  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 queueing
models. In this paper, we provide a sharp theoretical bound to the basis of
this exponential decay together with a tool for its estimation based on a
rational interpolation problem. Applications to solving $n\times n$ block
tridiagonal block Toeplitz systems with $n\times n$ semiseparable blocks and
certain generalized Sylvester equations in $O(n^2\log n)$ arithmetic operations
are shown.
