General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros
In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative me...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1599 |
| _version_ | 1797553482947887104 |
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| author | Stoil I. Ivanov |
| author_facet | Stoil I. Ivanov |
| author_sort | Stoil I. Ivanov |
| collection | DOAJ |
| description | In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros. |
| first_indexed | 2024-03-10T16:16:37Z |
| format | Article |
| id | doaj.art-f36d6d75e4714f3ea14271d41d90a9e5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:16:37Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f36d6d75e4714f3ea14271d41d90a9e52023-11-20T14:02:30ZengMDPI AGMathematics2227-73902020-09-0189159910.3390/math8091599General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial ZerosStoil I. Ivanov0Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, BulgariaIn this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.https://www.mdpi.com/2227-7390/8/9/1599iterative methodslocal convergenceerror estimatesnormed fieldsSuper-Halley methodpolynomial zeros |
| spellingShingle | Stoil I. Ivanov General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros Mathematics iterative methods local convergence error estimates normed fields Super-Halley method polynomial zeros |
| title | General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros |
| title_full | General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros |
| title_fullStr | General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros |
| title_full_unstemmed | General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros |
| title_short | General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros |
| title_sort | general local convergence theorems about the picard iteration in arbitrary normed fields with applications to super halley method for multiple polynomial zeros |
| topic | iterative methods local convergence error estimates normed fields Super-Halley method polynomial zeros |
| url | https://www.mdpi.com/2227-7390/8/9/1599 |
| work_keys_str_mv | AT stoiliivanov generallocalconvergencetheoremsaboutthepicarditerationinarbitrarynormedfieldswithapplicationstosuperhalleymethodformultiplepolynomialzeros |