Efficient cyclic reduction for QBDs with rank structured blocks

TL;DR

This paper introduces efficient algorithms for solving block tridiagonal systems and quadratic matrix equations with quasiseparable blocks, leveraging cyclic reduction and rank-structured matrix technology to achieve significant computational speedups.

Contribution

The paper develops novel algorithms based on cyclic reduction and rank-structured matrices for quasiseparable blocks, with proven exponential decay of singular values, enhancing computational efficiency.

Findings

Algorithms outperform general methods starting at block size ~100.

Exponential decay of singular values is formally proven in the Markovian context.

Numerical experiments demonstrate substantial speedups.

Abstract

We provide effective algorithms for solving block tridiagonal block Toeplitz systems with $m\times m$ quasiseparable blocks, as well as quadratic matrix equations with $m\times m$ quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size $m\approx 10^2$.