Quadratic vector equations

Abstract

We study in a unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses, by seeing them as special cases of the general problem M x = a + b (x, x), where a and the unknown x are componentwise nonnegative vectors, M is a nonsingular M-matrix, and b is a bilinear map from pairs of nonnegative vectors to nonnegative vectors. Specific cases of this equation have been studied extensively in the past by several authors, and include unilateral matrix equations from queuing problems [Bini, Latouche, Meini, 2005], nonsymmetric algebraic Riccati equations [Guo, Laub, 2000], and quadratic matrix equations encountered in neutron transport theory .We present a unified approach which treats the common aspects of their theoretical properties and basic iterative solution algorithms. This has interesting consequences: in some cases, we are able to derive in full generality theorems and proofs appeared in literature only for special cases of the problem; this broader view highlights the role of hypotheses such as the strict positivity of the minimal solution. In an example, we adapt an algorithm derived for one equation of the class to another, with computational advantage with respect to the existing methods. We discuss possible research lines, including the relationship among Newton-type methods and the cyclic reduction algorithm for unilateral quadratic equations.

Perron‐based algorithms for the multilinear PageRank

“…This problem originally appeared in Li and Ng, and is a variation of problems related to tensor eigenvalue problems and Perron–Frobenius theory for tensors (see, e.g., Lim, Chang et al, Friedland et al). However, it also fits in the framework of quadratic vector equations derived from Markovian binary tree models introduced by Hautphenne et al and later considered in Bini et al, Meini and Poloni, and Poloni . Indeed, Poloni considered a more general problem, which is essentially without the hypotheses .…”

“…However, it also fits in the framework of quadratic vector equations derived from Markovian binary tree models introduced by Hautphenne et al and later considered in Bini et al, Meini and Poloni, and Poloni . Indeed, Poloni considered a more general problem, which is essentially without the hypotheses . Hence, all of its results apply here and can be used in the context of multilinear PageRank.…”

“…Hence, all of its results apply here and can be used in the context of multilinear PageRank. In particular, Poloni considered the minimal nonnegative solution of in the componentwise sense, which is not necessarily stochastic as the one sought in Gleich et al…”

“…In this paper, we use the theory of quadratic vector equations in Bini et al, Meini and Poloni, and Poloni to better understand the behavior of the solutions of and suggest new algorithms for computing the stochastic solution. More specifically, we show that if one considers the minimal nonnegative solution of as well, the theoretical properties of become clearer, even if one is only interested in stochastic solutions.…”

Perturbation analysis of the extinction probability of a Markovian binary tree

“…P.-C. Guo et al / Linear Algebra and its Applications 475 (2015)[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] …”