Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

Abstract

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B(-) - XF(-) - F(+)X + XB(+)X = 0, where F(+/-) equivalent to I - (s) over cap PD(+/-), B(-) equivalent to (b) over capl + (s) over capP)D(-))D(-) and B(+) equivalent to (b) over capl + (s) over capP)D(+))D(+) with a nonnegative matrix P, positive diagonal matrices D, and nonnegative parameters f, (b) over cap equivalent to(1 - f) and (s) over cap equivalent to (1 - f). We prove the existence of the minimal nonnegative solution X* under the physically reasonable assumption f + b + s parallel to P(D(+) + D-)parallel to(infinity) < 1, and study its numerical computation by fixed-point iteration, Newton's method and doubling. We shall also study several special cases; e.g. when = 0 and P is low-ranked, then X* = <(s)over cap>/2 UV is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results. (C) 2010 Elsevier Inc. All rights reserved.