fzero.f
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