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 ± ≡ I - s ˆ PD ± , B - ≡ ( b ˆ I + s ˆ P ) D - and B + ≡ b ˆ I + s ˆ PD + with a nonnegative matrix P, positive diagonal matrices D ± , and nonnegative parameters f, b ˆ ≡ b / ( 1 - f ) and s ˆ ≡ s / ( 1 - f ) . We prove the existence of the minimal nonnegative solution X ∗ under the physically reasonable assumption f + b + s ‖ P ( D + + D - ) ‖ ∞ 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 b ˆ = 0 and P is low-ranked, then X ∗ = s ˆ 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.