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Jacobi eigenvalue algorithm
About: Jacobi eigenvalue algorithm is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 23521 citations.
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2,265 citations
TL;DR: Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices, and the methods and software that have led to these advances are surveyed.
Abstract: Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices. Krylov methods and variants of subspace iteration have been improved to the point that problems of the order of several million variables can be solved. The methods and software that have led to these advances are surveyed.
1,124 citations
TL;DR: This note gives the required Jacobi angles in close form for simultaneous diagonalization of several matrices.
Abstract: Simultaneous diagonalization of several matrices can be implemented by a Jacobi-like technique. This note gives the required Jacobi angles in close form.
903 citations
TL;DR: In this article, a new method for the iterative computation of a few extremal eigenvalues of a symmetric matrix and their associated eigenvectors is proposed, based on an old and almost unknown method of Jacobi.
Abstract: In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well.
900 citations
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01 Jan 1999TL;DR: In this paper, the Toda system and the Kac-van Moerbeke system are studied. But the initial value problem is not considered in this paper, as it is in the case of Jacobi operators with periodic coefficients.
Abstract: Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.
782 citations