The purpose of this research is to develop modular and parallelizable algorithms for problems in adaptive filtering, robust estimation, control, and structured matrix computations. A major motivation for this work is the need to properly identify, model, and exploit convenient structures that might exist in a particular problem in order to develop a computationally effective solution. An intrinsic and relevant part of the proposed work is to first identify common ingredients, as well as to explicitly clarify, the interrelations and interplays that exist among several problems in the disciplines of control, signal processing, and mathematics. In this regard, the proposed research develops a unifying point of view that simultaneously addresses a variety of seemingly unrelated problems in estimation, control, adaptive filtering, and matrix computations. It shows that the solutions to many applications in these areas turn out to share a key and surprisingly simple ingredient, namely, that of computing the triangular factors of matrices that exhibit structure. The significance of this particular fact and, more generally, of the proposed research as a whole, is that it enables several well-understood concepts from matrix theory and linear algebra to be applied in order to simplify the derivation of a variety of algorithms, to propose new computationally efficient variants, and to exploit new forms of structure. The proposed research consists of theoretical investigations and algorithm implementations. The techniques developed in this work have applications in several areas including, among others, new windowing schemes for adaptive RLS filtering, analysis of gradient- based algorithms, and modular and square-root algorithms for instrumental variable methods, robust estimation, and control. Equally important is the development of a systematic and unifying methodology.
