Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX-XD-AX+B=0XCX-XD-AX+B=0 from transport theory (Juang 1995), with M≡[D,-C;-B,A]∈R2n×2nM≡[D,-C;-B,A]∈R2n×2n being a nonsingular M-matrix. In addition, A,DA,D are rank-1 updates of diagonal matrices, with the products A-1u,A-⊤u,D-1vA-1u,A-⊤u,D-1v and D-⊤vD-⊤v computable in O(n)O(n) complexity, for some vectors u and v, and B, C   are rank 1. The structure-preserving doubling algorithm by Guo et al. (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)O(n) computational complexity and memory requirement per iteration and converges essentially quadratically, as illustrated by the numerical examples.