zbiry.f

Go to the documentation of this file.

      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         DEFINTIONS 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,XZABS,ZDIV,XZSQRT,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, XZABS
      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 = xzabs(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 xzsqrt(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 xzsqrt(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 xzsqrt(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