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On preconditioning and penalized matrices
Author(s) -
Dostál Zdeněk
Publication year - 1999
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199903)6:2<109::aid-nla150>3.0.co;2-0
Subject(s) - preconditioner , conjugate gradient method , mathematics , positive definite matrix , rank (graph theory) , convergence (economics) , rate of convergence , matrix (chemical analysis) , conjugate residual method , conjugate , spectrum (functional analysis) , mathematical analysis , linear system , algorithm , eigenvalues and eigenvectors , gradient descent , combinatorics , computer science , channel (broadcasting) , computer network , physics , materials science , quantum mechanics , economics , composite material , economic growth , machine learning , artificial neural network
An alternative approach to the preconditioning of a system of linear equations with a matrix A + ρ C T C that is the sum of a positive definite matrix A and a penalization term is proposed. After showing that there is a gap in the spectrum of A + ρ C T C provided ρ is sufficiently large, a preconditioner for A is applied to A + ρ C T C in such a way that it preserves the gap in the spectrum but still improves the convergence of the conjugate gradient method. A bound on the rate of convergence of the conjugate gradient method with our preconditioning based on the estimates by Axelsson is given that depends neither on ρ nor on the rank of C . Numerical experiments confirm the efficiency of the approach presented. Copyright © 1999 John Wiley & Sons, Ltd.