Decay bounds for the numerical quasiseparable preservation in matrix functions

TL;DR

This paper establishes bounds on how well quasiseparable structures are preserved in matrix functions, extending integral formulas and leveraging hierarchical matrices for efficient computation.

Contribution

It introduces new decay bounds for quasiseparable preservation in matrix functions and extends integral formulas to handle poles inside contours.

Findings

Bounds depend on spectrum and singularities

Guarantees quasiseparable structure preservation

Uses hierarchical matrices for computation

Abstract

Given matrices $A$ and $B$ such that $B=f(A)$, where $f(z)$ is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of $A$ and $B$. We provide family of bounds which depend on the interplay between the spectrum of the argument $A$ and the singularities of the function. In particular, these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses. We extend the Dunford-Cauchy integral formula to the case in which some poles are contained inside the contour of integration. We use this tool together with the technology of hierarchical matrices ($\mathcal H$-matrices) for the effective computation of matrix functions with quasiseparable arguments.