Date: 14 Jul 2005

Repeated matrix squaring for the parallel solution of linear systems

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Abstract

Given a n×n nonsingular linear system Ax=b, we prove that the solution x can be computed in parallel time ranging from Ω(log n) to O(log2 n), provided that the condition number, μ(A), of A is bounded by a polynomial in n. In particular, if μ(A) = O(1), a time bound O(log n) is achieved. To obtain this result, we reduce the computation of x to repeated matrix squaring and prove that a number of steps independent of n is sufficient to approximate x up to a relative error 2−d, d=O(1). This algorithm has both theoretical and practical interest, achieving the same bound of previously published parallel solvers, but being far more simple.

This work has been partly supported by the Italian National Research Council, under the “Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo”, subproject 2 “Processori dedicati”. Part of this work was done while the first author was with the Istituto di Elaborazione dell'Informazione, Consiglio Nazionale delle Ricerche, Pisa (Italy).