This concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic way. Some proofs of theoretical results have been simplified and a unified notation has been adopted.\ud \ud Readers will find:\ud - a unified discussion of doubling algorithms, which are effective in solving algebraic Riccati equations. \ud - a detailed description of all classical and advanced algorithms for solving algebraic Riccati equations and their MATLAB® codes. This will help the reader gain an understanding of the computational issues and provide ready-to-use implementation of the different solution techniques.\ud \ud This book is intended for researchers who work in the design and analysis of algorithms and for practitioners who are solving problems in applications and need to understand the available algorithms and software. It is also intended for students with no expertise in this area who wish to approach this subject from a theoretical or computational point of view. The book can be used in a semester course on algebraic Riccati equations or as a reference in a course on advanced numerical linear algebra and applications

Nonsymmetric algebraic Riccati equations for which the four coefficient matrices form an irreducible M-matrix M are considered. The emphasis is on the case where M is an irreducible singular M-matrix, which arises in the study of Markov models. The doubling algorithm is considered for finding the minimal nonnegative solution, the one of practical interest. The algorithm has been recently studied by others for the case where M is a nonsingular M-matrix. A shift technique is proposed to transform the original Riccati equation into a new Riccati equation for which the four coefficient matrices form a nonsingular matrix. The convergence of the doubling algorithm is accelerated when it is applied to the shifted Riccati equation.

We present and analyze an iterative method for approximating the Karcher\ud mean of a set of $n\times n$ positive definite matrices $A_i$, $i=1,\ldots,k$, defined as the\ud unique positive definite solution of the matrix equation $\sum_{i=1}^k \log(A_i^{-1}X)=0$

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The Barzilai-Borwein method, an effective gradient descent method with cleaver choice of the step-length, is adapted from nonlinear optimization to Riemannian manifold optimization. More generally, global convergence of a nonmonotone line-search strategy for Riemannian optimization algorithms is proved under some standard assumptions. By a set of numerical tests, the Riemannian Barzilai-Borwein method with nonmonotone line-search is shown to be competitive in several Riemannian optimization problems. When used to compute the matrix geometric mean, known as the Karcher mean of positive definite matrices, it notably outperforms existing first-order optimization methods. Riemannian optimization; manifold optimization; Barzilai-Borwein algorithm; nonmonotone line-search; Karcher mean; matrix geometric mean; positive definite matrix.