On the singular values of matrices with high displacement rank

TL;DR

This paper presents a new ADI-based low rank solver for matrix equations with rapidly decaying singular values, offering theoretical bounds, practical algorithms, and efficient Poisson solvers with spectral accuracy.

Contribution

It introduces novel bounds on singular values for high displacement rank matrices and develops practical low rank algorithms for Lyapunov, Sylvester, and Poisson equations.

Findings

Derived new bounds on singular values for high displacement rank matrices.

Developed practical algorithms for Lyapunov and Sylvester equations with high rank right-hand sides.

Created low rank Poisson solvers with spectral accuracy and optimal complexity.

Abstract

We introduce a new ADI-based low rank solver for $AX-XB=F$, where $F$ has rapidly decaying singular values. Our approach results in both theoretical and practical gains, including (1) the derivation of new bounds on singular values for classes of matrices with high displacement rank, (2) a practical algorithm for solving certain Lyapunov and Sylvester matrix equations with high rank right-hand sides, and (3) a collection of low rank Poisson solvers that achieve spectral accuracy and optimal computational complexity.