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On a stopping rule for iterative orthogonalization of symmetric matrices
Author(s) -
Popa Constantin
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810823
Subject(s) - orthogonalization , eigenvalues and eigenvectors , mathematics , discretization , convergence (economics) , matrix (chemical analysis) , eigenvalue algorithm , iterative method , matrix norm , norm (philosophy) , collocation (remote sensing) , mathematical optimization , fredholm integral equation , integral equation , symmetric matrix , algorithm , computer science , mathematical analysis , hamiltonian matrix , physics , materials science , quantum mechanics , machine learning , political science , law , economics , composite material , economic growth
Abstract In this paper we consider three versions of Kovarik's iterative orthogonalization algorithms, for approximating the minimal norm solution of symmetric least squares problems. Although the convergence of these algorithms is linear, in practical applications we observed that a too big number of iterations can dramatically deteriorate the already obtained approximation. In this respect we analyse the above mentioned Kovarik–like methods according to the modifications they make on the “machine zero” eigenvalues of the problem (symmetric) matrix. We establish a theoretical almost optimal formula for the number of iterations necessary to obtain an enough accurate approximation, as well as to avoid the above mentioned troubles. Experiments on collocation discretization of a Fredholm first kind integral equation ilustrate the efficiency of our considerations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)