Improving numerical accuracy of Grobner basis polynomial equation solvers
Byröd, Martin LU ; Josephson, Klas LU and Åström, Karl LU
(2007)
IEEE 11th International Conference on Computer Vision, 2007. ICCV 2007
p.449-456
- 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... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1407302
- author
- Byröd, Martin
LU
; Josephson, Klas
LU
and Åström, Karl
LU
- organization
- publishing date
- 2007
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- keywords
- numerical stability, Gröbner basis, polynomial equations
- host publication
- Proceedings of the IEEE 11th International Conference on Computer Vision
- pages
- 449 - 456
- publisher
- IEEE - Institute of Electrical and Electronics Engineers Inc.
- conference name
- IEEE 11th International Conference on Computer Vision, 2007. ICCV 2007
- conference location
- Rio de Janeiro, Brazil
- conference dates
- 2007-10-14 - 2007-10-21
- external identifiers
-
- wos:000255099300059
- scopus:50649098598
- ISSN
- 1550-5499
- ISBN
- 978-1-4244-1631-8
- DOI
- 10.1109/ICCV.2007.4408885
- language
- English
- LU publication?
- yes
- id
- 0928ebba-b346-4146-a0bc-9a8fcc7f3b66 (old id 1407302)
- date added to LUP
- 2016-04-01 17:12:39
- date last changed
- 2022-04-23 03:25:39
@inproceedings{0928ebba-b346-4146-a0bc-9a8fcc7f3b66,
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.}},
author = {{Byröd, Martin and Josephson, Klas and Åström, Karl}},
booktitle = {{Proceedings of the IEEE 11th International Conference on Computer Vision}},
isbn = {{978-1-4244-1631-8}},
issn = {{1550-5499}},
keywords = {{numerical stability; Gröbner basis; polynomial equations}},
language = {{eng}},
pages = {{449--456}},
publisher = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
title = {{Improving numerical accuracy of Grobner basis polynomial equation solvers}},
url = {{https://lup.lub.lu.se/search/files/8178282/1245112.pdf}},
doi = {{10.1109/ICCV.2007.4408885}},
year = {{2007}},
}