Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model

Tiexiang Li Eric King-Wah Chu Jong Juang Wen-Wei Lin

Numerical Analysis and Scientific Computing mathscidoc:1912.43741

IMA journal of numerical analysis, 31, (4), 1453-1467, 2011.5
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 <i>B</i><sup></sup> <i>XF</i><sup></sup> <i>F</i><sup>+</sup><i>X</i> + <i>XB</i><sup>+</sup><i>X</i> = 0, where <i>F</i><sup></sup> (<i>I</i> <i>F</i>)<i>D</i><sup></sup> and <i>B</i><sup></sup> <i>BD</i><sup></sup> with positive diagonal matrices <i>D</i><sup></sup> and possibly low-ranked matrices <i>F</i> and <i>B</i>. We prove the existence of the minimal positive solution <i>X</i><sup>*</sup> under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newtons method and the doubling algorithm. We shall also study several special cases. For example when <i>B</i> and <i>F</i> are low ranked then X*=(i=14UiViT) with low-ranked <i>U<sub>i</sub></i> and <i>V<sub>i</sub></i> that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.
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@inproceedings{tiexiang2011solution,
  title={Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model},
  author={Tiexiang Li, Eric King-Wah Chu, Jong Juang, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205429770194305},
  booktitle={IMA journal of numerical analysis},
  volume={31},
  number={4},
  pages={1453-1467},
  year={2011},
}
Tiexiang Li, Eric King-Wah Chu, Jong Juang, and Wen-Wei Lin. Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model. 2011. Vol. 31. In IMA journal of numerical analysis. pp.1453-1467. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205429770194305.
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