  In this work, we consider two types of large-scale quadratic matrix
equations: Continuous-time algebraic Riccati equations, which play a central
role in optimal and robust control, and unilateral quadratic matrix equations,
which arise from stochastic processes on 2D lattices and vibrating systems. We
propose a simple and fast way to update the solution to such matrix equations
under low-rank modifications of the coefficients. Based on this procedure, we
develop a divide-and-conquer method for quadratic matrix equations with
coefficients that feature a specific type of hierarchical low-rank structure,
which includes banded matrices. This generalizes earlier work on linear matrix
equations. Numerical experiments indicate the advantages of our newly proposed
method versus iterative schemes combined with hierarchical low-rank arithmetic.
