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Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

We present a critical point method based on a penalty function for finding certain solution (witness) points on real solutions components of general real polynomial systems. Unlike other existing numerical methods, the new method does not require the input polynomial system to have pure dimension or satisfy certain regularity conditions.

This method has two stages. In the first stage it finds approximate solution points of the input system such that there is at least one real point on each connected solution component. In the second stage it refines the points by a homotopy continuation or traditional Newton iteration. The singularities of the original system are removed by embedding it in a higher dimensional space.

In this paper we also analyze the convergence rate and give an error analysis of the method. Experimental results are also given and shown to be in close agreement with the theory.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their constructive comments that greatly helped improving the paper. This work is partially supported by the projects NSFC (11471307, 11671377, 61572024), cstc2015jcyjys40001, and the Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SYS026).

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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, Chongqing, China

    Wenyuan Wu & Changbo Chen

  2. University of Chinese Academy of Sciences, Beijing, China

    Wenyuan Wu & Changbo Chen

  3. Applied Mathematics Department, Western University, London, Canada

    Greg Reid

Authors
  1. Wenyuan Wu
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  2. Changbo Chen
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  3. Greg Reid
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Corresponding author

Correspondence to Changbo Chen .

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Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Wu, W., Chen, C., Reid, G. (2017). Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_27

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_27

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66319-7

  • Online ISBN: 978-3-319-66320-3

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