Solving nonlinear polynomial systems in the barycentric Bernstein basis

Martin Reuter, Tarjei S. Mikkelsen, Evan C. Sherbrooke, Takashi Maekawa, Nicholas M. Patrikalakis

Research output: Contribution to journalArticle

22 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)187-200
Number of pages14
JournalVisual Computer
Volume24
Issue number3
DOIs
Publication statusPublished - 2008 Mar 1
Externally publishedYes

Fingerprint

Polynomials
Tensors

Keywords

  • CAD
  • CAGD
  • CAM
  • Distance computation
  • Engineering design
  • Geometric modeling
  • Intersections
  • Solid modeling

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design

Cite this

Reuter, M., Mikkelsen, T. S., Sherbrooke, E. C., Maekawa, T., & Patrikalakis, N. M. (2008). Solving nonlinear polynomial systems in the barycentric Bernstein basis. Visual Computer, 24(3), 187-200. https://doi.org/10.1007/s00371-007-0184-x

Solving nonlinear polynomial systems in the barycentric Bernstein basis. / Reuter, Martin; Mikkelsen, Tarjei S.; Sherbrooke, Evan C.; Maekawa, Takashi; Patrikalakis, Nicholas M.

In: Visual Computer, Vol. 24, No. 3, 01.03.2008, p. 187-200.

Research output: Contribution to journalArticle

Reuter, M, Mikkelsen, TS, Sherbrooke, EC, Maekawa, T & Patrikalakis, NM 2008, 'Solving nonlinear polynomial systems in the barycentric Bernstein basis', Visual Computer, vol. 24, no. 3, pp. 187-200. https://doi.org/10.1007/s00371-007-0184-x
Reuter, Martin ; Mikkelsen, Tarjei S. ; Sherbrooke, Evan C. ; Maekawa, Takashi ; Patrikalakis, Nicholas M. / Solving nonlinear polynomial systems in the barycentric Bernstein basis. In: Visual Computer. 2008 ; Vol. 24, No. 3. pp. 187-200.
@article{a32ecc9afda640499977702def19ba91,
title = "Solving nonlinear polynomial systems in the barycentric Bernstein basis",
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.",
keywords = "CAD, CAGD, CAM, Distance computation, Engineering design, Geometric modeling, Intersections, Solid modeling",
author = "Martin Reuter and Mikkelsen, {Tarjei S.} and Sherbrooke, {Evan C.} and Takashi Maekawa and Patrikalakis, {Nicholas M.}",
year = "2008",
month = "3",
day = "1",
doi = "10.1007/s00371-007-0184-x",
language = "English",
volume = "24",
pages = "187--200",
journal = "Visual Computer",
issn = "0178-2789",
publisher = "Springer Verlag",
number = "3",

}

TY - JOUR

T1 - Solving nonlinear polynomial systems in the barycentric Bernstein basis

AU - Reuter, Martin

AU - Mikkelsen, Tarjei S.

AU - Sherbrooke, Evan C.

AU - Maekawa, Takashi

AU - Patrikalakis, Nicholas M.

PY - 2008/3/1

Y1 - 2008/3/1

N2 - 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.

AB - 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.

KW - CAD

KW - CAGD

KW - CAM

KW - Distance computation

KW - Engineering design

KW - Geometric modeling

KW - Intersections

KW - Solid modeling

UR - http://www.scopus.com/inward/record.url?scp=38749103575&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38749103575&partnerID=8YFLogxK

U2 - 10.1007/s00371-007-0184-x

DO - 10.1007/s00371-007-0184-x

M3 - Article

AN - SCOPUS:38749103575

VL - 24

SP - 187

EP - 200

JO - Visual Computer

JF - Visual Computer

SN - 0178-2789

IS - 3

ER -