Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

Jean-Paul Chehab and Marcos Raydan
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Jean-Paul Chehab: LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France
Marcos Raydan: Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Ap. 89000, Caracas 1080-A, Venezuela

Mathematics, 2016, vol. 4, issue 3, 1-20

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.

Keywords: preconditioning; cones of matrices; gradient method; minimal residual method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2016
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