On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems
نویسندگان
چکیده
منابع مشابه
On the Convergence of Q-ritz Pairs and Refined Q-ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the q-Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the q-Rayleigh-Ritz method defines the q-Ritz values and the q-Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the...
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This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (μ + τ, x̃) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value...
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After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss different extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approach, which are new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numeri...
متن کاملSome insights on restarting symmetric eigenvalue methods with Ritz and harmonic Ritz vectors
Eigenvalue iterative methods, such as Arnoldi and Jacobi-Davidson, are typically used with restarting. This has signiicant performance shortcomings, since important components of the invariant subspace may be discarded. One way of saving more information at restart is the idea of \thick" restarting which keeps more Ritz vectors than needed. Our previously proposed dynamic thick restarting choos...
متن کاملNew estimates for Ritz vectors
The following estimate for the Rayleigh–Ritz method is proved: |λ̃−λ||(ũ,u)| ≤ ‖Aũ− λ̃ũ‖sin∠{u;Ũ}, ‖u‖= 1. Here A is a bounded self-adjoint operator in a real Hilbert/euclidian space, {λ,u} one of its eigenpairs, Ũ a trial subspace for the Rayleigh–Ritz method, and {λ̃, ũ} a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh–Ritz method, in part...
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ژورنال
عنوان ژورنال: BIT Numerical Mathematics
سال: 2013
ISSN: 0006-3835,1572-9125
DOI: 10.1007/s10543-013-0438-0