# Low-rank updates and divide-and-conquer methods for quadratic matrix equations

## TL;DR

This paper introduces a fast low-rank update technique and a divide-and-conquer method for solving large-scale quadratic matrix equations, especially those with hierarchical low-rank structured coefficients, improving efficiency over iterative schemes.

## Contribution

It presents a novel low-rank update approach and a divide-and-conquer algorithm tailored for quadratic matrix equations with hierarchical low-rank structures, extending previous linear matrix methods.

## Key findings

- The proposed methods outperform iterative schemes in numerical experiments.
- The divide-and-conquer approach efficiently handles structured large-scale equations.
- Low-rank updates enable quick adjustments to solutions under coefficient modifications.

## Abstract

In this work, we consider two types of large-scale quadratic matrix equations: Continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems. We propose a simple and fast way to update the solution to such matrix equations under low-rank modifications of the coefficients. Based on this procedure, we develop a divide-and-conquer method for quadratic matrix equations with coefficients that feature a specific type of hierarchical low-rank structure, which includes banded matrices. This generalizes earlier work on linear matrix equations. Numerical experiments indicate the advantages of our newly proposed method versus iterative schemes combined with hierarchical low-rank arithmetic.