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Volume 11 number 1

Pages: 61-66

An Optimal Linear Estimation Approach to the Parallel Solution of Linear Algebraic Systems of Equations

Atair Rios Neto(1) e Wilson Rios Neto(2)

    1 - Universidade do Vale do Paraba-UNIVAP
    Instituto de Pesquisa e Desenvolvimento-IP&D
    12244-000 So Jose dos Campos, SP
    e-mail: [email protected]
    2 - Virginia Polytechnic Institute and State University
    Center for Intelligent Material Systems and Structures - CIMSS
    Blacksburg, Virginia 24061-0261
    e mail: [email protected]
Resumo:
Estimao linear tima estocstica de parmetros utilizada para gerar um novo mtodo iterativo, para a soluo paralela de sistemas algbricos de equaes. No limite, a abordagem proposta leva a algoritmos iterativos que resolvem em paralelo sistemas lineares, varivel por varivel. mostrado, tanto no caso de sistemas determinados como no de sistemas indeterminados, que o mtodo iterativo de soluo paralela desenvolvido equivalente, em cada iterao, a se aplicar a uma funo objetivo quadrtica, e definida positiva, um mtodo de Newton modificado, com convergncia garantida. A motivao se ter um mtodo que explore as possibilidades oferecidas por processamento paralelo, que pode ser til na soluo eficiente de sistemas de larga escala, especialmente aqueles envolvendo matrizes de coeficientes esparsas. mostrado tambm que a abordagem mais geral por estimao estocstica leva a um mtodo que generaliza e que, em consequncia, se espera ter melhor desempenho que o mtodo usual de Jacobi. Embora sejam apresentados exemplos numricos, este trabalho no trata do aspecto de testes e avaliao de desempenho numrico e est focado no desenvolvimento heurstico e na verificao de convergncia do mtodo proposto.
Palavras Chave: Sistemas Lineares, Soluo Paralela de Sistemas Lineares, Mtodo Generalizado de Jacobi.
Abstract:
Stochastic optimal linear estimation of parameters is used to generate a new iterative method for the parallel solution of systems of linear algebraic equations. In the limit the approach proposed leads to iterative algorithms which in parallel can solve linear systems, variable by variable. It is shown, for both determined and undetermined systems, that the parallel iterative method developed is in each iteration equivalent to apply to a quadratic positive definite functional a modified Newton method, with guaranteed convergence. The motivation is to have a method which explores the possibilities offered by parallel processing and that can be useful in the efficient solution of large scale systems, especially those with sparse coefficient matrices. It is shown also that the more general stochastic approach taken leads to a method which generalizes and which as a consequence is expected to perform better than the usual Jacobi method. Though numerical examples are presented, this paper does not yet address the aspect of testing and evaluation of numerical performance and is focused in presenting the heuristic development and verification of convergence of the proposed method.
Keywords: Algebraic linear systems; Parallel solution of linear systems; Generalized Jacobi method.

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