Showing papers on "Spectrum of a matrix published in 1984"
TL;DR: This paper presents a fast method for estimating dominant harmonics in a sequence of data and shows that this method is related to “lattice methods” for linear prediction and to Prony’s method for exponential approximation.
Abstract: This paper presents a fast method for estimating dominant harmonics in a sequence of data. In a stochastic sense, the proposed method finds the autoregressive scheme with a pure point spectrum that best describes the data, while from a deterministic point of view, the method is a special case of the Lanczos algorithm for finding eigenvalues of a symmetric matrix. Eigenvalue approximations come into play because every circulant matrix is diagonalized by the discrete Fourier transform matrix, and so using the Lanczos algorithm with the given data as the initial vector on a simple circulant matrix, the eigenvalues that are first approximated are the eigenvalues corresponding to eigenvectors which are dominant in the initial vector. It is shown that this method is related to “lattice methods” for linear prediction and to Prony’s method for exponential approximation.
24 citations
TL;DR: Explicit (computable) lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining (not necessarily real) eigenvalues of the matrix are developed.
8 citations
TL;DR: This paper gives a generalization of the well-known Frank matrix and shows how to compute its eigensystem accurately and attempts to explain the ill-condition of its eigenvalues by treating it as a perturbation of a defective matrix.
Abstract: In this paper, we give a generalization of the well-known Frank matrix and show how to compute its eigensystem accurately. As well, we attempt to explain the ill-condition of its eigenvalues by treating it as a perturbation of a defective matrix.
4 citations
TL;DR: In this paper, the problem of redundancy conditions among the internal coordinates and rows of G-matrix with zero eigenvalues was discussed, in short matrix form, and it was shown that the same relations are satisfied both for the coordinates and Gmatrix rows.
3 citations
TL;DR: In this article, an extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues is investigated, and sufficient (and necessary) conditions are given such that convergence is assured.
Abstract: An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.
3 citations
TL;DR: In this article, conditions are given for the spectrum in an eigenvalue problem of the form to be discrete, where and are operators that are odd-homogeneous of degree, acting from a reflexive Banach space into the dual.
Abstract: In this paper conditions are given for the spectrum in an eigenvalue problem of the form to be discrete, where and are operators that are odd-homogeneous of degree (), acting from a reflexive Banach space into the dual. It is proved that the eigenvalues vary monotonically as and vary in the normed linear space of homogeneous operators of degree . Explicit formulas for the eigenvalues and functions are obtained for the case where and are the gradients of the norms in the spaces and ( is a parallelepiped in ). Using these formulas the author obtains estimates for the eigenvalues in homogeneous and asymptotically homogeneous problems with variable coefficients in the space , where is an arbitrary bounded domain in .Bibliography: 12 titles.
06 Jun 1984
TL;DR: This work uses a canonical form of the bond graph, namely the gyrobondgraph, to estimate bounds on the eigenvalues of the associated system and results are obtained, when suitably automated, that can reduce the time required to locate the eigens.
Abstract: For a class of linear, time-invariant dynamic systems, information about the eigenvalues of the system can be obtained directly from a graphical model of the system, namely the bond graph model. In contrast to the standard approach, which starts from the state matrix of the system, we use a canonical form of the bond graph, namely the gyrobondgraph, to estimate bounds on the eigenvalues of the associated system. For some cases the the spectrum of the system can be obtained by comparing the graph structure to existing graphs with known spectra. For more general cases bounds are obtained on the largest real and imaginary parts of the eigenvalues as a function of the system's gyrobondgraph topology and its parameters. These results, when suitably automated, can reduce the time required to locate the eigenvalues.
TL;DR: In this paper, the discrete spectrum of the Rayleigh piston was investigated using a WKB method for integral operators and analytic formulae asymptotically valid for eigenvalues and eigenfunctions were obtained.
Abstract: The discrete spectrum of the Rayleigh piston is investigated using a WKB method for integral operators. Analytic formulae asymptotically valid for eigenvalues and eigenfunctions are obtained. The discrete spectrum is finite for 0< gamma <<1.