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

TL;DR

This paper reviews and compares iterative doubling algorithms for solving Riccati- and Lyapunov-type matrix equations, highlighting their connections, differences, and open theoretical issues.

Contribution

It provides a comprehensive comparative introduction to doubling algorithms for various Riccati and related matrix equations, emphasizing their interrelations and theoretical challenges.

Findings

Doubling algorithms construct iterates via repeated squaring.

Connections between doubling algorithms and subspace iteration are highlighted.

Open issues in the theoretical understanding of these algorithms are discussed.

Abstract

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{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 $X - A^*XA=Q$, Lyapunov equations $A^*X+XA+Q=0$, discrete-time algebraic Riccati equations $X=Q+A^*X(I+GX)^{-1}A$, continuous-time algebraic Riccati equations $Q+A^*X+XA-XGX=0$, palindromic quadratic matrix equations $A+QY+A^*Y^2=0$, 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.