Journal Article10.1137/S0895479800371529
A Krylov--Schur Algorithm for Large Eigenproblems
591
TL;DR: A general Krylov decomposition is introduced that solves both the problem of deflate converged Ritz vectors and the potential forward instability of the implicit QR algorithm in a natural and efficient manner.
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Abstract: Sorensen's implicitly restarted Arnoldi algorithm is one of the most successful and flexible methods for finding a few eigenpairs of a large matrix. However, the need to preserve the structure of the Arnoldi decomposition on which the algorithm is based restricts the range of transformations that can be performed on the decomposition. In consequence, it is difficult to deflate converged Ritz vectors from the decomposition. Moreover, the potential forward instability of the implicit QR algorithm can cause unwanted Ritz vectors to persist in the computation. In this paper we introduce a general Krylov decomposition that solves both problems in a natural and efficient manner.
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Citations
Anasazi software for the numerical solution of large-scale eigenvalue problems
TL;DR: Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of large-scale eigenvalue problems and provides implementations for some of the most recent eigensolver methods.
407
GMRES with Deflated Restarting
TL;DR: The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES and it is demonstrated that using harmonic Ritz vectors is important because then the whole subspace is a Krylov subspace that contains certain important smaller subspaces.
340
The nonlinear eigenvalue problem
Stefan Güttel,Françoise Tisseur +1 more
TL;DR: This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques.
Scalable matrix computations on large scale-free graphs using 2D graph partitioning
Erik G. Boman,Karen D. Devine,Sivasankaran Rajamanickam +2 more
- 17 Nov 2013
TL;DR: This work proposes a new two-dimensional partitioning algorithm that combines graph partitioning with 2D block distribution and demonstrates that this new 2D partitioning method consistently outperforms the other methods considered, for both SpMV and an eigensolver, on matrices with up to 1.6 billion nonzeros.
143
Generalized Rational Krylov Decompositions with an Application to Rational Approximation
Mario Berljafa,Stefan Güttel +1 more
TL;DR: A rational Krylov method for rational least squares fitting is developed and an implicit Q theorem forrational Krylov spaces is presented.
References
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Richard B. Lehoucq,Danny C. Sorensen,C. Yang +2 more
- 01 Jan 1998
TL;DR: The Arnoldi factorization, the implicitly restarted Arnoldi method: structure of the Eigenvalue problem Krylov subspaces and projection methods, and more.
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The principle of minimized iterations in the solution of the matrix eigenvalue problem
TL;DR: In this paper, an interpretation of Dr. Cornelius Lanczos' iteration method, which he has named ''minimized iterations'' is discussed, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form.
Implicit application of polynomial filters in a k-step Arnoldi method
TL;DR: The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence.
Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems
Kesheng Wu,Horst D. Simon +1 more
TL;DR: A restarted variant of the Lanczos method for symmetric eigenvalue problems named the thick-restart Lanczo method is proposed, able to retain an arbitrary number of Ritz vectors from the previous iterations with a minimal restarting cost.