Speaker
Description
The study of the asymptotic properties of the spectrum of a matrix sequence with a specific structure and increasing order has been a rich field of research and of great interest in applications, since they arise from approximation of integro-differential equations, also of fractional order (see e.g. [4, 5, 3] and references therein). This kind of approach leads most of the times to a huge linear system, whose dimension is related to the precision of the approximation. For this reason, researchers focused their attention on properties of clustering and symbol analysis of the eigenvalues of the sequence, since those are connected with fast convergence of iterative procedures for solving linear systems, such as Krylov-type methods [2, 6]. While classical results are often efficient in the case of a sequence of normal matrices, a wilder behavior should be expected in non-normal settings. In this poster, two such cases [9, 10] are presented and specific tools as long as complex analysis and quite new literature are involved to obtain the main results.
[1] G. Barbarino, S. Serra-Capizzano. Non-Hermitian perturbations of Hermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations. Numer. Linear Algebra Appl. 27 (2020), no. 3, e2286, 31 pp.
[2] B. Beckermann, A.B.J. Kuijlaars. Superlinear convergence of conjugate gradients. SIAM J. Numer. Anal. 39-1 (2001), 300–329.
[3] M. Donatelli, M. Mazza, S. Serra-Capizzano. Spectral analysis and preconditioning for variable coefficient fractional derivative operators. J. Comput. Phys. 307 (2016), 262–279.
[4] C. Garoni, S. Serra-Capizzano. The theory of Generalized Locally Toeplitz sequences: theory and applications - Vol I. SPRINGER - Springer Monographs in Mathematics, Berlin, (2017).
[5] C. Garoni, S. Serra-Capizzano. The theory of Generalized Locally Toeplitz sequences: theory and applications - Vol II. SPRINGER - Springer Monographs in Mathematics, Berlin, (2018).
[6] A.B.J. Kuijlaars. Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48-1 (2006), 3–40.
[7] D.S. Lubinsky. Distribution of eigenvalues of Toeplitz matrices with smooth entries. Linear Algebra Appl. 633 (2022), 332–365.
[8] W. Parry. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.
[9] A.J.A. Schiavoni-Piazza, S. Serra-Capizzano. Distribution results for a special class of matrix sequences: joining approximation theory and asymptotic linear algebra. Electron. Trans. Numer. Anal. 59 (2023), 1–8
[10] A.J.A. Schiavoni-Piazza, D. Meadon, S. Serra-Capizzano. The β maps and the strong clustering at the complex unit circle. Submitted for publication.
[11] J.-L. Verger-Gaugry. Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers. Unif. Distrib. Theory 3-2 (2008), 157–190.
