Momentary Symbol: Spectral Analysis of Structured Matrices
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A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of generalized locally Toeplitz (GLT) sequences. By the theory of GLT sequences one can derive a function, called the symbol, which describes the singular value or eigenvalue distribution of the sequence. However, for small value of the matrix size of the considered sequence, the approximations may not be as exact as is attainable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. Local Fourier analysis (LFA) can be used to construct symbols in a similar manner for discretizations where the geometric information is present, but here the small norm perturbations are retained, i.e., more information is kept in the symbol. The LFA is predominantly used in the analysis and design of multigrid methods, and to estimate the worst case spectral radius or 2-norm. The main focus of this paper is the introduction of the concept of a (singular value and spectral) "momentary symbol", associated with trucated Toeplitz-like matrices. We construct the symbol in the same way as in the theory of GLT sequences, but we keep the information of the small norm contributions. The low-rank contributions as still disregarded, and we give an idea on why this is negligible. Moreover, we highlight that the difference to the LFA symbols is that momentary symbols are applicable to a much larger class of matrices. We show the applicability of the symbol, and higher accuracy, when approximating the singular values and eigenvalues of trucated Toeplitz-like matrices using the momentary symbol, compared with the GLT symbol. Moreover, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful defintions, applicable for non-square momentary symbols.
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