Browsing by Author "Atya, Naima Ibrahim"
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Item NEW PRECONDITIONERS FOR STATIONARY ITERATIVE METHODS(2022-03-01) Atya, Naima Ibrahim; Özban, Ahmet YaşarThe convergence of an iterative method used for the solution of systems of linear equation depends on the properties of the spectrum of the matrix of the linear sys tem. So, to speed up the convergence, the given linear system is transformed into an equivalent one by linear transformations, known as preconditioners. In this thesis, we introduce two new preconditioners for Jacobi and Gauss-Seidel (GS) iterative methods for the solution of linear systems with strictly columnwise diago nally dominant (SCDD) L− matrices and SCDD positive matrices. The new precon ditioners can be applied on a single row, on a limited number of rows, called partial preconditioning, or on all rows, called complete preconditioning, of the system ma trix. First of all, the properties of the preconditioned matrices are determined for systems with SCDD L− matrices and SCDD positive matrices. Then convergence analysis of the Jacobi and GS iterative methods are performed for the preconditioned systems. For systems with SCDD L− matrices, it is shown that the spectral radii of Jacobi and GS iteration matrices for preconditioned systems are smaller than the ones associated with unpreconditioned systems. Although, for systems with SCDD positive matrices, we prove that the spectral radii of Jacobi iteration matrices for preconditioned systems are smaller than the ones associated with unpreconditioned systems, no such result is available for GS iteration matrices. Numerical results show that for systems with SCDD L−matrices, the new precondi tioners are quite competitive with the ones existing in the literature in the sense of spectral radii and the number of iterations. Nevertheless, for systems with SCDD positive matrices, numerical results demonstrate that although new preconditioners are still competitive with some other preconditioners, usually they are not preferable on many of the already existing ones. Finally, the performances of new precondition ers for CDD L−matrices and non-CDD L−matrices and even for non-SCDD positive matrices deserve further research.