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Iterative minimization of the Rayleigh quotient by block steepest descent iterations
Author(s) -
Neymeyr Klaus,
Zhou Ming
Publication year - 2014
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/nla.1915
Subject(s) - mathematics , rayleigh quotient , rayleigh quotient iteration , eigenvalues and eigenvectors , subspace topology , convergence (economics) , gradient descent , method of steepest descent , iterative method , positive definite matrix , matrix (chemical analysis) , quotient , eigendecomposition of a matrix , algorithm , combinatorics , mathematical optimization , power iteration , mathematical analysis , computer science , physics , materials science , quantum mechanics , machine learning , artificial neural network , economics , composite material , economic growth
SUMMARY The topic of this paper is the convergence analysis of subspace gradient iterations for the simultaneous computation of a few of the smallest eigenvalues plus eigenvectors of a symmetric and positive definite matrix pair ( A , M ). The methods are based on subspace iterations for A − 1 M and use the Rayleigh‐Ritz procedure for convergence acceleration. New sharp convergence estimates are proved by generalizing estimates, which have been presented for vectorial steepest descent iterations (see SIAM J. Matrix Anal. Appl., 32(2):443‐456, 2011). Copyright © 2013 John Wiley & Sons, Ltd.