## Rectangular GLT sequences

Giovanni Barbarino, Carlo Garoni, Mariarosa Mazza, and Stefano Serra-Capizzano

### Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of square matrices $A_n$ arising from the discretization of differential problems. Indeed, as the mesh fineness parameter $n$ increases to $\infty$, the sequence $\{A_n\}_n$ often turns out to be a GLT sequence. In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory of rectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also provide two examples of application as an illustration of the potential of the theory presented herein.

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### Key words

asymptotic distribution of singular values and eigenvalues, rectangular Toeplitz matrices, rectangular generalized locally Toeplitz matrices, discretization of differential equations, finite elements, tensor products, B-splines, multigrid methods

### AMS subject classifications

15A18, 15B05, 47B06, 65N30, 15A69, 65D07, 65N55

### Links to the cited ETNA articles

 [6] Vol. 53 (2020), pp. 28-112 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case [7] Vol. 53 (2020), pp. 113-216 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

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