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https://hdl.handle.net/2142/72056
Description
Title
Row Projection Methods for Linear Systems
Author(s)
Bramley, Randall Barry
Issue Date
1989
Doctoral Committee Chair(s)
Sameh, Ahmed H.
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Abstract
Row projection (RP) methods for solving linear systems of equations are algorithms that require orthogonal projections onto the set of solutions of subsets of the equations. RP methods are categorized and their convergence properties summarized and extended. An implementation approach is presented for nonsymmetric matrices arising from the seven point centered difference operator applied to three dimensional elliptic partial differential equations. This approach allows large scale parallelism in the computations, requires only a few extra vector of additional storage, provides good data locality, gives numerically well conditioned subproblems defining the projectors, and can be generalized to other matrices.
Three conjugate gradient (CG) accelerated RP methods based on this implementation are tested. The first two are generalizations of Kaczmarz's and Cimmino's methods, and the third is a new RP method which allows an explicit reduction in the problem size, minimizes the two norm of the error on every CG step, and requires less work per iteration than the equivalent Kaczmarz method. All three RP methods are shown to be more robust than some currently popular iterative nonsymmetric system solvers, and can solve systems with indefinite symmetric parts and eigenvalues not contained in a half plane of the complex plane. Additionally, the RP methods are as efficient as the best of the other methods. Explanations are given for these features, and the underlying relationship between the RP methods and CG applied to the normal equations is shown.
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