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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.