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14 Jul 2005
Repeated matrix squaring for the parallel solution of linear systems
 Bruno Codenotti,
 Mauro Leoncini,
 Giovanni Resta
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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(log^{2} 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).
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 Title
 Repeated matrix squaring for the parallel solution of linear systems
 Book Title
 PARLE '92 Parallel Architectures and Languages Europe
 Book Subtitle
 4th International PARLE Conference Paris, France, June 15–18, 1992 Proceedings
 Pages
 pp 725732
 Copyright
 1992
 DOI
 10.1007/3540555994_120
 Print ISBN
 9783540555995
 Online ISBN
 9783540472506
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 605
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
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 Editors
 Authors

 Bruno Codenotti ^{(1)}
 Mauro Leoncini ^{(2)}
 Giovanni Resta ^{(2)}
 Author Affiliations

 1. Int. Comp. Sci. Instit., 94704, Berkeley, Ca
 2. Ist. di Elaborazione dell'Informazione, Consiglio Nazionale delle Ricerche, Pisa, Italy
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