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.

Contains a new unifying approach to quadratic vector and matrix equations in applied probability

Gives new insight on the structured doubling algorithm which can be exploited to develop suitable modifications and generalizations

Contains a variety of new research results which, as of today, are only available in articles or preprints

Inhalt
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.