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Solving nonlinear polynomial systems in the barycentric Bernstein basis

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Abstract

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Martin Reuter, Tarjei S. Mikkelsen, Evan C. Sherbrooke & Nicholas M. Patrikalakis

  2. Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan

    Takashi Maekawa

Authors
  1. Martin Reuter
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  2. Tarjei S. Mikkelsen
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  3. Evan C. Sherbrooke
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  4. Takashi Maekawa
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  5. Nicholas M. Patrikalakis
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Correspondence to Martin Reuter.

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Reuter, M., Mikkelsen, T., Sherbrooke, E. et al. Solving nonlinear polynomial systems in the barycentric Bernstein basis. Visual Comput 24, 187–200 (2008). https://doi.org/10.1007/s00371-007-0184-x

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  • DOI: https://doi.org/10.1007/s00371-007-0184-x

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