### Abstract

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX-XD-AX+B=0 from transport theory (Juang 1995), with M≡[D,-C;-B,A]∈R2 ^{n×2n} being a nonsingular M-matrix. In addition, A,D are rank-1 updates of diagonal matrices, with the products A- ^{1}u,A- ^{⊤}u,D- ^{1}v and D- ^{⊤}v computable in 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) computational complexity and memory requirement per iteration and converges essentially quadratically, as illustrated by the numerical examples.

Original language | English |
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Pages (from-to) | 729-740 |

Number of pages | 12 |

Journal | Applied Mathematics and Computation |

Volume | 219 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Oct 1 |

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### Keywords

- Doubling algorithm
- Krylov subspace
- M-matrix
- Nonsymmetric algebraic Riccati equation

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Mathematics and Computation*,

*219*(2), 729-740. https://doi.org/10.1016/j.amc.2012.06.066