Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root-finding method of Weierstrass, also known as the Durand\u2013Kerner-method: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130\u00a0years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties.<\/p>\n\n

We show that the Weierstrass method, like the well-known Newton method, is not generally convergent: there are open sets of polynomials\u00a0

We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree

Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems.<\/p>","DOI":"10.1090\/mcom\/3783","type":"journal-article","created":{"date-parts":[[2022,9,14]],"date-time":"2022-09-14T14:32:59Z","timestamp":1663165979000},"page":"839-866","source":"Crossref","is-referenced-by-count":2,"title":["The Weierstrass\u2013Durand\u2013Kerner root finder is not generally convergent"],"prefix":"10.1090","volume":"92","author":[{"given":"Bernhard","family":"Reinke","sequence":"first","affiliation":[]},{"given":"Dierk","family":"Schleicher","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Stoll","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,11,10]]},"reference":[{"issue":"298","key":"1","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1090\/mcom\/2985","article-title":"On the speed of convergence of Newton\u2019s method for complex polynomials","volume":"85","author":"Bilarev, Todor","year":"2016","journal-title":"Math. 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