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

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(i=1UiViT)-U-r) 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.