Quadratic Vector Equations

TL;DR

This paper unifies the study of various quadratic vector and matrix equations, generalizing and extending existing results across multiple specific problem types through an abstracted framework.

Contribution

It introduces a unified framework for quadratic vector and matrix equations, generalizing previous results and removing some restrictive hypotheses.

Findings

Unified approach to quadratic vector and matrix equations

Generalization of existing results across multiple problem types

Simplification and extension of proofs for key equations

Abstract

We study in an unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses. Specific cases of such problems (QBD equations, nonsymmetric algebraic Riccati equations, Lu's simple equation, Markovian binary trees equations) have been studied extensively in the past by several authors. Many of the results appearing here have already been proved for one or more of the single instances of the problems, resorting to specific characteristics of the problem. In some cases the proofs we present here are mere rewriting of the original proofs with a little change of notation to adapt them to our framework, but in some cases we are effectively able to remove some hypotheses and generalize the results by abstracting the specific aspects of each problem.