Efficient Algorithms for Large-Scale Quadratic Matrix Equations

Abstract

Quadratic matrix equations and in particular symmetric algebraic Riccati equations play a fundamental role in systems and control theory. Classically, they are solved via methods using their connection to Hamiltonian eigenproblems. Due to the ever-increasing complexity of the models describing the underlying control problems, new methods are needed that can be used for large-scale problems. In particular, sparsity of the coefficient matrices, obtained, e.g., from semi-discretizing partial differential equations to describe the physical process to be controlled, need to be addressed. We briefly review recent approaches based on Krylov subspace methods and discuss a new approach employing a sparse implementation of Newton's method for algebraic Riccati equations.