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 alwaysconverge 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) |
, giving | (2) |
 |  |  | (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.
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