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Brent's method

For Brent's cycle-detection algorithm, see Cycle detection § Brent's algorithm.

In numerical analysis, Brent's method is a hybrid root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent's method is due to Richard Brent[1] and builds on an earlier algorithm by Theodorus Dekker.[2] Consequently, the method is also known as the Brent–Dekker method.

Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots;[3][4] Ridders' method, which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the ITP method which is a hybrid between regula-falsi and bisection that achieves optimal worst-case and asymptotic guarantees.

  1. ^ Brent 1973
  2. ^ Dekker 1969
  3. ^ Chandrupatla, Tirupathi R. (1997). "A new hybrid quadratic/Bisection algorithm for finding the zero of a nonlinear function without using derivatives". Advances in Engineering Software. 28 (3): 145–149. doi:10.1016/S0965-9978(96)00051-8.
  4. ^ "Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method - Jason Sachs".