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Ritz and harmonic Ritz values and the convergence of FOM and GMRES
Author(s) -
Goossens Serge,
Roose Dirk
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(199906)6:4<281::aid-nla158>3.0.co;2-b
Subject(s) - generalized minimal residual method , ritz method , mathematics , krylov subspace , eigenvalues and eigenvectors , convergence (economics) , norm (philosophy) , mathematical analysis , linear system , boundary value problem , physics , quantum mechanics , political science , law , economics , economic growth
The Ritz and harmonic Ritz values are approximate eigenvalues, which can be computed cheaply within the FOM and GMRES Krylov subspace iterative methods for solving non‐symmetric linear systems. They are also the zeros of the residual polynomials of FOM and GMRES, respectively. In this paper we show that the Walker–Zhou interpretation of GMRES enables us to formulate the relation between the harmonic Ritz values and GMRES in the same way as the relation between the Ritz values and FOM. We present an upper bound for the norm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The differences between the Ritz and harmonic Ritz values enable us to describe the breakdown of FOM and stagnation of GMRES. Copyright © 1999 John Wiley & Sons, Ltd.