Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1991"
Book•
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: The learing time of a simple neural-network model is obtained through an analytic computation of the eigenvalue spectrum for the Hessian matrix, which describes the second-order properties of the objective function in the space of coupling coefficients.
Abstract: The learing time of a simple neural-network model is obtained through an analytic computation of the eigenvalue spectrum for the Hessian matrix, which describes the second-order properties of the objective function in the space of coupling coefficients. The results are generic for symmetric matrices obtained by summing outer products of random vectors. The form of the eigenvalue distribution suggests new techniques for accelerating the learning process, and provides a theoretical justification for the choice of centered versus biased state variables.
159 citations
TL;DR: In this article, a relatively obscure eigenvalue inequality due to Wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix.
Abstract: A relatively obscure eigenvalue inequality due to Wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix. The asymptotic distributions are obtained under a fairly general setting. An application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given. 1. Introduction and summary. The derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis. Although the main idea behind most of the derivations is quite basic, i.e., the expansion of the sample roots about the population roots, the derivations themselves are often quite involved. These complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem. One of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices. The method is based upon a relatively obscure eigenvalue inequality
127 citations
TL;DR: In this paper, the Schur parameter pencil problem is solved in an O(n 3 )-time process using Householder eliminations, and it is backward stable in the sense that the condensed form is preserved throughout the process.
90 citations
TL;DR: Application of the two-sided Lanczos recursion to the unsymmetric generalized eigenvalue problem is presented and the results are compared with the eigenfrequencies extracted by an unsyMMetric subspace iteration procedure presented in the literature.
Abstract: Application of the two-sided Lanczos recursion to the unsymmetric generalized eigenvalue problem is presented. The system matrices are real and unsymmetric. Therefore, the recursions are performed in real arithmetic and complex arithmetic is employed in the QR algorithm used to extract the eigenvalues of the transformed tridiagonal matrix. The biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in Appendices. Issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed. The example problems solved demonstrate the working of the algorithm in extracting the complex and/or real eignevalues of an unsymmetric system of matrices. Also, the algorithm is applied to extract a few of the eigenvalues of a large fluid-structure interaction problem, and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature.
86 citations
TL;DR: In this paper, an oscillation method is presented for finding the eigenvalues of a fourth-order, self-adjoint, two-point boundary value problem, which may occur nonlinearly in the differential equation.
Abstract: An oscillation method is presented for finding the eigenvalues of a fourth-order, self-adjoint, two-point boundary value problem. The eigenvalue may occur nonlinearly in the differential equation, ...
50 citations
TL;DR: In this article, a method is presented to find a selected set of eigenvalues and the respective eigenvectors of the generalised eigenvalue problem Ax = λBx for large, sparse, real or complex, non-Hermitian matrices.
Abstract: A method is presented to find a selected set of eigenvalues and the respective eigenvectors of the generalised eigenvalue problem Ax = λBx for large, sparse, real or complex, non-Hermitian matrices. Although the method is clearly applicable to other problems, results are given of application to the vectorial finite element analysis of dielectric waveguides.
39 citations
TL;DR: In this article, the minimum eigenvalue of the associated clamped drum was characterized for an open bounded connected set and a prescribed amount of two homogeneous materials of different density.
Abstract: Given an open bounded connected set δ ⊂ R2 and a prescribed amount of two homogeneous materials of different density we characterize that distribution which minimizes the least eigenvalue of the associated clamped drum. We establish geometric conditions on δ under which the interface separating the two materials is an analytic Jordan curve. We bound the length of this interface and construct and test an algorithm for its calculation in the case of square δ. Our numerical results depict the dependence of this minimum least eigenvalue on the volume fractions of the two phases and suggest possible candidates for the two phase drum with the least second eigenvalue.
33 citations
TL;DR: SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm Liouville boundary value problems, modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej.
Abstract: SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm Liouville boundary value problems, The package is a modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej which is described in ACM Z’OMS 4 (1978), 193-208. The modifications and extensions include (1) a restructuring of the FORTRAN program, (2) the coverage of problems with semidefi nite weight functions, and (3) the coverage of problems with indefinite weight functions.
32 citations
01 Mar 1991
TL;DR: Surprisingly simple corollaries from the Courant-Fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set S interleaving the set E of the eigenvalues of a real symmetric tridiagonal matrix Tn.
