Introduction

Carlo Garoni () and Stefano Serra-Capizzano ()
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Carlo Garoni: University of Insubria, Department of Science and High Technology
Stefano Serra-Capizzano: University of Insubria, Department of Science and High Technology

Chapter Chapter 1 in Generalized Locally Toeplitz Sequences: Theory and Applications, 2017, pp 1-5 from Springer

Abstract: Abstract The theory of generalized locally ToeplitzGeneralized Locally Toeplitz (GLT) (GLT) sequences stems from Tilli’s workLocally Toeplitz (LT) on locally Toeplitz (LT) sequences [120] and from the theory of Toeplitz matrices [5, 23, 24, 28, 29, 70, 89, 119, 123, 124, 129], and was developed by the authors in [60–62, 106, 107]. It was devised in order to solve a specific application problem, namely the problem of computing/analyzing the spectral distribution of matrices arising from the numerical discretization of integral equationsIEs (IEs) and, especially, differential equationsDEs (DEs). A final goal of this spectral analysis is the design of efficient numerical methods for computing the related numerical solutions. As we shall see in Chap. 10 , the theory of GLT sequences finds applications also in other areas of science, but the computation of the spectral distribution of DE discretization matrices remains the main application. Section 1.1 is therefore devoted to a general description of this application. Then, in Sect. 1.2, we provide an overview of the theory of GLT sequences. After reading this introductory chapter, one is ready to enter into the mathematical details with an already-enough-precise idea of what we are going to deal with.

Keywords: Related Numerical Solutions; Local Toeplitz (LT); Specific Application Problem; Toeplitz Matrices; Asymptotic Spectral Distribution (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-53679-8_1

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