Analytic and empirical study of the rate of convergence of some iterative methods
We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods \[ x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots, x_{{i_{k-1}+n-k+1}}),\,n\geq k-1, \] where \(T\colon X^k\rightarrow X\) is a mapping satisfying some Presic type c...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Publishing House of the Romanian Academy
2015-12-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://ictp.acad.ro/jnaat/journal/article/view/1059 |
| _version_ | 1818108429787463680 |
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| author | Vasile Berinde Abdul Rahim Khan Mădălina Păcurar |
| author_facet | Vasile Berinde Abdul Rahim Khan Mădălina Păcurar |
| author_sort | Vasile Berinde |
| collection | DOAJ |
| description |
We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods
\[
x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots, x_{{i_{k-1}+n-k+1}}),\,n\geq k-1,
\]
where \(T\colon X^k\rightarrow X\) is a mapping satisfying some Presic type contraction conditions and \((i_0,i_1,\dots,i_{k-1})\) is a permutation of \((0,1,\dots,k-1)\).
We also consider the Picard iteration associated to the fixed point problem \(x=T(x,\dots,x)\) and compare analytically and empirically the rate and speed of convergence of the three iterative methods. Our approach opens a new perspective on the study of the rate of convergence / speed of convergence of fixed point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments. |
| first_indexed | 2024-12-11T02:15:13Z |
| format | Article |
| id | doaj.art-18d98f0c71884a17a0ae24588478dd2d |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-12-11T02:15:13Z |
| publishDate | 2015-12-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-18d98f0c71884a17a0ae24588478dd2d2022-12-22T01:24:11ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2015-12-01441Analytic and empirical study of the rate of convergence of some iterative methodsVasile Berinde0Abdul Rahim Khan1Mădălina Păcurar2North University of Baia MareKing Fahd University of Petroleum and MineralsBabeş-Bolyai University We study analytically and empirically the rate of convergence of two \(k\)-step fixed point iterative methods in the family of methods \[ x_{n+1}=T(x_{i_0+n-k+1},x_{i_1+n-k+1},\dots, x_{{i_{k-1}+n-k+1}}),\,n\geq k-1, \] where \(T\colon X^k\rightarrow X\) is a mapping satisfying some Presic type contraction conditions and \((i_0,i_1,\dots,i_{k-1})\) is a permutation of \((0,1,\dots,k-1)\). We also consider the Picard iteration associated to the fixed point problem \(x=T(x,\dots,x)\) and compare analytically and empirically the rate and speed of convergence of the three iterative methods. Our approach opens a new perspective on the study of the rate of convergence / speed of convergence of fixed point iterative methods and also illustrates the essential difference between them by means of some concrete numerical experiments.https://ictp.acad.ro/jnaat/journal/article/view/1059metric spacecontractive mappingfixed pointk-step fixed pointiterative methodrate of convergence |
| spellingShingle | Vasile Berinde Abdul Rahim Khan Mădălina Păcurar Analytic and empirical study of the rate of convergence of some iterative methods Journal of Numerical Analysis and Approximation Theory metric space contractive mapping fixed point k-step fixed point iterative method rate of convergence |
| title | Analytic and empirical study of the rate of convergence of some iterative methods |
| title_full | Analytic and empirical study of the rate of convergence of some iterative methods |
| title_fullStr | Analytic and empirical study of the rate of convergence of some iterative methods |
| title_full_unstemmed | Analytic and empirical study of the rate of convergence of some iterative methods |
| title_short | Analytic and empirical study of the rate of convergence of some iterative methods |
| title_sort | analytic and empirical study of the rate of convergence of some iterative methods |
| topic | metric space contractive mapping fixed point k-step fixed point iterative method rate of convergence |
| url | https://ictp.acad.ro/jnaat/journal/article/view/1059 |
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