Please use this identifier to cite or link to this item: http://hdl.handle.net/10553/72746
Title: Orthogonal projections of the identity: spectral analysis and applications to approximate inverse preconditioning
Authors: GonzáLez SáNchez, Luis 
UNESCO Clasification: 1206 Análisis numérico
Keywords: Frobenius norm
Orthogonal projection
Eigenvalues
Singular values
Approximate inverse preconditioning
Issue Date: 2006
Journal: SIAM Review 
Abstract: Many strategies for constructing different structures of sparse approximate inverse preconditioners for large linear systems have been proposed in the literature. In a more general framework, this paper analyzes the theoretical effectiveness of the optimal preconditioner (in the Frobenius norm) of a linear system over an arbitrary subspace of M-n(R). For this purpose, the spectral analysis of the Frobenius orthogonal projections of the identity matrix onto the linear subspaces of M-n(R) is performed. This analysis leads to a simple, general criterion: The effectiveness of the optimal approximate inverse preconditioners (parametrized by any vectorial structure) improves at the same time as the smallest singular value (or the smallest eigenvalue's modulus) of the corresponding preconditioned matrices increases to 1.
URI: http://hdl.handle.net/10553/72746
ISSN: 0036-1445
DOI: 10.1137/S0036144504431905
Source: Siam Review [ISSN 0036-1445], v. 48 (1), p. 66-75, (Marzo 2006)
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