## Abstract

For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^{-} - XF^{-} - F^{+}X + XB^{+}X = 0, where F ^{±} ≡ (I - F)D ^{±} and B ^{±} ≡ BD ^{±} with positive diagonal matrices D ^{±} and possibly low-ranked matrices F and B. We prove the existence of the minimal positive solution X ^{*} under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newton's method and the doubling algorithm. We shall also study several special cases. For example when B and F are low ranked then X*=Γ(∑i=14UiViT) with low-ranked Ui and Vi that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.

Original language | English |
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Pages (from-to) | 1453-1467 |

Number of pages | 15 |

Journal | IMA Journal of Numerical Analysis |

Volume | 31 |

Issue number | 4 |

DOIs | |

State | Published - 1 Oct 2011 |

## Keywords

- Newton's method
- algebraic Riccati equation
- doubling algorithm
- fixed-point iteration
- reflection
- transport theory