src/numerical/slatec/fortran/zbiry.f

   1 *DECK ZBIRY
   2       SUBROUTINE ZBIRY (ZR, ZI, ID, KODE, BIR, BII, IERR)
   3 C***BEGIN PROLOGUE  ZBIRY
   4 C***PURPOSE  Compute the Airy function Bi(z) or its derivative dBi/dz
   5 C            for complex argument z.  A scaling option is available
   6 C            to help avoid overflow.
   7 C***LIBRARY   SLATEC
   8 C***CATEGORY  C10D
   9 C***TYPE      COMPLEX (CBIRY-C, ZBIRY-C)
  10 C***KEYWORDS  AIRY FUNCTION, BESSEL FUNCTION OF ORDER ONE THIRD,
  11 C             BESSEL FUNCTION OF ORDER TWO THIRDS
  12 C***AUTHOR  Amos, D. E., (SNL)
  13 C***DESCRIPTION
  14 C
  15 C                      ***A DOUBLE PRECISION ROUTINE***
  16 C         On KODE=1, ZBIRY computes the complex Airy function Bi(z)
  17 C         or its derivative dBi/dz on ID=0 or ID=1 respectively.
  18 C         On KODE=2, a scaling option exp(abs(Re(zeta)))*Bi(z) or
  19 C         exp(abs(Re(zeta)))*dBi/dz is provided to remove the
  20 C         exponential behavior in both the left and right half planes
  21 C         where zeta=(2/3)*z**(3/2).
  22 C
  23 C         The Airy functions Bi(z) and dBi/dz are analytic in the
  24 C         whole z-plane, and the scaling option does not destroy this
  25 C         property.
  26 C
  27 C         Input
  28 C           ZR     - DOUBLE PRECISION real part of argument Z
  29 C           ZI     - DOUBLE PRECISION imag part of argument Z
  30 C           ID     - Order of derivative, ID=0 or ID=1
  31 C           KODE   - A parameter to indicate the scaling option
  32 C                    KODE=1  returns
  33 C                            BI=Bi(z)  on ID=0
  34 C                            BI=dBi/dz on ID=1
  35 C                            at z=Z
  36 C                        =2  returns
  37 C                            BI=exp(abs(Re(zeta)))*Bi(z)  on ID=0
  38 C                            BI=exp(abs(Re(zeta)))*dBi/dz on ID=1
  39 C                            at z=Z where zeta=(2/3)*z**(3/2)
  40 C
  41 C         Output
  42 C           BIR    - DOUBLE PRECISION real part of result
  43 C           BII    - DOUBLE PRECISION imag part of result
  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
  48 C                            (Re(Z) too large with KODE=1)
  49 C                    IERR=3  Precision warning - COMPUTATION COMPLETED
  50 C                            (Result has less than half precision)
  51 C                    IERR=4  Precision error   - NO COMPUTATION
  52 C                            (Result has no precision)
  53 C                    IERR=5  Algorithmic error - NO COMPUTATION
  54 C                            (Termination condition not met)
  55 C
  56 C *Long Description:
  57 C
  58 C         Bi(z) and dBi/dz are computed from I Bessel functions by
  59 C
  60 C                Bi(z) =  c*sqrt(z)*( I(-1/3,zeta) + I(1/3,zeta) )
  61 C               dBi/dz =  c*   z   *( I(-2/3,zeta) + I(2/3,zeta) )
  62 C                    c =  1/sqrt(3)
  63 C                 zeta =  (2/3)*z**(3/2)
  64 C
  65 C         when abs(z)>1 and from power series when abs(z)<=1.
  66 C
  67 C         In most complex variable computation, one must evaluate ele-
  68 C         mentary functions.  When the magnitude of Z is large, losses
