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*=Γ∘(∑i=14UiViT) with low-ranked Ui and Vi that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.