Next Article in Journal
Sharing of Nonlocality of a Single Member of an Entangled Pair of Qubits Is Not Possible by More than Two Unbiased Observers on the Other Wing
Next Article in Special Issue
Data Clustering with Quantum Mechanics
Previous Article in Journal
Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations
Article Menu

Export Article

Open AccessArticle
Mathematics 2016, 4(3), 46;

Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France
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)
Full-Text   |   PDF [702 KB, uploaded 9 July 2016]   |  


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

Graphical abstract

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).

Share & Cite This Article

MDPI and ACS Style

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

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top