28 citations
TL;DR: In this article, a lower bound for the maximum eigenvalue of symmetric positive semidefinite matrices was obtained for the non-symmetric case, where the angle of the matrix is known.
Book•
01 Jan 1991
TL;DR: In this article, Richardson extrapolation method was used for root-finding and extrapolation of the matrix eigenvalue problem, as well as the matrix inverse and generalized inverse problems.
Abstract: General introduction Root-finding methods and their application The Richardson extrapolation method Some interpolation and extrapolation methods The matrix inverse and generalized inverse The matrix eigenvalue problem Two perturbation methods Finite difference eigenvalue calculations Recurrence relation methods Two research problems Bibliography Index
TL;DR: Numerical experiments showed that the preconditioned CG like methods required much smaller number of iterations and computational time compared with the conventional inner-outer iterative scheme for both one- group and few-group eigenvalue problems.
TL;DR: In this article, an iterative procedure for computing the eigenvalues and eigenvectors of a class of specially structured Hermitian Toeplitz matrices is proposed.
01 Sep 1991
TL;DR: This paper describes a parallel algorithm for computing the eigenvalues and eigenvectors of a non-symmetric matrix based on a divide-and-conquer procedure and uses an iterative refinement technique.
Abstract: This paper describes a parallel algorithm for computing the eigenvalues and eigenvectors of a non-symmetric matrix. The algorithm is based on a divide-and-conquer procedure and uses an iterative refinement technique.
TL;DR: In this article, it was shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigen value parameter γ can be linearized by making use of the theory of operator colligations.
Abstract: It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter γ can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on γ and Sturm-Liouville problems with γ-holomorphic boundary conditions are considered.
TL;DR: In this article, the remainder estimate for the eigenvalue distribution of the elliptic operator of order 2m with Holder continuous coefficients of top order was improved for a symmetric integro-differential sesquilinear form.
Abstract: This is the continuation of the previous paper [11], in which we attempted the improvement of the remainder estimate for the eigenvalue distribution of the elliptic operator of order 2m with Holder continuous coefficients of top order. We use the same notation as in [11] if not specified. Let us recall the situation. Let ίl be a bounded domain in R. We consider a symmetric integro-differential sesquilinear form
TL;DR: In this article, a numerical algorithm for the inverse eigenvalue problem for symmetric matrices is developed, based on continually updating the eigenvector matrix using plane rotations, and two criteria of closeness to a solution are defined, either of which makes it possible to monitor progress towards a solution.
TL;DR: The standard theory describing the result of the QR algorithm with k shifts on a Hessenberg matrix A is extended to the case where some of the shifts can be eigenvalues, which has a practical value in special cases such as eigenvalue allocation.
Abstract: A new approach is suggested for deriving the theory of implicit shifting in the QR algorithm applied to a Hessenberg matrix. This is less concise than Francis’ original approach ([Comput. J., 4(1961), pp. 265–271], [Comput. J., 4(1962), pp. 332–345]) but is more instructive, and extends easily to more general cases. For example, it enables us to design implicitly shifted QR algorithms for band and block Hessenberg matrices. It can also be applied to related algorithms such as the LR algorithm, and to algorithms which do not produce triangular matrices in the factorization step. The approach provides details that can be useful in designing numerically effective algorithms in various areas.In addition to the above, the standard theory describing the result of the QR algorithm with k shifts on a Hessenberg matrix A is extended to the case where some of the shifts can be eigenvalues. This has a practical value in special cases such as eigenvalue allocation. The extension is given for both the explicitly and i...
TL;DR: In this article, a subspace iteration scheme for the Hermitian eigenvalue problem arising from analysis of cyclic symmetric structures is presented, which operates in the semi-complex domain and its computational efficiency is demonstrated.
Abstract: A subspace iteration scheme for the Hermitian eigenvalue problem arising from analysis of cyclic symmetric structures is presented. This operates in the semi-complex domain and its computational efficiency is demonstrated. A modified convergence checking criterion which gives better error estimates, and a new trial vector selection scheme which accelerates convergence for the Hermitian eigenvalue problem based on the Ritz method, are presented.
TL;DR: In this article, the F-test is applied to a succession of solutions, each containing an incremental number of eigenvalues, to determine statistical significance of the data variance reduction, which is widely used for multiple variable regression analysis, but has not been applied to eigenvalue problems.
