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Mathematics 2016, 4(3), 46; doi:10.3390/math4030046

Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

1
LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France
2
Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Ap. 89000, Caracas 1080-A, Venezuela
*
Author to whom correspondence should be addressed.
Academic Editor: Khalide Jbilou
Received: 11 May 2016 / Revised: 29 June 2016 / Accepted: 1 July 2016 / Published: 9 July 2016
(This article belongs to the Special Issue Numerical Linear Algebra with Applications)
View Full-Text   |   Download PDF [702 KB, uploaded 9 July 2016]   |  

Abstract

We focus on inverse preconditioners based on minimizing F ( X ) = 1 cos ( X A , I ) , where X A 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 and sparse approximations are included. View Full-Text
Keywords: preconditioning; cones of matrices; gradient method; minimal residual method preconditioning; cones of matrices; gradient method; minimal residual method
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Chehab, J.-P.; Raydan, M. Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices. Mathematics 2016, 4, 46.

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