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 \emph{doubling}: they construct the iterate $Qk = X{2^k}$ 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 $A+QY+A^*Y2=0$, 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.