Parallel Dichotomy Algorithm for solving tridiagonal system of linear equations with multiple right-hand sides
Andrew V. Terekhov
- 01 Aug 2010
- Vol. 36, Iss: 8, pp 423-438
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TL;DR: A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides and an original algorithm for calculating share components of the solution vector is proposed and studied.
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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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Citations
A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method
TL;DR: This study develops a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides and proposes a parallel realization of the domain decomposition method (the Schur complement method).
High-performance modeling acoustic and elastic waves using the parallel Dichotomy Algorithm
TL;DR: The proposed parallel fine-mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy and was demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors.
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A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method
Andrew V. Terekhov
- 01 Jun 2013
TL;DR: In this paper, a parallel algorithm for solving block-tridiagonal systems of equations is presented, which is an effective and simple set of procedures for solving engineering tasks on a supercomputer.
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A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations
TL;DR: A modification of the "Dichotomy Algorithm" (Terekhov, 2010) is proposed, aimed at parallel realization of a broad class of numerical methods including the inversion of Toeplitz and quasi-Toeplitzer tridiagonal matrices.
The Laguerre finite difference one-way equation solver
TL;DR: A new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations using the integral Laguerre transform instead of Fourier transform is presented.
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