  69 C         of significance by argument reduction occur.  Consequently, if
  70 C         the magnitude of ZETA=(2/3)*Z**(3/2) exceeds U1=SQRT(0.5/UR),
  71 C         then losses exceeding half precision are likely and an error
  72 C         flag IERR=3 is triggered where UR=MAX(D1MACH(4),1.0D-18) is
  73 C         double precision unit roundoff limited to 18 digits precision.
  74 C         Also, if the magnitude of ZETA is larger than U2=0.5/UR, then
  75 C         all significance is lost and IERR=4.  In order to use the INT
  76 C         function, ZETA must be further restricted not to exceed
  77 C         U3=I1MACH(9)=LARGEST INTEGER.  Thus, the magnitude of ZETA
  78 C         must be restricted by MIN(U2,U3).  In IEEE arithmetic, U1,U2,
  79 C         and U3 are approximately 2.0E+3, 4.2E+6, 2.1E+9 in single
  80 C         precision and 4.7E+7, 2.3E+15, 2.1E+9 in double precision.
  81 C         This makes U2 limiting is single precision and U3 limiting
  82 C         in double precision.  This means that the magnitude of Z
  83 C         cannot exceed approximately 3.4E+4 in single precision and
  84 C         2.1E+6 in double precision.  This also means that one can
  85 C         expect to retain, in the worst cases on 32-bit machines,
  86 C         no digits in single precision and only 6 digits in double
  87 C         precision.
  88 C
  89 C         The approximate relative error in the magnitude of a complex
  90 C         Bessel function can be expressed as P*10**S where P=MAX(UNIT
  91 C         ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
  92 C         sents the increase in error due to argument reduction in the
  93 C         elementary functions.  Here, S=MAX(1,ABS(LOG10(ABS(Z))),
  94 C         ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
  95 C         ABS(Z),ABS(EXPONENT OF FNU)) ).  However, the phase angle may
  96 C         have only absolute accuracy.  This is most likely to occur
  97 C         when one component (in magnitude) is larger than the other by
  98 C         several orders of magnitude.  If one component is 10**K larger
  99 C         than the other, then one can expect only MAX(ABS(LOG10(P))-K,
 100 C         0) significant digits; or, stated another way, when K exceeds
 101 C         the exponent of P, no significant digits remain in the smaller
 102 C         component.  However, the phase angle retains absolute accuracy
 103 C         because, in complex arithmetic with precision P, the smaller
 104 C         component will not (as a rule) decrease below P times the
 105 C         magnitude of the larger component. In these extreme cases,
 106 C         the principal phase angle is on the order of +P, -P, PI/2-P,
 107 C         or -PI/2+P.
 108 C
 109 C***REFERENCES  1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
 110 C                 matical Functions, National Bureau of Standards
 111 C                 Applied Mathematics Series 55, U. S. Department
 112 C                 of Commerce, Tenth Printing (1972) or later.
 113 C               2. D. E. Amos, Computation of Bessel Functions of
 114 C                 Complex Argument and Large Order, Report SAND83-0643,
 115 C                 Sandia National Laboratories, Albuquerque, NM, May
 116 C                 1983.
