Algorithms for Quadratic Matrix and Vector Equations

This book is devoted to studying algorithms for the solution of a class of quadratic matrix and vector equations. These equations appear, in different forms, in several practical applications, especially in applied probability and control theory. The equations are first presented using a novel unifying approach; then, specific numerical methods are presented for the cases most relevant for applications, and new algorithms and theoretical results developed by the author are presented. The book focuses on "matrix multiplication-rich" iterations such as cyclic reduction and the structured doubling algorithm (SDA) and contains a variety of new research results which, as of today, are only available in articles or preprints.

Contenu
Linear algebra preliminaries. Quadratic vector equations. A Perron vector iteration for QVEs. Unilateral quadratic matrix equations. Nonsymmetric algebraic Riccati equations. Transforming NAREs into UQMEs. Storage optimal algorithms for Cauchy-like matrices. Newton method for rank-structured algebraic Riccati equations. Lur'e equations. Generalized SDA. An effective matrix geometric mean. Constructing other matrix geometric means.