Convergence of algorithms of decomposition type for the eigenvalue problem
David S. Watkins,Ludwig Elsner +1 more
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TLDR
The theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR,LR,SR, and other algorithms as special cases is developed and it is shown that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate.About:Â
This article is published in Linear Algebra and its Applications.The article was published on 1991-01-01 and is currently open access. It has received 107 citations till now. The article focuses on the topics: Compact convergence & Convergence tests.read more
Citations
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Book ChapterDOI
Implicitly restarted arnoldi/lanczos methods for large scale eigenvalue calculations
TL;DR: The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems by focusing on a class of methods called Krylov subspace projection methods.
Journal ArticleDOI
Parallel Numerical Linear Algebra
TL;DR: This work discusses basic principles of paralled processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms, and presents direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigene value problem, and the singular value decomposition.
Journal ArticleDOI
Some perspectives on the eigenvalue problem
TL;DR: This expository paper explores the relationships among a number of algorithms for solving eigenvalue problems, including the power method, subspace iteration, the $QR$ algorithm, and the Arnoldi and symmetric Lanczos algorithms.
Journal ArticleDOI
The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance
TL;DR: This paper presents a small-bulge multishift variation of the multishIFT QR algorithm that avoids the phenomenon of shift blurring, which retards convergence and limits the number of simultaneous shifts.
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The Multishift QR Algorithm. Part II: Aggressive Early Deflation
TL;DR: A new deflation strategy that takes advantage of matrix perturbations outside of the subdiagonal entries of the Hessenberg QR iterate and identifies and deflates converged eigenvalues long before the classic small-subdiagonal strategy would.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
Book
The Symmetric Eigenvalue Problem
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.