{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:12:10Z","timestamp":1760238730179,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,24]],"date-time":"2020-08-24T00:00:00Z","timestamp":1598227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.<\/jats:p>","DOI":"10.3390\/sym12091408","type":"journal-article","created":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T09:24:56Z","timestamp":1598347496000},"page":"1408","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Convergence Analysis of a Modified Weierstrass Method for the Simultaneous Determination of Polynomial Zeros"],"prefix":"10.3390","volume":"12","author":[{"given":"Plamena I.","family":"Marcheva","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0082-7791","authenticated-orcid":false,"given":"Stoil I.","family":"Ivanov","sequence":"additional","affiliation":[{"name":"Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,24]]},"reference":[{"key":"ref_1","first-page":"1085","article-title":"Neuer Beweis des Satzes, dass jede ganze rationale Function einer Ver\u00e4nderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Ver\u00e4nderlichen","volume":"Volume II","author":"Weierstrass","year":"1891","journal-title":"Sitzungsber. K\u00f6nigl. Preuss. Akad. Wiss. Berlinn"},{"key":"ref_2","first-page":"1625","article-title":"Numerical Solution of Polynomial Equations","volume":"3","author":"Sendov","year":"1994","journal-title":"Handb. Numer. Anal."},{"key":"ref_3","unstructured":"Kjurkchiev, N.V. (1998). Initial Approximations and Root Finding Methods. Mathematical Research, Wiley."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M. (2008). Point Estimation of Root Finding Methods. Lecture Notes in Mathematics, Springer.","DOI":"10.1007\/978-3-540-77851-6"},{"key":"ref_5","first-page":"136","article-title":"Modified Newton method for simultaneous approximation of all roots of a given algebraic equation","volume":"5","author":"Dochev","year":"1962","journal-title":"J. Bulg. Acad. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.jco.2015.10.001","article-title":"General convergence theorems for iterative processes and applications to the Weierstrass root- finding method","volume":"33","author":"Proinov","year":"2016","journal-title":"J. Complex."},{"key":"ref_7","first-page":"131","article-title":"A new semilocal convergence theorem for the Weierstrass method from data at one point","volume":"59","author":"Proinov","year":"2006","journal-title":"C. R. Acad. Bulg. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"809","DOI":"10.7546\/CR-2013-66-6-13101331-5","article-title":"Convergence of the Weierstrass method for simultaneous approximation of polynomial zeros","volume":"66","author":"Proinov","year":"2013","journal-title":"C. R. Acad. Bulg. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1016\/j.jco.2013.11.002","article-title":"A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously","volume":"30","author":"Proinov","year":"2014","journal-title":"J. Complex."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s00009-020-01545-z","article-title":"Local and semilocal convergence of a family of multi-point Weierstrass-type root-finding methods","volume":"17","author":"Proinov","year":"2020","journal-title":"Mediterr. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"74","DOI":"10.5899\/2016\/cna-00261","article-title":"Convergence of the modified inverse Weierstrass method for simultaneous approximation of polynomial zeros","volume":"2016","author":"Nedzhibov","year":"2016","journal-title":"Commun. Numer. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.12988\/ijma.2016.69110","article-title":"Local Convergence of the Inverse Weierstrass Method for Simultaneous Approximation of Polynomial Zeros","volume":"10","author":"Nedzhibov","year":"2016","journal-title":"Int. J. Math. Anal."},{"key":"ref_13","unstructured":"Nedzhibov, G.H. (, January October). Improved local convergence analysis of the Inverse Weierstrass method for simultaneous approximation of polynomial zeros. Proceedings of the MATTEX 2018 Conference, Targovishte, Bulgaria."},{"key":"ref_14","first-page":"247","article-title":"On semilocal convergence analysis of the Inverse Weierstrass method for simultaneous computing of polynomial zeros","volume":"11","author":"Nedzhibov","year":"2019","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"key":"ref_15","unstructured":"Nedzhibov, G.H. (2016, January 11\u201313). On local convergence analysis of the Inverse WDK method. Proceedings of the MATTEX 2016 Conference, Shumen, Bulgaria."},{"key":"ref_16","first-page":"266","article-title":"New local convergence theorems for the Inverse Weierstrass method for simultaneous approximation of polynomial zeros","volume":"10","author":"Nedzhibov","year":"2018","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.aml.2015.08.016","article-title":"Relationships between different types of initial conditions for simultaneous root finding methods","volume":"52","author":"Proinov","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.jco.2008.05.006","article-title":"General local convergence theory for a class of iterative processes and its applications to Newton\u2019s method","volume":"25","author":"Proinov","year":"2009","journal-title":"J. Complex."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.jco.2009.05.001","article-title":"New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems","volume":"26","author":"Proinov","year":"2010","journal-title":"J. Complex."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10092-018-0251-x","article-title":"Unified convergence analysis for Picard iteration in n-dimensional vector spaces","volume":"55","author":"Proinov","year":"2018","journal-title":"Calcolo"},{"key":"ref_21","first-page":"1081","article-title":"Local convergence of Chebyshev-like method for simultaneous finding polynomial zeros","volume":"66","author":"Cholakov","year":"2013","journal-title":"C. R. Acad. Bulg. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cholakov, S.I. (2019). Local and semilocal convergence of Wang-Zheng\u2019s method for simultaneous finding polynomial zeros. Symmetry, 11.","DOI":"10.3390\/sym11060736"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.cam.2017.02.038","article-title":"A convergence analysis of a fourth-order method for computing all zeros of a polynomial simultaneously","volume":"321","author":"Cholakov","year":"2017","journal-title":"J. Comp. Appl. Math."},{"key":"ref_24","first-page":"705","article-title":"Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros","volume":"59","author":"Proinov","year":"2006","journal-title":"C. R. Acad. Bulg. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s10092-015-0155-y","article-title":"On the local convergence of Ehrlich method for numerical computation of polynomial zeros","volume":"53","author":"Proinov","year":"2016","journal-title":"Calcolo"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s13160-014-0138-4","article-title":"Convergence of the two-point Weierstrass root-finding method","volume":"31","author":"Proinov","year":"2014","journal-title":"Jpn. J. Indust. Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1186\/s13660-015-0855-5","article-title":"On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously","volume":"2015","author":"Proinov","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_28","first-page":"957","article-title":"On a family of Weierstrass-type root-finding methods with accelerated convergence","volume":"273","author":"Proinov","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.22436\/jnsa.011.09.03","article-title":"On the local convergence of Gargantini-Farmer-Loizou method for simultaneous approximation of multiple polynomial zeros","volume":"11","author":"Proinov","year":"2018","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_30","first-page":"669","article-title":"Semilocal convergence of Chebyshev-like root-finding method for simultaneous approximation of polynomial zeros","volume":"236","author":"Proinov","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/S0377-0427(99)00375-1","article-title":"A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic","volume":"130","author":"Sun","year":"2001","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1090\/S0025-5718-1973-0329236-7","article-title":"Iteration methods for finding all zeros of a polynomial simultaneously","volume":"27","author":"Aberth","year":"1973","journal-title":"Math. Comp."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1408\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:05:42Z","timestamp":1760177142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1408"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,24]]},"references-count":32,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091408"],"URL":"https:\/\/doi.org\/10.3390\/sym12091408","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,24]]}}}