Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

Abstract

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B =0 , withM ≡ (D, −C; −B, A) ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A −1 u, A −� u, D −1 v ,a ndD −� v computable in O(n )c omplexity (withn =m ax{n1 ,n 2}), for some vectors u and v ,a ndB, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 392-412) 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 and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.