Editing Bisection method (section)

{{short description|Algorithm for finding a zero of a function}}
{{about|searching zeros of continuous functions|searching a finite sorted array|binary search algorithm|the method of determining what software change caused a change in behavior|Bisection (software engineering)}}
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[[Image:Bisection method.svg|250px|thumb|A few steps of the bisection method applied over the starting range [a<sub>1</sub>;b<sub>1</sub>]. The bigger red dot is the root of the function.]]

In [[mathematics]], the '''bisection method''' is a [[Root-finding algorithm|root-finding method]] that applies to any [[continuous function]] for which one knows two values with opposite signs. The method consists of repeatedly [[Bisection|bisecting]] the [[Interval (mathematics)|interval]] defined by these values and then selecting the  subinterval in which the function changes sign, and therefore must contain a [[Root of a function|root]]. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.<ref>{{Harvnb|Burden|Faires|2014|p=51}}</ref> The method is also called the '''interval halving''' method,<ref>{{cite web |url=http://siber.cankaya.edu.tr/NumericalComputations/ceng375/node32.html |title=Interval Halving (Bisection) |access-date=2013-11-07 |url-status=dead |archive-url=https://web.archive.org/web/20130519092250/http://siber.cankaya.edu.tr/NumericalComputations/ceng375/node32.html |archive-date=2013-05-19 }}</ref> the '''[[Binary search algorithm|binary search method]]''',<ref>{{Harvnb|Burden|Faires|2014|p=28}}</ref> or the '''dichotomy method'''.<ref>{{Cite web|title = Dichotomy method - Encyclopedia of Mathematics|url = https://www.encyclopediaofmath.org/index.php/Dichotomy_method|website = www.encyclopediaofmath.org|access-date = 2015-12-21}}</ref>

For [[polynomial]]s, more elaborate methods exist for testing the existence of a root in an interval ([[Descartes' rule of signs]], [[Sturm's theorem]], [[Budan's theorem]]). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see [[Real-root isolation]].