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 Q k = X 2 k of another naturally-arising fixed-point iteration (X h) via a sort of repeated squaring. The equations we consider are Stein equations X − A * X A = Q, Lyapunov equations A * X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A * X(I + G X) −1 A, continuous-time algebraic Riccati equations Q + A * X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A * Y 2 = 0, and nonlinear matrix equations X + A * X −1 A = Q. 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.