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{{Short description|Algorithms for zeros of functions}}
<noinclude>{{User:RMCD bot/subject notice|1=Root-finding algorithm|2=Talk:Root-finding algorithms#Requested move 21 July 2024}}
</noinclude>In [[numerical analysis]], a '''root-finding algorithm''' is an [[algorithm]] for finding [[Zero of a function|zeros]], also called "roots", of [[continuous function]]s. A [[zero of a function]] {{math|''f''}}, from the [[real number]]s to real numbers or from the [[complex number]]s to the complex numbers, is a number {{math|''x''}} such that {{math|1=''f''(''x'') = 0}}. As, generally, the zeros of a function cannot be computed exactly nor expressed in [[closed form expression|closed form]], root-finding algorithms provide approximations to zeros, expressed either as [[floating-point arithmetic|floating-point]] numbers or as small isolating [[interval (mathematics)|intervals]], or [[disk (mathematics)|disks]] for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).<ref>{{Cite book |last1=Press |first1=W. H. |title=Numerical Recipes: The Art of Scientific Computing |last2=Teukolsky |first2=S. A. |last3=Vetterling |first3=W. T. |last4=Flannery |first4=B. P. |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-88068-8 |edition=3rd |publication-place=New York |chapter=Chapter 9. Root Finding and Nonlinear Sets of Equations |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=442}}</ref>
[[Equation solving|Solving an equation]] {{math|1=''f''(''x'') = ''g''(''x'')}} is the same as finding the roots of the function {{math|1=''h''(''x'') = ''f''(''x'') – ''g''(''x'')}}. Thus root-finding algorithms allow solving any [[equation (mathematics)|equation]] defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists.
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