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

Abstract

For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B(-) -XF(-) -F(+) X + XB(+)X = 0, where F(+/-) = (I -F) D(+/-) and B(+/-) = BD(+/-) with positive diagonal matrices D(+/-) and possibly low-ranked matrices F and B. We prove the existence of the minimal positive solution X* under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newton's method and the doubling algorithm. We shall also study several special cases. For example when B and F are low ranked then X* = Gamma circle(Sigma(r)(i=1)U(i)V(i)(T)) with low-ranked U(i) and V(i) that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.