TY - JOUR

T1 - Polytope-based computation of polynomial ranges

AU - Fünfzig, Christoph

AU - Michelucci, Dominique

AU - Foufou, Sebti

N1 - Funding Information:
This research work has been funded in part by Conseil Régional de Bourgogne and by NPRP grant number NPRP 09-906-1-137 from the Qatar National Research Fund (a member of The Qatar Foundation).

PY - 2012/1

Y1 - 2012/1

N2 - Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n≤10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

AB - Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n≤10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

KW - Bernstein polynomials

KW - Multivariate polynomials

KW - Polynomial ranges

KW - Polytopes

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U2 - 10.1016/j.cagd.2011.09.001

DO - 10.1016/j.cagd.2011.09.001

M3 - Article

AN - SCOPUS:81855221800

VL - 29

SP - 18

EP - 29

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

SN - 0167-8396

IS - 1

ER -