Spectral distributions of structured matrix-sequences: tools and applications.

This Ph.D. thesis provides a systematic and unifying treatment of the spectral analysis of matrix sequences collecting the state of the art of the research in this field and providing new advanced results which are interesting both from the theoretical and from the algorithmic point of view. The first part, made up by Chapters 1–5 concerns the description and analysis of the available tools, together with their generalization including the development of new effective tools of analysis. The basic concept of “approximating class of sequences” is introduced and analyzed in depth. Some new results of great interest have been obtained. It is worth citing an extension of a perturbation result concerning the eigenvalues of Hermitian matrix sequences which are perturbed with non-Hermitian perturbations. These results have been applied to Toeplitz sequences, and to the Tilli class. The classes of g-circulant and g-Toeplitz matrix sequences are introduced as well, and their spectral analysis is carried out. The second part, formed by Chapters 6–8 is addressed to applications. It contains several new and interesting results. A relevant role is played by the preconditioning of g-Toeplitz matrix sequences by means of g-circulant matrices with a specific attention to the regularizing properties which take their importance in the applications concerning image processing, filtering and restoration. Another interesting application concerns two-grid and multi-grid techniques. A third application concerns the saddle point problem where once again the spectral approximation of a suitable matrix plays a fundamental role.