Riccati ADI: Existence, uniqueness and new iterative methods

Abstract: The approximate solution of large-scale algebraic Riccati equations is considered. We are interested in approximate solutions which yield a Riccati residual matrix of a particular small rank. It is assumed that such approximate solutions can be written in factored form $ZYZ^*$ with a rectangular matrix $Z$ and a small quadratic matrix $Y$. We propose to choose $Z$ such that its columns span a certain rational Krylov subspace. Conditions under which such an approximate solution exists and is unique are determined. It is shown that the proposed method can be interpreted as an oblique projection method. Two new iterative procedures with efficient updates of the solution and the residual factor are derived. With our approach complex system matrices can be handled, realification is provided and parallelization is introduced.