Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

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−1v, A−>v, D−1v and D−>v computable in O(n1) or O(n2) complexity, for some vector v) and B,C are low-ranked. 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 sparseplus-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.