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A Subspace-Projected Approximate Matrix Method for Systems of Linear Equations

Published online by Cambridge University Press:  28 May 2015

Jan Brandts  and
Ricardo R. da Silva
Jan Brandts*
Affiliation:
University of Amsterdam, Faculty of Science, Korteweg-de Vries Institute for Mathematics, Science Park 904, Amsterdam, Netherlands
Ricardo R. da Silva*
Affiliation:
University of Amsterdam, Faculty of Science, Korteweg-de Vries Institute for Mathematics, Science Park 904, Amsterdam, Netherlands
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

Given two n × n matrices A and A0 and a sequence of subspaces with dim the k-th subspace-projected approximated matrix Ak is defined as Ak = A + Πk(A0 − A)Πk, where Πk is the orthogonal projection on . Consequently, Akν = Aν and ν*Ak = ν*A for all Thus is a sequence of matrices that gradually changes from A0 into An = A. In principle, the definition of may depend on properties of Ak, which can be exploited to try to force Ak+1 to be closer to A in some specific sense. By choosing A0 as a simple approximation of A, this turns the subspace-approximated matrices into interesting preconditioners for linear algebra problems involving A. In the context of eigenvalue problems, they appeared in this role in Shepard et al. (2001), resulting in their Subspace Projected Approximate Matrix method. In this article, we investigate their use in solving linear systems of equations Ax = b. In particular, we seek conditions under which the solutions xk of the approximate systems Akxk = b are computable at low computational cost, so the efficiency of the corresponding method is competitive with existing methods such as the Conjugate Gradient and the Minimal Residual methods. We also consider how well the sequence (xk)k≥0 approximates x, by performing some illustrative numerical tests.

Keywords

65F1065F08Linear systemGalerkin methodsubspace projected approximate matrixminimal residual method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 2 , May 2013 , pp. 120 - 137
Copyright
Copyright © Global-Science Press 2013

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