Editing Brent's method (section)

{{Short description|Root-finding algorithm}}
{{For|Brent's cycle-detection algorithm|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 (scientist)|Richard Brent]]<ref>{{harvnb|Brent|1973}}</ref> and builds on an earlier algorithm by [[Theodorus Dekker]].<ref>{{harvnb|Dekker|1969}}</ref> 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;<ref>{{Cite journal |doi = 10.1016/S0965-9978(96)00051-8|title = A new hybrid quadratic/Bisection algorithm for finding the zero of a nonlinear function without using derivatives|journal = Advances in Engineering Software|volume = 28|issue = 3|pages = 145–149|year = 1997|last1 = Chandrupatla|first1 = Tirupathi R.}}</ref><ref>{{Cite web | url=https://www.embeddedrelated.com/showarticle/855.php | title=Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method - Jason Sachs}}</ref> [[Ridders' method]], which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the [[ITP Method|ITP method]] which is a hybrid between regula-falsi and bisection that achieves optimal worst-case and asymptotic guarantees.