Brent's Method

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 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 $x_1$, $x_2$, and $x_3$, Brent's method fits $x$ as a quadratic function of $y$, then uses the interpolation formula

$x={[y-f(x_1)][y-f(x_2)]x_3\over [f(x_3)-f(x_1)][f(x_3)-f(x_2)]}+{[y-f(x_2)][y-f(x_3)]x_1\over [f(x_1)-f(x_2)][f(x_1)-f(x_3)]}$
$+{[y-f(x_3)][y-f(x_1)]x_2\over [f(x_2)-f(x_3)][f(x_2)-f(x_1)]}.\quad$	(1)

Subsequent root estimates are obtained by setting $y=0$, giving

$$x=x_2+{P\over Q},$$ (2)

where

$$P = S[R(R-T)(x_3-x_2)-(1-R)(x_2-x_1)]$$ (3)

$$Q = (T-1)(R-1)(S-1)$$ (4)

with

$$R \equiv {f(x_2)\over f(x_3)}$$ (5)

$$S \equiv {f(x_2)\over f(x_1)}$$ (6)

$$T \equiv {f(x_1)\over f(x_3)}$$ (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.