Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems
A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/6/978 |
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| author | Ioannis K. Argyros Stepan Shakhno Halyna Yarmola |
| author_facet | Ioannis K. Argyros Stepan Shakhno Halyna Yarmola |
| author_sort | Ioannis K. Argyros |
| collection | DOAJ |
| description | A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the solution. In this article, such methods of higher convergence order are studied. In particular, we develop a two-step iterative method to solve large scale systems that does not require finding an inverse operator. Instead of the operator’s inverting, it uses a two-step Schultz approximation. The convergence is investigated using Lipschitz condition on the first-order derivatives. The cubic order of convergence is established and the results of the numerical experiment are given to determine the real benefits of the proposed method. |
| first_indexed | 2024-03-10T19:18:49Z |
| format | Article |
| id | doaj.art-59337b0043e540618eeea47d864b3945 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T19:18:49Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-59337b0043e540618eeea47d864b39452023-11-20T03:10:56ZengMDPI AGSymmetry2073-89942020-06-0112697810.3390/sym12060978Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale SystemsIoannis K. Argyros0Stepan Shakhno1Halyna Yarmola2Department of Mathematics, Cameron University, Lawton, OK 73505, USADepartment of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, UkraineDepartment of Computational Mathematics, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, UkraineA vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the solution. In this article, such methods of higher convergence order are studied. In particular, we develop a two-step iterative method to solve large scale systems that does not require finding an inverse operator. Instead of the operator’s inverting, it uses a two-step Schultz approximation. The convergence is investigated using Lipschitz condition on the first-order derivatives. The cubic order of convergence is established and the results of the numerical experiment are given to determine the real benefits of the proposed method.https://www.mdpi.com/2073-8994/12/6/978nonlinear equationiterative methodapproximation of inverse operatorlocal convergenceorder of convergenceLipschitz condition |
| spellingShingle | Ioannis K. Argyros Stepan Shakhno Halyna Yarmola Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems Symmetry nonlinear equation iterative method approximation of inverse operator local convergence order of convergence Lipschitz condition |
| title | Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems |
| title_full | Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems |
| title_fullStr | Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems |
| title_full_unstemmed | Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems |
| title_short | Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems |
| title_sort | method of third order convergence with approximation of inverse operator for large scale systems |
| topic | nonlinear equation iterative method approximation of inverse operator local convergence order of convergence Lipschitz condition |
| url | https://www.mdpi.com/2073-8994/12/6/978 |
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