TY - JOUR

T1 - Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

AU - Li, Tiexiang

AU - Chu, Eric King Wah

AU - Jonq, Juang

AU - Lin, Wen-Wei

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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-ŝPD±, B-≡(b̂I+ŝP)D- and B+≡b̂I+ŝPD+ with a nonnegative matrix P, positive diagonal matrices D±, and nonnegative parameters f, b̂≡b/(1-f) and ŝ≡s/(1-f). We prove the existence of the minimal nonnegative solution X under the physically reasonable assumption f+b+sP(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 b̂=0 and P is low-ranked, then X=ŝ2UV 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.

AB - 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-ŝPD±, B-≡(b̂I+ŝP)D- and B+≡b̂I+ŝPD+ with a nonnegative matrix P, positive diagonal matrices D±, and nonnegative parameters f, b̂≡b/(1-f) and ŝ≡s/(1-f). We prove the existence of the minimal nonnegative solution X under the physically reasonable assumption f+b+sP(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 b̂=0 and P is low-ranked, then X=ŝ2UV 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.

KW - Algebraic Riccati equation

KW - Doubling algorithm

KW - Fixed-point iteration

KW - Newton's method

KW - Reflection kernel

KW - Transport theory

UR - http://www.scopus.com/inward/record.url?scp=78049442911&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.09.006

DO - 10.1016/j.laa.2010.09.006

M3 - Article

AN - SCOPUS:78049442911

SN - 0024-3795

VL - 434

SP - 201

EP - 214

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1

ER -