On the solution of a quadratic vector equation arising in Markovian Binary Trees

TL;DR

This paper advances the theoretical understanding and computational methods for solving a quadratic vector equation in Markovian Binary Trees, introducing new convergence conditions and a quadratic convergence Newton method.

Contribution

It relaxes irreducibility assumptions, characterizes minimal solutions via Jacobian properties, and proposes a Newton-based algorithm with quadratic convergence for the QVE.

Findings

Perron iteration convergence condition established

Newton method achieves quadratic convergence

Algorithm modifications affect convergence speed

Abstract

We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a suitable function. From the computational point of view, we elaborate on the Perron vector-based iteration proposed in [http://arxiv.org/abs/1006.0577]. In particular we provide a condition which ensures that the Perron iteration converges to the sought solution of the QVE. Moreover we introduce a variant of the algorithm which consists in applying the Newton method instead of a fixed-point iteration. This method has the same convergence behaviour as the Perron iteration, since its convergence speed increases for close-to-critical problems. Moreover, unlike the Perron iteration, the method has a quadratic convergence. Finally, we show that it is possible to alter the bilinear form defining the QVE in several ways without changing the solution. This modification has an impact on convergence speed of the algorithms.