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A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems


Authors: Bo Dong, Bo Yu and Yan Yu
Journal: Math. Comp. 83 (2014), 1847-1868
MSC (2010): Primary 13P15, 65H20; Secondary 65H10, 14Q99, 68W30
DOI: https://doi.org/10.1090/S0025-5718-2013-02763-9
Published electronically: November 7, 2013
MathSciNet review: 3194132
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Abstract: A mixed trigonometric polynomial system, which rather frequently occurs in applications, is a polynomial system where every monomial is a mixture of some variables and sine and cosine functions applied to the other variables. Polynomial systems transformed from the mixed trigonometric polynomial systems have a special structure. Based on this structure, a hybrid polynomial system solving method, which is more efficient than random product homotopy method and polyhedral homotopy method in solving this class of systems, has been presented. Furthermore, the transformed polynomial system has an inherent partially symmetric structure, which cannot be adequately exploited to reduce the computation by the existing methods for solving polynomial systems. In this paper, a symmetric homotopy is constructed and, combining homotopy methods, decomposition, and elimination techniques, an efficient symbolic-numerical method for solving this class of polynomial systems is presented. Preservation of the symmetric structure assures us that only part of the homotopy paths have to be traced, and more important, the computation work can be reduced due to the existence of the inconsistent subsystems, which need not to be solved at all. Exploiting the new hybrid method, some problems from the literature and a challenging practical problem, which cannot be solved by the existing methods, are resolved. Numerical results show that our method has an advantage over the polyhedral homotopy method, hybrid method and regeneration method, which are considered as the state-of-art numerical methods for solving highly deficient polynomial systems of high dimension.


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Additional Information

Bo Dong
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, People’s Republic of China
Email: dongbo@dlut.edu.cn

Bo Yu
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, People’s Republic of China
Email: yubo@dlut.edu.cn

Yan Yu
Affiliation: Faculty of Sciences, Shenyang Agricultural University, Shenyang, Liaoning, 110866, People’s Republic of China
Email: yuyan1015@syau.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-2013-02763-9
Keywords: Mixed trigonometric polynomial system, polynomial system, symmetry, homotopy method, hybrid algorithm, symbolic-numeric computation
Received by editor(s): October 21, 2011
Received by editor(s) in revised form: July 3, 2012
Published electronically: November 7, 2013
Additional Notes: The first author was supported in part by the National Natural Science Foundation of China (Grant No. 11101067), TianYuan Special Funds of the National Natural Science Foundation of China (Grant No. 11026164) and the Fundamental Research Funds for the Central Universities
The second author’s research was supported by Major Research Plan of the National Natural Science Foundation of China (Grant No. 91230103) and the National Natural Science Foundation of China (Grant No. 11171051)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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