Volume 10, Issue 6
Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation

Adv. Appl. Math. Mech., 10 (2018), pp. 1327-1343.

Published online: 2018-09

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• Abstract

We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with $M$-matrix. It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm (SDA) with the shift-and-shrink transformation or the generalized Cayley transformation. In this paper, we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases. Meanwhile, the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined. Moreover, the convergence result and the comparison theorem on convergent rate are established. Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with $M$-matrix.

• Keywords

Shift-and-shrink transformation, generalized Cayley transformation, doubling algorithm, nonsymmetric algebraic Riccati equation.

65F50, 15A24

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@Article{AAMM-10-1327, author = {Bo and Tang and and 19041 and and Bo Tang and Yunqing and Huang and and 19042 and and Yunqing Huang and Ning and Dong and and 19043 and and Ning Dong}, title = {Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {1327--1343}, abstract = {

We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with $M$-matrix. It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm (SDA) with the shift-and-shrink transformation or the generalized Cayley transformation. In this paper, we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases. Meanwhile, the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined. Moreover, the convergence result and the comparison theorem on convergent rate are established. Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with $M$-matrix.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0012}, url = {http://global-sci.org/intro/article_detail/aamm/12713.html} }
TY - JOUR T1 - Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation AU - Tang , Bo AU - Huang , Yunqing AU - Dong , Ning JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1327 EP - 1343 PY - 2018 DA - 2018/09 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0012 UR - https://global-sci.org/intro/article_detail/aamm/12713.html KW - Shift-and-shrink transformation, generalized Cayley transformation, doubling algorithm, nonsymmetric algebraic Riccati equation. AB -

We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with $M$-matrix. It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm (SDA) with the shift-and-shrink transformation or the generalized Cayley transformation. In this paper, we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases. Meanwhile, the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined. Moreover, the convergence result and the comparison theorem on convergent rate are established. Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with $M$-matrix.

Bo Tang, Yunqing Huang & Ning Dong. (1970). Doubling Algorithm for Nonsymmetric Algebraic Riccati Equations Based on a Generalized Transformation. Advances in Applied Mathematics and Mechanics. 10 (6). 1327-1343. doi:10.4208/aamm.OA-2018-0012
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