On the decay of the off-diagonal singular values in cyclic reduction
Provides a theoretical foundation for an observed phenomenon, enabling faster algorithms for specific matrix equations in queueing analysis.
This paper provides a sharp theoretical bound explaining the exponential decay of off-diagonal singular values in Cyclic Reduction, enabling O(n² log n) solutions for block tridiagonal Toeplitz systems and generalized Sylvester equations.
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.