Solving nonlinear polynomial systems in the barycentric Bernstein basis
- 138 Downloads
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.
KeywordsCAD CAGD CAM Geometric modeling Solid modeling Intersections Distance computation Engineering design
Unable to display preview. Download preview PDF.
- 3.Elber, G., Kim, M.S.: Geometric constraint solver using multivariate rational spline functions. In: SMA ’01: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 1–10. ACM Press, New York, NY, USA (2001) (DOI http://doi.acm.org/10.1145/376957.376958)Google Scholar
- 9.Goldman, R.N., Lyche, T. (eds.): Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces. SIAM, Philadelphia (1993)Google Scholar
- 11.Hanniel, I., Elber, G.: Subdivision termination criteria in subdivision multivariate solvers using dual hyperplanes representations. Comput. Aided Des. 39(5), 369–378 (2007) (DOI http://dx.doi.org/10.1016/j.cad.2007.02.004)Google Scholar
- 12.Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A.K. Peters, Wellesley, MA (1993) (Translated by L.L. Schumaker)Google Scholar
- 15.Mourrain, B., Pavone, J.P.: Subdivision Methods for Solving Polynomial Equations. Tech. Rep. 5658, INRIA Sophia-Antipolis (2005)Google Scholar
- 20.Wilkinson, J.H.: Rounding Errors in Algebraic Processs. Prentice-Hall, Englewood Cliffs (1963)Google Scholar