Polytope-based computation of polynomial ranges

Christoph Fünfzig, Dominique Michelucci, Sebti Foufou

    Research output: Contribution to journalArticle

    Abstract

    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.

    Original languageEnglish (US)
    Pages (from-to)18-29
    Number of pages12
    JournalComputer Aided Geometric Design
    Volume29
    Issue number1
    DOIs
    StatePublished - Jan 1 2012

    Fingerprint

    Polytope
    Bernstein Basis
    Polynomials
    Polynomial
    Range of data
    Polytopes
    Convex Hull
    Bernstein Polynomials
    Polynomial Systems
    Half-space
    Linear programming
    Interval
    Coefficient

    Keywords

    • Bernstein polynomials
    • Multivariate polynomials
    • Polynomial ranges
    • Polytopes

    ASJC Scopus subject areas

    • Modeling and Simulation
    • Automotive Engineering
    • Aerospace Engineering
    • Computer Graphics and Computer-Aided Design

    Cite this

    Polytope-based computation of polynomial ranges. / Fünfzig, Christoph; Michelucci, Dominique; Foufou, Sebti.

    In: Computer Aided Geometric Design, Vol. 29, No. 1, 01.01.2012, p. 18-29.

    Research output: Contribution to journalArticle

    Fünfzig, Christoph ; Michelucci, Dominique ; Foufou, Sebti. / Polytope-based computation of polynomial ranges. In: Computer Aided Geometric Design. 2012 ; Vol. 29, No. 1. pp. 18-29.
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