Solving large-scale nonsymmetric algebraic Riccati equations from two-dimensional transport models by doubling

Abstract

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX−XD−AX+B=0, with M≡[D,−C;−B,A]∈R(n1+n2)×(n1+n2) being a nonsingular M-matrix, and A,D being sparse-like, with the products A−1u, A−⊤u, D−1v and D−⊤v computable in O(n1) or O(n2) complexity, for some vectors u and v. In the nonsymmetric algebraic Riccati equation arose from a two-dimensional transport model, B,C are low-ranked corrections of some invertible diagonal matrices. The structure-preserving doubling algorithm by Guo, Lin and Xu (2006) is adapted, with the appropriate applications of the Sherman–Morrison–Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration (with n=max{n1,n2}) and converges essentially quadratically, as illustrated by the numerical examples.