Solving nonlinear polynomial systems in the barycentric Bernstein basis

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

Research output: Contribution to journalArticle

22 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)187-200
Number of pages14
JournalVisual Computer
Volume24
Issue number3
DOIs
Publication statusPublished - 2008 Mar 1
Externally publishedYes

Keywords

  • CAD
  • CAGD
  • CAM
  • Distance computation
  • Engineering design
  • Geometric modeling
  • Intersections
  • Solid modeling

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design

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  • Cite this

    Reuter, M., Mikkelsen, T. S., Sherbrooke, E. C., Maekawa, T., & Patrikalakis, N. M. (2008). Solving nonlinear polynomial systems in the barycentric Bernstein basis. Visual Computer, 24(3), 187-200. https://doi.org/10.1007/s00371-007-0184-x