  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.
