Some Hyperbolic Iterative Methods for Linear Systems
The indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite i...
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Wiley
2020-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2020/9874162 |
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| author | K. Niazi Asil M. Ghasemi Kamalvand |
| author_facet | K. Niazi Asil M. Ghasemi Kamalvand |
| author_sort | K. Niazi Asil |
| collection | DOAJ |
| description | The indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J-orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J-Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J-Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations. |
| format | Article |
| id | doaj-art-570bd371d36e407eb4dc206ef3aa182d |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-570bd371d36e407eb4dc206ef3aa182d2025-08-20T02:05:59ZengWileyJournal of Applied Mathematics1110-757X1687-00422020-01-01202010.1155/2020/98741629874162Some Hyperbolic Iterative Methods for Linear SystemsK. Niazi Asil0M. Ghasemi Kamalvand1Department of Mathematics, Lorestan University, Khorramabad, IranDepartment of Mathematics, Lorestan University, Khorramabad, IranThe indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J-orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J-Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J-Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations.http://dx.doi.org/10.1155/2020/9874162 |
| spellingShingle | K. Niazi Asil M. Ghasemi Kamalvand Some Hyperbolic Iterative Methods for Linear Systems Journal of Applied Mathematics |
| title | Some Hyperbolic Iterative Methods for Linear Systems |
| title_full | Some Hyperbolic Iterative Methods for Linear Systems |
| title_fullStr | Some Hyperbolic Iterative Methods for Linear Systems |
| title_full_unstemmed | Some Hyperbolic Iterative Methods for Linear Systems |
| title_short | Some Hyperbolic Iterative Methods for Linear Systems |
| title_sort | some hyperbolic iterative methods for linear systems |
| url | http://dx.doi.org/10.1155/2020/9874162 |
| work_keys_str_mv | AT kniaziasil somehyperboliciterativemethodsforlinearsystems AT mghasemikamalvand somehyperboliciterativemethodsforlinearsystems |