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The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matricesToeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.
Document ID
19930013640
Acquisition Source
Legacy CDMS
Document Type
Technical Memorandum (TM)
Authors
Beam, Richard M.
(NASA Ames Research Center Moffett Field, CA, United States)
Warming, Robert F.
(NASA Ames Research Center Moffett Field, CA, United States)
Date Acquired
September 6, 2013
Publication Date
November 1, 1991
Subject Category
Numerical Analysis
Report/Patent Number
NASA-TM-103900
A-92006
NAS 1.15:103900
Accession Number
93N22829
Funding Number(s)
PROJECT: RTOP 505-59-53
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.
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