Quadratic vector equations

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 Mx=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, and Meini, 2005 [7]), nonsymmetric algebraic Riccati equations (Guo and Laub, 2000 [14]), and quadratic matrix equations encountered in neutron transport theory (Lu, 2005 [23,24]). 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. © 2011 Elsevier Inc. All rights reserved.