Parallel Dichotomy Algorithm for solving tridiagonal system of linear equations with multiple right-hand sides
A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is calculating some rows of the inverse matrix of system...
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| Vydáno v: | Parallel computing Ročník 36; číslo 8; s. 423 - 438 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier B.V
01.08.2010
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| ISSN: | 0167-8191, 1872-7336 |
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| Abstract | A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is calculating some rows of the inverse matrix of system of linear algebraic equations. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved the main Gaussian Parallel Elimination Theorem for tridiagonal system of equations, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the MPI paradigm. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel Dichotomy Algorithm. |
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| AbstractList | A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is calculating some rows of the inverse matrix of system of linear algebraic equations. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved the main Gaussian Parallel Elimination Theorem for tridiagonal system of equations, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the MPI paradigm. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel Dichotomy Algorithm. |
| Author | Terekhov, Andrew V. |
| Author_xml | – sequence: 1 givenname: Andrew V. surname: Terekhov fullname: Terekhov, Andrew V. email: andrew.terekhov@mail.ru organization: Institute of Computational Mathematics and Mathematical Geophysics, 630090 Novosibirsk, Russia |
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| Keywords | Alternating direction method Poisson equation Parallel algorithm Thomas algorithm Fourier method Tridiagonal matrix algorithm (TDMA) |
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| SubjectTerms | Alternating direction method Fourier method Parallel algorithm Poisson equation Thomas algorithm Tridiagonal matrix algorithm (TDMA) |
| Title | Parallel Dichotomy Algorithm for solving tridiagonal system of linear equations with multiple right-hand sides |
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