Efficient Algorithms for Large‐Scale Quadratic Matrix Equations

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.