Find a root of a function in given interval.
Return float, a zero of f between a and b. f must be a continuous function, and [a,b] must be a sign changing interval.
Description: Uses the classic Brent (1973) method to find a zero of the function f on the sign changing interval [a , b]. Generally considered the best of the rootfinding routines here. It is a safe version of the secant method that uses inverse quadratic extrapolation. Brent’s method combines root bracketing, interval bisection, and inverse quadratic interpolation. It is sometimes known as the van Wijngaarden-Deker-Brent method. Brent (1973) claims convergence is guaranteed for functions computable within [a,b].
[Brent1973] provides the classic description of the algorithm. Another description can be found in a recent edition of Numerical Recipes, including [PressEtal1992]. Another description is at http://mathworld.wolfram.com/BrentsMethod.html. It should be easy to understand the algorithm just by reading our code. Our code diverges a bit from standard presentations: we choose a different formula for the extrapolation step.
f : function
a : number
b : number
xtol : number, optional
maxiter : number, optional
args : tuple, optional
full_output : bool, optional
disp : bool, optional
x0 : float
r : RootResults (present if full_output = True)
f must be continuous. f(a) and f(b) must have opposite signs.
|[Brent1973]||(1, 2) Brent, R. P., Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973. Ch. 3-4.|
|[PressEtal1992]||(1, 2) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 352-355, 1992. Section 9.3: “Van Wijngaarden-Dekker-Brent Method.”|