The Algebraic Eigenvalue Problem
About: This article is published in Mathematics of Computation.The article was published on 1966-10-01. It has received 2408 citations till now. The article focuses on the topics: Algebraic number & Eigenvalues and eigenvectors.
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TL;DR: In this paper, the problem of finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one is considered.
Abstract: We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.
615 citations
TL;DR: Several methods are described for modifying Cholesky factors and a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case.
Abstract: In recent years several algorithms have appeared for modifying the factors of a matrix following a rank-one change. These methods have always been given in the context of specific applications and this has probably inhibited their use over a wider field. In this report several methods are described for modifying Cholesky factors. Some of these have been published previously while others appear for the first time. In addition, a new algorithm is presented for modifying the complete orthogonal factorization of a general matrix, from which the conventional QR factors are obtained as a special case. A uniform notation has been used and emphasis has been placed on illustrating the similarity between different methods.
562 citations
TL;DR: This report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified Newton iteration.
Abstract: A modified Newton method for unconstrained minimization is presented and analyzed. The modification is based upon the model trust region approach. This report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified Newton iteration. Several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis. Very strong convergence results are presented concerning the minimization algorithm. In particular, the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization. With the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives.
561 citations
Cites background from "The Algebraic Eigenvalue Problem"
...A (k) ~ Aik\ which is the smallest eigenvalue of B k , the next theorem follows easily from the boundedness of IIJkll, IIJk1 11 together with Lemma 4.10....
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TL;DR: In this article, the problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed.
548 citations
01 Jan 2009
TL;DR: Algorithms for the Cholesky, LU and QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data are presented.
Abstract: As multicore systems continue to gain ground in the high performance computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents algorithms for the Cholesky, LU and QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in out of order execution of tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with LAPACK algorithms where parallelism can only be exploited at the level of the BLAS operations and vendor implementations.
546 citations
Cites background from "The Algebraic Eigenvalue Problem"
...When b = n (a 1{by{1 tiled matrix with n{by{n tiles), GETWP reduces to GEPP. GEWP dates back to Wilkinson’s work [ 39 ]....
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