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Volume 14, Issue 3
A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations

Liqiang Dong, Jicheng Li & Xuenian Liu

DOI: 10.4208/nmtma.OA-2019-0140

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 650-691.

Published online: 2021-06

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  • Abstract

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.

  • Keywords

Complex nonsymmetric algebraic Riccati equation, extremal solution, numerical method, doubling algorithm, complex parameter selection strategy.

  • AMS Subject Headings

65F15, 65F18, 65F30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-650, author = {Dong , LiqiangLi , Jicheng and Liu , Xuenian}, title = {A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {3}, pages = {650--691}, abstract = {

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0140}, url = {http://global-sci.org/intro/article_detail/nmtma/19193.html} }
TY - JOUR T1 - A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations AU - Dong , Liqiang AU - Li , Jicheng AU - Liu , Xuenian JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 650 EP - 691 PY - 2021 DA - 2021/06 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2019-0140 UR - https://global-sci.org/intro/article_detail/nmtma/19193.html KW - Complex nonsymmetric algebraic Riccati equation, extremal solution, numerical method, doubling algorithm, complex parameter selection strategy. AB -

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.

Dong , LiqiangLi , Jicheng and Liu , Xuenian. (2021). A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations. Numerical Mathematics: Theory, Methods and Applications. 14 (3). 650-691. doi:10.4208/nmtma.OA-2019-0140
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