Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction

Abstract

We review a family of algorithms for Lyapunov-and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate = 2 of another naturally-arising fixed-point iteration ( ℎ ) via a sort of repeated squaring. The equations we consider are Stein equations − * = , Lyapunov equations * + + = 0, discrete-time algebraic Riccati equations = + * ( + ) −1 , continuous-time algebraic Riccati equations + * + − = 0, palindromic quadratic matrix equations + + * 2 = 0, and nonlinear matrix equations + * −1 = . We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.