### 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 |
---|---|

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 |

### Fingerprint

### 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

**Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory.** / Weng, Peter Chang Yi; Fan, Hung-Yuan; Chu, Eric King Wah.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 219, no. 2, pp. 729-740. https://doi.org/10.1016/j.amc.2012.06.066

}

TY - JOUR

T1 - Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory

AU - Weng, Peter Chang Yi

AU - Fan, Hung-Yuan

AU - Chu, Eric King Wah

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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- 1u,A- ⊤u,D- 1v 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.

AB - 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- 1u,A- ⊤u,D- 1v 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.

KW - Doubling algorithm

KW - Krylov subspace

KW - M-matrix

KW - Nonsymmetric algebraic Riccati equation

UR - http://www.scopus.com/inward/record.url?scp=84864983285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864983285&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2012.06.066

DO - 10.1016/j.amc.2012.06.066

M3 - Article

AN - SCOPUS:84864983285

VL - 219

SP - 729

EP - 740

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 2

ER -