Solving Rank-Structured Sylvester and Lyapunov Equations

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied in [A. Haber and M. Verhaegen, Automatica J. IFAC, 73 (2016), pp. 256--268; D. Palitta and V. Simoncini, Numerical Methods for Large-Scale Lyapunov Equations with Symmetric Banded Data, preprint, arxiv, 1711.04187, 2017], which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allo...