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.
@article{de124e5e4b5d4cbb98910e501ee8fc1a,
title = "Polytope-based computation of polynomial ranges",
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.",
keywords = "Bernstein polynomials, Multivariate polynomials, Polynomial ranges, Polytopes",
author = "Christoph F{\"u}nfzig and Dominique Michelucci and Sebti Foufou",
year = "2012",
month = "1",
day = "1",
doi = "10.1016/j.cagd.2011.09.001",
language = "English (US)",
volume = "29",
pages = "18--29",
journal = "Computer Aided Geometric Design",
issn = "0167-8396",
publisher = "Elsevier",
number = "1",

}

TY - JOUR

T1 - Polytope-based computation of polynomial ranges

AU - Fünfzig, Christoph

AU - Michelucci, Dominique

AU - Foufou, Sebti

PY - 2012/1/1

Y1 - 2012/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=81855221800&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81855221800&partnerID=8YFLogxK

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 -