# 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^*XA=Q, Lyapunov equations A^*X+XA+Q=0, discrete-time algebraic Riccati equations X=Q+A^*X(I+GX)^-1A, 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^-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.

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