Abstract
A new algorithm for real root isolation of zero-dimensional nonsingular square polynomial systems based on hybrid computation is presented in this paper. First, approximate the (complex) roots of the given polynomial equations via homotopy continuation method. Then, for each approximate root, an initial box relying on the Kantorovich theorem is constructed, which contains the corresponding accurate root. Finally, the Krawczyk interval iteration with interval arithmetic is applied to the initial boxes so as to check whether or not the corresponding approximate roots are real and to obtain the real root isolation boxes. Moreover, an empirical construction of initial box is provided for speeding-up the computation in practice. Our experiments on many benchmarks show that the new hybrid method is very efficient. The method can find all real roots of any given zero-dimensional nonsingular square polynomial systems provided that the homotopy continuation method can find all complex roots of the equations.
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Notes
- 1.
See reference [26], Sect. 11, Automatic differentiation.
- 2.
In Matlab2008b that we do the experiments, the zero threshold is 2.2204e\(-\)016.
- 3.
As mentioned before, the zero threshold in Matlab2008b is 2.2204e\(-\)016, which is almost the same order of magnitude of those radiuses.
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Acknowledgments
The work is partly supported by the ANR-NSFC project EXACTA (ANR-09-BLAN-0371-01/60911130369), NSFC-11001040, NSFC-11271034 and the project SYSKF1207 from ISCAS. The authors especially thank professor Dongming Wang for the early discussion on this topic in 2010 and also thank professor T.Y. Li for his helpful suggestions and his team’s work on Hom4ps2-Matlab interface. Thanks also go to Ting Gan who computed the 15 systems in Table 3 and provided us suggestion on possible improvements on our program. Thank Zhenyi Ji who shared with us his insights on our hybrid method. Thank the referees for their valuable constructive comments which help improve the presentation greatly.
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Shen, F., Wu, W., Xia, B. (2014). Real Root Isolation of Polynomial Equations Based on Hybrid Computation. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_26
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