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Improving numerical accuracy of Grobner basis polynomial equation solvers

In English

Författare

Summary, in English

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.

Avdelning/ar

Publiceringsår

2007

Språk

Engelska

Sidor

449-456

Publikation/Tidskrift/Serie

Proceedings of the IEEE 11th International Conference on Computer Vision

Dokumenttyp

Konferensbidrag

Förlag

IEEE - Institute of Electrical and Electronics Engineers Inc.

Ämne

  • Computer Vision and Robotics (Autonomous Systems)
  • Mathematics

Nyckelord

  • numerical stability
  • Gröbner basis
  • polynomial equations

Conference name

IEEE 11th International Conference on Computer Vision, 2007. ICCV 2007

Conference date

2007-10-14 - 2007-10-21

Conference place

Rio de Janeiro, Brazil

Status

Published

Forskningsgrupp

  • Mathematical Imaging Group

ISBN/ISSN/Övrigt

  • ISSN: 1550-5499
  • ISBN: 978-1-4244-1631-8