SLATEC Routines --- FZERO ---

*DECK FZERO
      SUBROUTINE FZERO (F, B, C, R, RE, AE, IFLAG)
C***BEGIN PROLOGUE  FZERO
C***PURPOSE  Search for a zero of a function F(X) in a given interval
C            (B,C).  It is designed primarily for problems where F(B)
C            and F(C) have opposite signs.
C***LIBRARY   SLATEC
C***CATEGORY  F1B
C***TYPE      SINGLE PRECISION (FZERO-S, DFZERO-D)
C***KEYWORDS  BISECTION, NONLINEAR EQUATIONS, ROOTS, ZEROS
C***AUTHOR  Shampine, L. F., (SNLA)
C           Watts, H. A., (SNLA)
C***DESCRIPTION
C
C     FZERO searches for a zero of a REAL function F(X) between the
C     given REAL values B and C until the width of the interval (B,C)
C     has collapsed to within a tolerance specified by the stopping
C     criterion,
C        ABS(B-C) .LE. 2.*(RW*ABS(B)+AE).
C     The method used is an efficient combination of bisection and the
C     secant rule and is due to T. J. Dekker.
C
C     Description Of Arguments
C
C   F     :EXT   - Name of the REAL external function.  This name must
C                  be in an EXTERNAL statement in the calling program.
C                  F must be a function of one REAL argument.
C
C   B     :INOUT - One end of the REAL interval (B,C).  The value
C                  returned for B usually is the better approximation
C                  to a zero of F.
C
C   C     :INOUT - The other end of the REAL interval (B,C)
C
C   R     :OUT   - A (better) REAL guess of a zero of F which could help
C                  in speeding up convergence.  If F(B) and F(R) have
C                  opposite signs, a root will be found in the interval
C                  (B,R); if not, but F(R) and F(C) have opposite signs,
C                  a root will be found in the interval (R,C);
C                  otherwise, the interval (B,C) will be searched for a
C                  possible root.  When no better guess is known, it is
C                  recommended that r be set to B or C, since if R is
C                  not interior to the interval (B,C), it will be
C                  ignored.
C
C   RE    :IN    - Relative error used for RW in the stopping criterion.
C                  If the requested RE is less than machine precision,
C                  then RW is set to approximately machine precision.
C
C   AE    :IN    - Absolute error used in the stopping criterion.  If
C                  the given interval (B,C) contains the origin, then a
C                  nonzero value should be chosen for AE.
C
C   IFLAG :OUT   - A status code.  User must check IFLAG after each
C                  call.  Control returns to the user from FZERO in all
C                  cases.
C
C                1  B is within the requested tolerance of a zero.
C                   The interval (B,C) collapsed to the requested
C                   tolerance, the function changes sign in (B,C), and
C                   F(X) decreased in magnitude as (B,C) collapsed.
C
C                2  F(B) = 0.  However, the interval (B,C) may not have
C                   collapsed to the requested tolerance.
C
C                3  B may be near a singular point of F(X).
C                   The interval (B,C) collapsed to the requested tol-
C                   erance and the function changes sign in (B,C), but
C                   F(X) increased in magnitude as (B,C) collapsed, i.e.
C                     ABS(F(B out)) .GT. MAX(ABS(F(B in)),ABS(F(C in)))
C
C                4  No change in sign of F(X) was found although the
C                   interval (B,C) collapsed to the requested tolerance.
C                   The user must examine this case and decide whether
C                   B is near a local minimum of F(X), or B is near a
C                   zero of even multiplicity, or neither of these.
C
C                5  Too many (.GT. 500) function evaluations used.
C
C***REFERENCES  L. F. Shampine and H. A. Watts, FZERO, a root-solving
C                 code, Report SC-TM-70-631, Sandia Laboratories,
C                 September 1970.
C               T. J. Dekker, Finding a zero by means of successive
C                 linear interpolation, Constructive Aspects of the
C                 Fundamental Theorem of Algebra, edited by B. Dejon
C                 and P. Henrici, Wiley-Interscience, 1969.
C***ROUTINES CALLED  R1MACH
C***REVISION HISTORY  (YYMMDD)
C   700901  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890531  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  FZERO