSQR
 341       S1R = STR
 342       AIR = S1R/SFAC
 343       AII = S1I/SFAC
 344       RETURN
 345   160 CONTINUE
 346       STR = -(S1R*ZR-S1I*ZI)
 347       S1I = -(S1R*ZI+S1I*ZR)
 348       S1R = STR
 349       AIR = S1R/SFAC
 350       AII = S1I/SFAC
 351       RETURN
 352   170 CONTINUE
 353       AA = 1.0D+3*D1MACH(1)
 354       S1R = ZEROR
 355       S1I = ZEROI
 356       IF (ID.EQ.1) GO TO 190
 357       IF (AZ.LE.AA) GO TO 180
 358       S1R = C2*ZR
 359       S1I = C2*ZI
 360   180 CONTINUE
 361       AIR = C1 - S1R
 362       AII = -S1I
 363       RETURN
 364   190 CONTINUE
 365       AIR = -C2
 366       AII = 0.0D0
 367       AA = DSQRT(AA)
 368       IF (AZ.LE.AA) GO TO 200
 369       S1R = 0.5D0*(ZR*ZR-ZI*ZI)
 370       S1I = ZR*ZI
 371   200 CONTINUE
 372       AIR = AIR + C1*S1R
 373       AII = AII + C1*S1I
 374       RETURN
 375   210 CONTINUE
 376       NZ = 1
 377       AIR = ZEROR
 378       AII = ZEROI
 379       RETURN
 380   270 CONTINUE
 381       NZ = 0
 382       IERR=2
 383       RETURN
 384   280 CONTINUE
 385       IF(NN.EQ.(-1)) GO TO 270
 386       NZ=0
 387       IERR=5
 388       RETURN
 389   260 CONTINUE
 390       IERR=4
 391       NZ=0
 392       RETURN
 393       END