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A Numerical Method for Computing Border Curves of Bi-parametric Real Polynomial Systems and Applications

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Book cover Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.

The border curve enables us to construct a so-called “solution map”. For a given value u of the parameters inside the rectangle but not on the border, the solution map tells the subset that u belongs to together with a connected path from the corresponding sample point w to u. Consequently, all the real solutions of the system at u (which are isolated) can be obtained by tracking a real homotopy starting from all the real roots at w throughout the path. The effectiveness of the proposed method is illustrated by some examples.

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Acknowledgements

This work is partially supported by NSFC (11301524, 11471307, 61572024) and CSTC (cstc2015jcyjys40001).

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Authors and Affiliations

  1. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Beijing, China

    Changbo Chen & Wenyuan Wu

Authors
  1. Changbo Chen
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  2. Wenyuan Wu
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Correspondence to Wenyuan Wu .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Chen, C., Wu, W. (2016). A Numerical Method for Computing Border Curves of Bi-parametric Real Polynomial Systems and Applications. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_11

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45640-9

  • Online ISBN: 978-3-319-45641-6

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