## 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 , , 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.