Brent's Method

Brent's method is a root-finding algorithm which combines root bracketing, bisection, and inverse quadratic interpolation. It is sometimes known as the van Wijngaarden-Deker-Brent method. Brent's method is implemented in the Wolfram Language as the undocumented option Method -> Brent in FindRoot[eqn, x, x0, x1].

Brent's method uses a Lagrange interpolating polynomial of degree 2. Brent (1973) claims that this method will always converge as long as the values of the function are computable within a given region containing a root. Given three points , , and , Brent's method fits  as a quadratic function of , then uses the interpolation formula

(1)

Subsequent root estimates are obtained by setting , giving

(2)

where

			(3)
			(4)

with

			(5)
			(6)
			(7)

(Press et al. 1992).

REFERENCES:

Brent, R. P. Ch. 3-4 in Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973.

Forsythe, G. E.; Malcolm, M. A.; and Moler, C. B. §7.2 in Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Van Wijngaarden-Dekker-Brent Method." §9.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 352-355, 1992.