 117 C               3. D. E. Amos, A Subroutine Package for Bessel Functions
 118 C                 of a Complex Argument and Nonnegative Order, Report
 119 C                 SAND85-1018, Sandia National Laboratory, Albuquerque,
 120 C                 NM, May 1985.
 121 C               4. D. E. Amos, A portable package for Bessel functions
 122 C                 of a complex argument and nonnegative order, ACM
 123 C                 Transactions on Mathematical Software, 12 (September
 124 C                 1986), pp. 265-273.
 125 C
 126 C***ROUTINES CALLED  D1MACH, I1MACH, ZABS, ZBINU, ZDIV, ZSQRT
 127 C***REVISION HISTORY  (YYMMDD)
 128 C   830501  DATE WRITTEN
 129 C   890801  REVISION DATE from Version 3.2
 130 C   910415  Prologue converted to Version 4.0 format.  (BAB)
 131 C   920128  Category corrected.  (WRB)
 132 C   920811  Prologue revised.  (DWL)
 133 C   930122  Added ZSQRT to EXTERNAL statement.  (RWC)
 134 C***END PROLOGUE  ZBIRY
 135 C     COMPLEX BI,CONE,CSQ,CY,S1,S2,TRM1,TRM2,Z,ZTA,Z3
 136       DOUBLE PRECISION AA, AD, AK, ALIM, ATRM, AZ, AZ3, BB, BII, BIR,
 137      * BK, CC, CK, COEF, CONEI, CONER, CSQI, CSQR, CYI, CYR, C1, C2,
 138      * DIG, DK, D1, D2, EAA, ELIM, FID, FMR, FNU, FNUL, PI, RL, R1M5,
 139      * SFAC, STI, STR, S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I,
 140      * TRM2R, TTH, ZI, ZR, ZTAI, ZTAR, Z3I, Z3R, D1MACH, ZABS
 141       INTEGER ID, IERR, K, KODE, K1, K2, NZ, I1MACH
 142       DIMENSION CYR(2), CYI(2)
 143       EXTERNAL ZABS, ZSQRT
 144       DATA TTH, C1, C2, COEF, PI /6.66666666666666667D-01,
 145      * 6.14926627446000736D-01,4.48288357353826359D-01,
 146      * 5.77350269189625765D-01,3.14159265358979324D+00/
 147       DATA CONER, CONEI /1.0D0,0.0D0/
 148 C***FIRST EXECUTABLE STATEMENT  ZBIRY
 149       IERR = 0
 150       NZ=0
 151       IF (ID.LT.0 .OR. ID.GT.1) IERR=1
 152       IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
 153       IF (IERR.NE.0) RETURN
 154       AZ = ZABS(ZR,ZI)
 155       TOL = MAX(D1MACH(4),1.0D-18)
 156       FID = ID
 157       IF (AZ.GT.1.0E0) GO TO 70
 158 C-----------------------------------------------------------------------
 159 C     POWER SERIES FOR ABS(Z).LE.1.
 160 C-----------------------------------------------------------------------
 161       S1R = CONER
 162       S1I = CONEI
 163       S2R = CONER
 164       S2I = CONEI
 165       IF (AZ.LT.TOL) GO TO 130
 166       AA = AZ*AZ
 167       IF (AA.LT.TOL/AZ) GO TO 40
 168       TRM1R = CONER
 169       TRM1I = CONEI
 170       TRM2R = CONER
 171       TRM2I = CONEI
 172       ATRM = 1.0D0
 173       STR = ZR*ZR - ZI*ZI
 174       STI = ZR*ZI + ZI*ZR
 175       Z3R = STR*ZR - STI*ZI
 176       Z3I = STR*ZI + STI*ZR
 177       AZ3 = AZ*AA
 178       AK = 2.0D0 + FID
 179       BK = 3.0D0 - FID - FID
 180       CK = 4.0D0 - FID
 181       DK = 3.0D0 + FID + FID
 182       D1 = AK*DK
 183       D2 = BK*CK
 184       AD = MIN(D1,D2)
 185       AK = 24.0D0 + 9.0D0*FID
 186       BK = 30.0D0 - 9.0D0*FID
 187       DO 30 K=1,25
 188         STR = (TRM1R*Z3R-TRM1I*Z3I)/D1
 189         TRM1I = (TRM1R*Z3I+TRM1I*Z3R)/D1
 190         TRM1R = STR
 191         S1R = S1R + TRM1R
 192         S1I = S1I + TRM1I
 193         STR = (TRM2R*Z3R-TRM2I*Z3I)/D2
 194         TRM2I = (TRM2R*Z3I+TRM2I*Z3R)/D2
