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

TL;DR

This paper provides a theoretical explanation and practical tools for the exponential decay of off-diagonal singular values in matrices from cyclic reduction, enabling more efficient solutions to certain matrix equations.

Contribution

It offers a sharp theoretical bound and an estimation tool for the exponential decay of off-diagonal singular values in cyclic reduction matrices.

Findings

Exponential decay of off-diagonal singular values is theoretically bounded.

Application of decay bounds improves efficiency in solving block tridiagonal systems.

Achieves $O(n^2 \, \log n)$ complexity for certain matrix equations.

Abstract

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.