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The method is not generally convergent |
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{{refstyle|date=November 2020}}
In [[numerical analysis]], the '''Weierstrass method''' or '''Durand–Kerner method''', discovered by [[Karl Weierstrass]] in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a [[root-finding algorithm]] for solving [[polynomial]] [[equation (mathematics)|equations]].<ref name="Petković">{{cite book |last1=Petković |first1=Miodrag |title=Iterative methods for simultaneous inclusion of polynomial zeros |date=1989 |publisher=Springer |___location=Berlin [u.a.] |isbn=978-3-540-51485-5 |pages=31–32}}</ref> In other words, the method can be used to solve numerically the equation
: ''f''(''x'') = 0,
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each containing exactly one zero of ''f''.
==Failing general convergence==
The Weierstrass / Durand-Kerner method is not generally convergent: in other words, it is not true that for every polynomial, the set of initial vectors that eventually converges to roots is open and dense. In fact, there are open sets of polynomials that have open sets of initial vectors that converge to periodic cycles other than roots (see Reinke et al.)
==References==
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* {{cite journal|first= Arnold|last= Neumaier|title= Enclosing clusters of zeros of polynomials|journal= Journal of Computational and Applied Mathematics|volume= 156 |year=2003|url=https://www.mat.univie.ac.at/~neum/papers.html#polzer|doi= 10.1016/S0377-0427(03)00380-7|pages= 389|doi-access= free}}
* Jan Verschelde, ''[http://www2.math.uic.edu/~jan/mcs471f03/Project_Two/proj2/node2.html The method of Weierstrass (also known as the Durand–Kerner method)]'', 2003.
* Bernhard Reinke, Dierk Schleicher, and Michael Stoll, ``[https://arxiv.org/abs/2004.04777 The Weierstrass root finder is not generally convergent]'', 2020
==External links==
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