 195         TRM2R = STR
 196         S2R = S2R + TRM2R
 197         S2I = S2I + TRM2I
 198         ATRM = ATRM*AZ3/AD
 199         D1 = D1 + AK
 200         D2 = D2 + BK
 201         AD = MIN(D1,D2)
 202         IF (ATRM.LT.TOL*AD) GO TO 40
 203         AK = AK + 18.0D0
 204         BK = BK + 18.0D0
 205    30 CONTINUE
 206    40 CONTINUE
 207       IF (ID.EQ.1) GO TO 50
 208       BIR = C1*S1R + C2*(ZR*S2R-ZI*S2I)
 209       BII = C1*S1I + C2*(ZR*S2I+ZI*S2R)
 210       IF (KODE.EQ.1) RETURN
 211       CALL ZSQRT(ZR, ZI, STR, STI)
 212       ZTAR = TTH*(ZR*STR-ZI*STI)
 213       ZTAI = TTH*(ZR*STI+ZI*STR)
 214       AA = ZTAR
 215       AA = -ABS(AA)
 216       EAA = EXP(AA)
 217       BIR = BIR*EAA
 218       BII = BII*EAA
 219       RETURN
 220    50 CONTINUE
 221       BIR = S2R*C2
 222       BII = S2I*C2
 223       IF (AZ.LE.TOL) GO TO 60
 224       CC = C1/(1.0D0+FID)
 225       STR = S1R*ZR - S1I*ZI
 226       STI = S1R*ZI + S1I*ZR
 227       BIR = BIR + CC*(STR*ZR-STI*ZI)
 228       BII = BII + CC*(STR*ZI+STI*ZR)
 229    60 CONTINUE
 230       IF (KODE.EQ.1) RETURN
 231       CALL ZSQRT(ZR, ZI, STR, STI)
 232       ZTAR = TTH*(ZR*STR-ZI*STI)
 233       ZTAI = TTH*(ZR*STI+ZI*STR)
 234       AA = ZTAR
 235       AA = -ABS(AA)
 236       EAA = EXP(AA)
 237       BIR = BIR*EAA
 238       BII = BII*EAA
 239       RETURN
 240 C-----------------------------------------------------------------------
 241 C     CASE FOR ABS(Z).GT.1.0
 242 C-----------------------------------------------------------------------
 243    70 CONTINUE
 244       FNU = (1.0D0+FID)/3.0D0
 245 C-----------------------------------------------------------------------
 246 C     SET PARAMETERS RELATED TO MACHINE CONSTANTS.
 247 C     TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
 248 C     ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
 249 C     EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL    AND
 250 C     EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL       ARE INTERVALS NEAR
 251 C     UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
 252 C     RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
 253 C     DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
 254 C     FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU.
 255 C-----------------------------------------------------------------------
 256       K1 = I1MACH(15)
 257       K2 = I1MACH(16)
 258       R1M5 = D1MACH(5)
 259       K = MIN(ABS(K1),ABS(K2))
 260       ELIM = 2.303D0*(K*R1M5-3.0D0)
 261       K1 = I1MACH(14) - 1
 262       AA = R1M5*K1
 263       DIG = MIN(AA,18.0D0)
 264       AA = AA*2.303D0
 265       ALIM = ELIM + MAX(-AA,-41.45D0)
 266       RL = 1.2D0*DIG + 3.0D0
 267       FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0)
 268 C-----------------------------------------------------------------------
 269 C     TEST FOR RANGE
 270 C-----------------------------------------------------------------------
 271       AA=0.5D0/TOL
 272       BB=I1MACH(9)*0.5D0
 273       AA=MIN(AA,BB)
 274       AA=AA**TTH
 275       IF (AZ.GT.AA) GO TO 260
 276       AA=SQRT(AA)
 277       IF (AZ.GT.AA) IERR=3
 278       CALL ZSQRT(ZR, ZI, CSQR, CSQI)
 279       ZTAR = TTH*(ZR*CSQR-ZI*CSQI)
 280       ZTAI = TTH*(ZR*CSQI+ZI*CSQR)
