Improving numerical accuracy of Grobner basis polynomial equation solvers

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceeding

Abstract

This paper presents techniques for improving the numerical stability of Grobner basis solvers for polynomial equations. Recently Grobner basis methods have been used succesfully to solve polynomial equations arising in global optimization e.g. three view triangulation and in many important minimal cases of structure from motion. Such methods work extremely well for problems of reasonably low degree, involving a few variables. Currently, the limiting factor in using these methods for larger and more demanding problems is numerical difficulties. In the paper we (i) show how to change basis in the quotient space R[x]/I and propose a strategy for selecting a basis which improves the conditioning of a crucial elimination step, (ii) use this technique to devise a Grobner basis with improved precision and (iii) show how solving for the eigenvalues instead of eigenvectors can be used to improve precision further while retaining the same speed. We study these methods on some of the latest reported uses of Grobner basis methods and demonstrate dramatically improved numerical precision using these new techniques making it possible to solve a larger class of problems than previously.

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Authors
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Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Computer Vision and Robotics (Autonomous Systems)
  • Mathematics

Keywords

  • numerical stability, Gröbner basis, polynomial equations
Original languageEnglish
Title of host publicationProceedings of the IEEE 11th International Conference on Computer Vision
PublisherIEEE - Institute of Electrical and Electronics Engineers Inc.
Pages449-456
ISBN (Print)978-1-4244-1631-8
Publication statusPublished - 2007
Publication categoryResearch
Peer-reviewedYes
EventIEEE 11th International Conference on Computer Vision, 2007. ICCV 2007 - Rio de Janeiro, Brazil
Duration: 2007 Oct 142007 Oct 21

Publication series

Name
ISSN (Print)1550-5499

Conference

ConferenceIEEE 11th International Conference on Computer Vision, 2007. ICCV 2007
CountryBrazil
CityRio de Janeiro
Period2007/10/142007/10/21

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