Fast singular value decay for Lyapunov solutions with nonnormal coefficients

TL;DR

This paper investigates the decay of singular values in Lyapunov solutions with nonnormal coefficients, revealing that beyond a certain point, increased nonnormality can lead to faster decay, contrary to previous beliefs.

Contribution

It provides new bounds showing that larger departure from normality can accelerate singular value decay beyond a certain threshold.

Findings

Larger nonnormality can lead to faster singular value decay after a threshold.

Previous bounds suggested decay slows with increased nonnormality, but this is only true up to a point.

The numerical range's extent into the right-half plane limits decay speed.

Abstract

Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities that depend quadratically on the departure of the coefficient matrix from normality: these bounds suggest that the larger the departure from normality, the slower the singular values will decay. We show this is only true up to a threshold, beyond which a larger departure from normality can actually correspond to faster decay of singular values: if the singular values decay slowly, the numerical range cannot extend far into the right-half plane.