 281 C-----------------------------------------------------------------------
 282 C     RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL
 283 C-----------------------------------------------------------------------
 284       SFAC = 1.0D0
 285       AK = ZTAI
 286       IF (ZR.GE.0.0D0) GO TO 80
 287       BK = ZTAR
 288       CK = -ABS(BK)
 289       ZTAR = CK
 290       ZTAI = AK
 291    80 CONTINUE
 292       IF (ZI.NE.0.0D0 .OR. ZR.GT.0.0D0) GO TO 90
 293       ZTAR = 0.0D0
 294       ZTAI = AK
 295    90 CONTINUE
 296       AA = ZTAR
 297       IF (KODE.EQ.2) GO TO 100
 298 C-----------------------------------------------------------------------
 299 C     OVERFLOW TEST
 300 C-----------------------------------------------------------------------
 301       BB = ABS(AA)
 302       IF (BB.LT.ALIM) GO TO 100
 303       BB = BB + 0.25D0*LOG(AZ)
 304       SFAC = TOL
 305       IF (BB.GT.ELIM) GO TO 190
 306   100 CONTINUE
 307       FMR = 0.0D0
 308       IF (AA.GE.0.0D0 .AND. ZR.GT.0.0D0) GO TO 110
 309       FMR = PI
 310       IF (ZI.LT.0.0D0) FMR = -PI
 311       ZTAR = -ZTAR
 312       ZTAI = -ZTAI
 313   110 CONTINUE
 314 C-----------------------------------------------------------------------
 315 C     AA=FACTOR FOR ANALYTIC CONTINUATION OF I(FNU,ZTA)
 316 C     KODE=2 RETURNS EXP(-ABS(XZTA))*I(FNU,ZTA) FROM CBESI
 317 C-----------------------------------------------------------------------
 318       CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, RL, FNUL, TOL,
 319      * ELIM, ALIM)
 320       IF (NZ.LT.0) GO TO 200
 321       AA = FMR*FNU
 322       Z3R = SFAC
 323       STR = COS(AA)
 324       STI = SIN(AA)
 325       S1R = (STR*CYR(1)-STI*CYI(1))*Z3R
 326       S1I = (STR*CYI(1)+STI*CYR(1))*Z3R
 327       FNU = (2.0D0-FID)/3.0D0
 328       CALL ZBINU(ZTAR, ZTAI, FNU, KODE, 2, CYR, CYI, NZ, RL, FNUL, TOL,
 329      * ELIM, ALIM)
 330       CYR(1) = CYR(1)*Z3R
 331       CYI(1) = CYI(1)*Z3R
 332       CYR(2) = CYR(2)*Z3R
 333       CYI(2) = CYI(2)*Z3R
 334 C-----------------------------------------------------------------------
 335 C     BACKWARD RECUR ONE STEP FOR ORDERS -1/3 OR -2/3
 336 C-----------------------------------------------------------------------
 337       CALL ZDIV(CYR(1), CYI(1), ZTAR, ZTAI, STR, STI)
 338       S2R = (FNU+FNU)*STR + CYR(2)
 339       S2I = (FNU+FNU)*STI + CYI(2)
 340       AA = FMR*(FNU-1.0D0)
 341       STR = COS(AA)
 342       STI = SIN(AA)
 343       S1R = COEF*(S1R+S2R*STR-S2I*STI)
 344       S1I = COEF*(S1I+S2R*STI+S2I*STR)
 345       IF (ID.EQ.1) GO TO 120
 346       STR = CSQR*S1R - CSQI*S1I
 347       S1I = CSQR*S1I + CSQI*S1R
 348       S1R = STR
 349       BIR = S1R/SFAC
 350       BII = S1I/SFAC
 351       RETURN
 352   120 CONTINUE
 353       STR = ZR*S1R - ZI*S1I
 354       S1I = ZR*S1I + ZI*S1R
 355       S1R = STR
 356       BIR = S1R/SFAC
 357       BII = S1I/SFAC
 358       RETURN
 359   130 CONTINUE
 360       AA = C1*(1.0D0-FID) + FID*C2
 361       BIR = AA
 362       BII = 0.0D0
 363       RETURN
 364   190 CONTINUE
 365       IERR=2
 366       NZ=0
 367       RETURN
 368   200 CONTINUE
 369       IF(NZ.EQ.(-1)) GO TO 190
 370       NZ=0
 371       IERR=5
 372       RETURN
 373   260 CONTINUE
 374       IERR=4
 375       NZ=0
 376       RETURN
 377       END