### Abstract

We consider the solution of the large-scale nonlinear matrix equation X+BX-^{1}A-Q=0, with A,B,Q,X∈Cn×^{n}, and in some applications B=^{A} (=⊤ or H). The matrix Q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system Qv=r in O(n) computational complexity. Furthermore, B and A are respectively of ranks ^{ra},^{rb}≪n. The type 2 structure-preserving doubling algorithm by Lin and Xu (2006) [24] is adapted, with the appropriate applications of the Sherman-Morrison-Woodbury formula and the low-rank updates of various iterates. Two resulting large-scale doubling algorithms have an O((^{ra}+^{rb})^{3}) computational complexity per iteration, after some pre-processing of data in O(n) computational complexity and memory requirement, and converge quadratically. These are illustrated by the numerical examples.

Original language | English |
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Pages (from-to) | 914-932 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2013 |

### Keywords

- Doubling algorithm
- Green's function
- Krylov subspace
- Leaky surface wave
- Nano research
- Nonlinear matrix equation
- Surface acoustic wave

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## Cite this

*Linear Algebra and Its Applications*,

*439*(4), 914-932. https://doi.org/10.1016/j.laa.2012.08.008