This monograph, which is part of SIAM’s important “Fundamentals of Algorithms” series, is dedicated to obtaining solutions, analytical or approximate, of algebraic matrix equations of the form C + X A + D X - X B X = 0, as well as many other matrix equations with similar features. In this equation, matrices A, B, C, and D are known and the challenge is to compute X. Equations of this type appear in many applications, ranging from optimal control to queueing models and differential games, and are referred to as algebraic Riccati equations because of their similarity and close connection to the classical differential equation x′ = ax2 + bx + c.
The monograph is mainly self-contained, beginning with a short introduction to the different ways algebraic Riccati equations come about, including a discussion on special properties of Sylvester, Lyapunov, and Stein equations. At the same time, the authors introduce the mathematical techniques from linear algebra and optimization theory that one must be familiar with in order to follow the proofs of theorems. A substantial part of the manuscript is devoted to developing computational algorithms, especially the doubling algorithms, whose detailed development seems to be a novel contribution of this text.
The text is augmented with a large number of MATLAB programs, which appear more frequently toward the end of the book; experimenting with specific examples has the potential of enhancing the reader’s appreciation of how powerful some of these algorithms are. As the authors point out in the introduction, the MATLAB programs are not meant to replace the commercial software packages that are designed and intended for algebraic Riccati equations; rather, they are meant to illustrate the ease with which these algorithms can be implemented, with perhaps the caveat of sacrificing the efficiency and robustness one may get from commercial software.
With some effort on the part of the reader, this book will serve as an excellent introduction to an important topic in applied mathematics.
