research-article

Computing cylindrical algebraic decomposition via triangular decomposition

Authors:

Marc Moreno Maza,

Lu YangAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 95 - 102

https://doi.org/10.1145/1576702.1576718

Published: 28 July 2009 Publication History

Abstract

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ [y₁,...,y_n] we apply comprehensive triangular decomposition in order to obtain an F-invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F-invariant cylindrical algebraic decomposition of the n-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

References

[1]

D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput., 13(4):865--877, 1984.

Digital Library

[2]

P. Aubry, D. Lazard, and M. Moreno Maza. On the theories of triangular sets. J. Symb. Comp., 28(1-2):105--124, 1999.

Digital Library

[3]

S. Basu, R. Pollack, and M. F. Roy. Algorithms in real algebraic geometry, volume 10 of Algorithms and Computations in Mathematics. Springer-Verlag, 2006.

Digital Library

[4]

F. Boulier, F. Lemaire, and M. Moreno Maza. Well known theorems on triangular systems and the D5 principle. In Proc. of Transgressive Computing 2006, Granada, Spain, 2006.

[5]

C. W. Brown. Improved projection for cylindrical algebraic decomposition. J. Symb. Comput., 32(5):447--465, 2001.

Digital Library

[6]

C. W. Brown. Simple cad construction and its applications. J. Symb. Comput., 31(5):521--547, 2001.

Digital Library

[7]

C. W. Brown and J. H. Davenport. The complexity of quantifier elimination and cylinrical algebraic decomposition. In Proc. ISSAC'07, pages 54--60, 2007.

Digital Library

[8]

B. Caviness and J. Johnson, editors. Quantifier Elimination and Cylindical Algebraic Decomposition, Texts and Mongraphs in Symbolic Computation. Springer, 1998.

[9]

C. Chen, O. Golubitsky, F. Lemaire, M. Moreno Maza, and W. Pan. Comprehensive Triangular Decomposition, volume 4770 of LNCS, pages 73--101. Springer, 2007.

Digital Library

[10]

J. S. Cheng, X. S. Gao, and C. K. Yap. Complete numerical isolation of real zeros in zero-dimensional triangular systems. In Proc. ISSAC'07, pages 92--99, 2007.

Digital Library

[11]

G. E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Springer Lecture Notes in Computer Science, 33:515--532, 1975.

Digital Library

[12]

G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition. Journal of Symbolic Computation, 12(3):299--328, 1991.

Digital Library

[13]

G. E. Collins, J. R. Johnson, and W. Krandick. Interval arithmetic in cylindrical algebraic decomposition. J. Symb. Comput., 34(2):145--157, 2002.

Digital Library

[14]

X. Dahan, M. Moreno Maza, &#201;. Schost, W. Wu, and Y. Xie. Lifting techniques for triangular decompositions. In ISSAC'05, pages 108--115, 2005.

Digital Library

[15]

A. Dolzmann, A. Seidl, and T. Sturm. Efficient projection orders for cad. In Proc. ISSAC '04, pages 111--118. ACM, 2004.

Digital Library

[16]

A. Dolzmann, T. Sturm, and V. Weispfenning. Real quantifier elimination in practice. In Algorithmic Algebra and Number Theory, pages 221--247, 1998.

[17]

H. Hong. An improvement of the projection operator in cylindrical algebraic decomposition. In Proc. ISSAC '90, pages 261--264. ACM, 1990.

Digital Library

[18]

H. Hong. