A New Family of High-Order Ehrlich-Type Iterative Methods
One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. W...
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| Format: | Article |
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MDPI AG
2021-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1855 |
| _version_ | 1827684835001368576 |
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| author | Petko D. Proinov Maria T. Vasileva |
| author_facet | Petko D. Proinov Maria T. Vasileva |
| author_sort | Petko D. Proinov |
| collection | DOAJ |
| description | One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods <i>Ehrlich’s methods with correction</i>. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems. |
| first_indexed | 2024-03-10T08:37:44Z |
| format | Article |
| id | doaj.art-08bd1d00477a4c43a2596d529ce5783b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:37:44Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-08bd1d00477a4c43a2596d529ce5783b2023-11-22T08:32:57ZengMDPI AGMathematics2227-73902021-08-01916185510.3390/math9161855A New Family of High-Order Ehrlich-Type Iterative MethodsPetko D. Proinov0Maria T. Vasileva1Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaFaculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaOne of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods <i>Ehrlich’s methods with correction</i>. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.https://www.mdpi.com/2227-7390/9/16/1855iterative methodssimultaneous methodsEhrlich methodpolynomial zerosaccelerated convergencelocal convergence |
| spellingShingle | Petko D. Proinov Maria T. Vasileva A New Family of High-Order Ehrlich-Type Iterative Methods Mathematics iterative methods simultaneous methods Ehrlich method polynomial zeros accelerated convergence local convergence |
| title | A New Family of High-Order Ehrlich-Type Iterative Methods |
| title_full | A New Family of High-Order Ehrlich-Type Iterative Methods |
| title_fullStr | A New Family of High-Order Ehrlich-Type Iterative Methods |
| title_full_unstemmed | A New Family of High-Order Ehrlich-Type Iterative Methods |
| title_short | A New Family of High-Order Ehrlich-Type Iterative Methods |
| title_sort | new family of high order ehrlich type iterative methods |
| topic | iterative methods simultaneous methods Ehrlich method polynomial zeros accelerated convergence local convergence |
| url | https://www.mdpi.com/2227-7390/9/16/1855 |
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