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Brent's Method

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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 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)/([f(x_3)-f(x_1)][f(x_3)-f(x_2)])+([y-f(x_2)][y-f(x_3)]x_1)/([f(x_1)-f(x_2)][f(x_1)-f(x_3)])+([y-f(x_3)][y-f(x_1)]x_2)/([f(x_2)-f(x_3)][f(x_2)-f(x_1)]).
(1)

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

x=x_2+P/Q,
(2)

where

P=S[T(R-T)(x_3-x_2)-(1-R)(x_2-x_1)]
(3)
Q=(T-1)(R-1)(S-1)
(4)

with

R=(f(x_2))/(f(x_3))
(5)
S=(f(x_2))/(f(x_1))
(6)
T=(f(x_1))/(f(x_3))
(7)

(Press et al. 1992).

See also

Bisection, Brent's Factorization Method, Root Bracketing, Root-Finding Algorithm

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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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Brent's Method

Cite this as:

Weisstein, Eric W. "Brent's Method." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrentsMethod.html

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