On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queuing models. In this paper, we provide a theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Numerical experiments show that the bound is often accurate in practice. Applications to solving × block tridiagonal block Toeplitz systems with × semiseparable blocks and certain generalized Sylvester equations in (²log ) arithmetic operations are shown.status: publishe