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

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)⁻¹A, continuous-time algebraic Riccati equations Q + A^∗ X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A^∗Y² = 0, and nonlinear matrix equations X + A^∗ X⁻¹A = 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.