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: Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.
937 citations
Proceedings Article•
01 Dec 1997TL;DR: An expectation-maximization (EM) algorithm for principal component analysis (PCA) which allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data and defines a proper density model in the data space.
Abstract: I present an expectation-maximization (EM) algorithm for principal component analysis (PCA). The algorithm allows a few eigenvectors and eigenvalues to be extracted from large collections of high dimensional data. It is computationally very efficient in space and time. It also naturally accommodates missing information. I also introduce a new variant of PCA called sensible principal component analysis (SPCA) which defines a proper density model in the data space. Learning for SPCA is also done with an EM algorithm. I report results on synthetic and real data showing that these EM algorithms correctly and efficiently find the leading eigenvectors of the covariance of datasets in a few iterations using up to hundreds of thousands of datapoints in thousands of dimensions.
928 citations
Cites background from "The Algebraic Eigenvalue Problem"
...Fortunately, several techniques exist for efficient matrix diagonalization when only the first few leading eigenvectors and eigenvalues are required (for example the power method [10] which is only )....
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TL;DR: A comprehensive survey of all issues associated with PageRank, covering the basic PageRank model, available and recommended solution methods, storage issues, existence, uniqueness, and convergence properties, possible alterations to the basic model, and suggested alternatives to the traditional solution methods.
Abstract: This paper serves as a companion or extension to the "Inside PageRank" paper by Bianchini et al. [Bianchini et al. 03]. It is a comprehensive survey of all issues associated with PageRank, covering the basic PageRank model, available and recommended solution methods, storage issues, existence, uniqueness, and convergence properties, possible alterations to the basic model, suggested alternatives to the traditional solution methods, sensitivity and conditioning, and finally the updating problem. We introduce a few new results, provide an extensive reference list, and speculate about exciting areas of future research.
910 citations
Cites background from "The Algebraic Eigenvalue Problem"
...The ill-conditioning of the linear system does not imply that the corresponding eigensystem is ill-conditioned, a fact documented by Wilkinson [115] (with respect to the inverse iteration method)....
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TL;DR: In this article, the Lanczos tridiagonal construction has been used to diagonalize matrices in determinantal spaces of dimensionality up to 10^9 using the Shell Model.
Abstract: The last decade has witnessed both quantitative and qualitative progresses in Shell Model studies, which have resulted in remarkable gains in our understanding of the structure of the nucleus. Indeed, it is now possible to diagonalize matrices in determinantal spaces of dimensionality up to 10^9 using the Lanczos tridiagonal construction, whose formal and numerical aspects we will analyze. Besides, many new approximation methods have been developed in order to overcome the dimensionality limitations. Furthermore, new effective nucleon-nucleon interactions have been constructed that contain both two and three-body contributions. The former are derived from realistic potentials (i.e., consistent with two nucleon data). The latter incorporate the pure monopole terms necessary to correct the bad saturation and shell-formation properties of the realistic two-body forces. This combination appears to solve a number of hitherto puzzling problems. In the present review we will concentrate on those results which illustrate the global features of the approach: the universality of the effective interaction and the capacity of the Shell Model to describe simultaneously all the manifestations of the nuclear dynamics either of single particle or collective nature. We will also treat in some detail the problems associated with rotational motion, the origin of quenching of the Gamow Teller transitions, the double beta-decays, the effect of isospin non conserving nuclear forces, and the specificities of the very neutron rich nuclei. Many other calculations--that appear to have ``merely'' spectroscopic interest--are touched upon briefly, although we are fully aware that much of the credibility of the Shell Model rests on them.
884 citations
Cites methods from "The Algebraic Eigenvalue Problem"
...zation of the matrix. A. The Lanczos method FIG. 8 m-scheme dimensions and total number of non-zero matrix elements in the pf-shell for nuclei with M = Tz = 0 In the standard diagonalization methods (Wilkinson, 1965) the CPU time increases as N3, N being the dimension of the matrix. Therefore, they cannot be used in large scale shell model (SM) calculations. Nuclear SM calculations have two specific features. The ...
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TL;DR: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems, applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenavectors.
Abstract: A simplified procedure is presented for the determination of the derivatives of eigenvectors of nth order algebraic eigensystems. The method is applicable to symmetric or nonsymmetric systems, and requires knowledge of only one eigenvalue and its associated right and left eigenvectors. In the procedure, the matrix of the original eigensystem of rank (/?-!) is modified to convert it to a matrix of rank /?, which then is solved directly for a vector which, together with the eigenvector, gives the eigenvector derivative to within an arbitrary constant. The norm of the eigenvector is used to determine this constant and complete the calculation. The method is simple, since the modified n rank matrix is formed without matrix multiplication or extensive manipulation. Since the matrix has the same bandedness as the original eigensystems, it can be treated efficiently using the same banded equation solution algorithms that are used to find the eigenvectors.
878 citations