Topic
Divide-and-conquer eigenvalue algorithm
About: Divide-and-conquer eigenvalue algorithm is a research topic. Over the lifetime, 2877 publications have been published within this topic receiving 81838 citations.
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01 Jan 1991TL;DR: This paper focuses on Gaussian Elimination as a model for Iterative Methods for Linear Systems, and its applications to Singular Value Decomposition and Sparse Eigenvalue Problems.
Abstract: Gaussian Elimination and its Variants Sensitivity of Linear Systems Effects of Roundoff Errors Orthogonal Matrices and the Least Squares Problem Eigenvalues, Eigenvectors and Invariant Subspaces Other Methods for the Symmetric Eigenvalue Problem The Singular Value Decomposition Appendices Bibliography
1,077 citations
TL;DR: A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B with particular attention to the degeneracies which result when B is singular.
Abstract: A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned.
1,038 citations
TL;DR: In this paper, a new discrete eigenvalue problem has been introduced to obtain and solve certain classes of nonlinear differential-difference equations, which can be obtained by inverse scattering.
Abstract: A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction of a new discrete eigenvalue problem allows the exact solution of the self−dual network equations to be found by inverse scattering. The eigenvalue problem has as its singular limit the continuous eigenvalue equations of Zakharov and Shabat. Some interesting differences arise both in the scattering analysis and in the time dependence from previous work.
818 citations
724 citations
TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).
667 citations