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Approximate solutions and eigenvalue bounds from Krylov subspaces
Author(s) -
Paige Chris C.,
Parlett Beresford N.,
van der Vorst Henk A.
Publication year - 1995
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.1680020205
Subject(s) - mathematics , krylov subspace , conjugate gradient method , lanczos resampling , eigenvalues and eigenvectors , linear subspace , generalized minimal residual method , ritz method , conjugate residual method , residual , polynomial , convergence (economics) , matrix (chemical analysis) , mathematical analysis , computation , linear system , mathematical optimization , pure mathematics , algorithm , materials science , gradient descent , boundary value problem , economic growth , computer science , composite material , quantum mechanics , machine learning , artificial neural network , physics , economics
Approximations to the solution of a large sparse symmetric system of equations are considered. The conjugate gradient and minimum residual approximations are studied without reference to their computation. Several different bases for the associated Krylov subspace are used, including the usual Lanczos basis. The zeros of the iteration polynomial for the minimum residual approximation ( harmonic Ritz values) are characterized in several ways and, in addition, attractive convergence properties are established. The connection of these harmonic Ritz values to Lehmann's optimal intervals for eigenvalues of the original matrix appears to be new.