2016
DOI: 10.3390/math4030046
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Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

Abstract: Abstract:We focus on inverse preconditioners based on minimizing F(X) = 1 − cos(XA, I), where XA is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods to minimize F(X) on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of F(X) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense a… Show more

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Cited by 1 publication
(9 citation statements)
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“…We have extended the MinCos iterative method, originally developed in [12] for symmetric and positive definite matrices, to approximate the inverse of the matrices associated with linear least-squares problems, and we have also described and adapted three different possible acceleration schemes to the generated convergent matrix sequences. Our experiments have shown that the geometrical MinCos scheme is also a robust option to approximate the inverse of matrices of the form A T A, associated with least-squares problems, without requiring the explicit knowledge of A T .…”
Section: Discussionmentioning
confidence: 99%
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“…We have extended the MinCos iterative method, originally developed in [12] for symmetric and positive definite matrices, to approximate the inverse of the matrices associated with linear least-squares problems, and we have also described and adapted three different possible acceleration schemes to the generated convergent matrix sequences. Our experiments have shown that the geometrical MinCos scheme is also a robust option to approximate the inverse of matrices of the form A T A, associated with least-squares problems, without requiring the explicit knowledge of A T .…”
Section: Discussionmentioning
confidence: 99%
“…In here PSD refers to the positive semi-definite closed cone of square matrices which possesses a rich geometrical structure; see, e.g., [1,11]. In Remark 2.1 we summarize the most important properties of Algorithm 2 (see [12]).…”
Section: The Mincos Methodsmentioning
confidence: 99%
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