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Approximate inverse computation using Frobenius inner product
Author(s) -
Montero G.,
González L.,
Flórez E.,
García M. D.,
Suárez A.
Publication year - 2002
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.269
Subject(s) - mathematics , subspace topology , krylov subspace , computation , inverse , product (mathematics) , matrix (chemical analysis) , inner product space , algorithm , combinatorics , iterative method , discrete mathematics , mathematical analysis , geometry , materials science , composite material
Parallel preconditioners are presented for the solution of general linear systems of equations. The computation of these preconditioners is achieved by orthogonal projections related to the Frobenius inner product. So, min M ∈ ∥ AM − I ∥ F and matrix M 0 ∈ corresponding to this minimum ( being any vectorial subspace of ℳ n (ℝ)) are explicitly computed using accumulative formulae in order to reduce computational cost when subspace is extended to another one containing it. Every step, the computation is carried out taking advantage of the previous one, what considerably reduces the amount of work. These general results are illustrated with the subspace of matrices M such that AM is symmetric. The main application is developed for the subspace of matrices with a given sparsity pattern which may be constructed iteratively by augmenting the set of non‐zero entries in each column. Finally, the effectiveness of the sparse preconditioners is illustrated with some numerical experiments. Copyright © 2002 John Wiley & Sons, Ltd.