Solving nonlinear polynomial systems in the barycentric Bernstein basis |
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Authors: | Martin Reuter Tarjei S Mikkelsen Evan C Sherbrooke Takashi Maekawa Nicholas M Patrikalakis |
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Affiliation: | (1) Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA;(2) Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan |
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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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Keywords: | CAD CAGD CAM Geometric modeling Solid modeling Intersections Distance computation Engineering design |
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