On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros
In the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to it...
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MDPI AG
2025-04-01
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| Online Access: | https://www.mdpi.com/1999-4893/18/4/205 |
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| author | Plamena I. Marcheva Ivan K. Ivanov Stoil I. Ivanov |
| author_facet | Plamena I. Marcheva Ivan K. Ivanov Stoil I. Ivanov |
| author_sort | Plamena I. Marcheva |
| collection | DOAJ |
| description | In the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to its computable initial condition. The obtained theorems improve and complement all existing such kind of convergence results about this method. At the end of the paper, we provide three numerical examples to show the applicability of our semilocal theorem to some physics problems. Within the examples, we propose a new algorithm for the experimental study of the dynamics of the simultaneous methods and compare the convergence and dynamical behaviors of the modified and the classical Weierstrass methods. |
| format | Article |
| id | doaj-art-3c9043c015d54f4ebb085c76de9bb3f1 |
| institution | DOAJ |
| issn | 1999-4893 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj-art-3c9043c015d54f4ebb085c76de9bb3f12025-08-20T03:14:16ZengMDPI AGAlgorithms1999-48932025-04-0118420510.3390/a18040205On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial ZerosPlamena I. Marcheva0Ivan K. Ivanov1Stoil I. Ivanov2Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaFaculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaFaculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaIn the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to its computable initial condition. The obtained theorems improve and complement all existing such kind of convergence results about this method. At the end of the paper, we provide three numerical examples to show the applicability of our semilocal theorem to some physics problems. Within the examples, we propose a new algorithm for the experimental study of the dynamics of the simultaneous methods and compare the convergence and dynamical behaviors of the modified and the classical Weierstrass methods.https://www.mdpi.com/1999-4893/18/4/205iteration methodspolynomial zerossimultaneous methodsmodified Weierstrass methodlocal convergencesemilocal convergence |
| spellingShingle | Plamena I. Marcheva Ivan K. Ivanov Stoil I. Ivanov On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros Algorithms iteration methods polynomial zeros simultaneous methods modified Weierstrass method local convergence semilocal convergence |
| title | On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros |
| title_full | On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros |
| title_fullStr | On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros |
| title_full_unstemmed | On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros |
| title_short | On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros |
| title_sort | on the q convergence and dynamics of a modified weierstrass method for the simultaneous extraction of polynomial zeros |
| topic | iteration methods polynomial zeros simultaneous methods modified Weierstrass method local convergence semilocal convergence |
| url | https://www.mdpi.com/1999-4893/18/4/205 |
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