Abstract: A fundamental problem using linear inverse theory to solve geophysical problems using eigenvalue decomposition algorithms is to determine how many eigenvalues to include in the solution. If very small eigenvalues are included, the solution variance increases rapidly, particularly if zero-values eigenvalues are computed as positive small numbers and are misidentified. F-tests can be applied to a succession of solutions, each containing an incremental number of eigenvalues, to determine statistical significance of the data variance reduction. This methodology is widely used for multiple variable regression analysis, but has not been applied to eigenvalue problems. The F-test is a statistical criterion for choosing an ‘optimal’ solution along the trade-off curve of model resolution and model variance for a particular model parameterization.
TL;DR: In this article, a comparison of several popular approaches for solving eigenvalue problems for linear boundary value ODES is given, and the choice of numerical methods used here is motivated by the desire to solve eigen value problems for stiff ODES.
TL;DR: In this paper, a first-order system with unknown gain controlled by an adaptive controller using the MIT rule was investigated by computing the eigenvalues of the transition matrix of the system, which depends on a gain in the controller and the frequency of the sinusoidal input.
Abstract: Conclusions In this Note we have investigated a first-order system with unknown gain controlled by an adaptive controller using the MIT rule. The stability of the system has been investigated by computing the eigenvalues of the transition matrix of the system. The stability of the system depends on a gain in the controller and the frequency of the sinusoidal input to the system. The regions of stability exhibit very complicated behavior.
TL;DR: In this paper, sufficient conditions for the solvability of additive and multiplicative inverse eigenvalue problems for Hermitian matrices are considered and the notion of majorization, Cauchy-Poincare interlacing inequalities, and Brouwer's fixed-point theorem are applied.
TL;DR: This paper presents an iterative method for the numerical solution of discreted two-parameter eigenvalue problems and analyses the accuracy of finite difference approximations to the eigenvalues of two- parameter S-L eigenproblems.
Abstract: Two-parameter Sturm-Liouville (S-L) eigenvalue problems arise in solving the Helmholtz equation (also the Laplace equation) by separation of variables. This paper presents an iterative method for the numerical solution of discreted two-parameter eigenvalue problems and analyses the accuracy of finite difference approximations to the eigenvalues of two-parameter S-L eigenproblems. Some numerical results are also reported.
TL;DR: In this article, it was shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system and that the largest eigen value of a non-negative irreducible matrix is increased whenever any matrix element is increased.
Abstract: It is shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system. This follows from the increase in the largest eigenvalue of a non-negative irreducible matrix whenever any matrix element is increased. This result of matrix analysis is generalised to more general Krein-Rutman operators that leave a cone of vectors invariant.
01 Jan 1991
TL;DR: In this paper, two methods are developed for finding probabilistic characteristics of the eigenvalues and eigenvectors of a stochastic matrix based on the mean zero-crossing rate of the characteristic polynomial of this matrix and a perturbation approach.
Abstract: Two methods are developed for finding probabilistic characteristics of the eigenvalues and eigenvectors of a stochastic matrix They are based on the mean zero-crossings rate of the characteristic polynomial of this matrix and a perturbation approach The methods are applied to characterize probabilistically the natural frequencies of an uncertain dynamic system and to find the first two moments of the displacement of a simple oscillator with random damping and stiffness that is subject to white noise
TL;DR: The necessary conditions of the first and second order for problems of first eigenvalue maximization are presented in this paper, where the second-order condition is formulated as completeness condition for a system of functions in Banach space.
Abstract: In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initial-boundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.
TL;DR: Sufficient conditions for convergence, error bounds and a procedure to improve the stability are discussed and some numerical examples are given to illustrate the effectiveness of the proposed Prufer method.
Abstract: In reference [19], the authors developed a shooting algorithm for Sturm-Liouville eigenvalue problems associated with periodic and semi-periodic boundary conditions. The technique is based on the application of the Floquet theory, and it has proven to be efficient for computing eigenvalues. However, the performance of this technique depends upon the choice of the starting eigenvalues. In the present paper, we continue our study and employ the Prufer method. An attractive property of this method is that eigenvalues can usually be accurately computed even when no information on the eigenvalue distribution is provided. Sufficient conditions for convergence, error bounds and a procedure to improve the stability are discussed. Some numerical examples are given to illustrate the effectiveness of the proposed method.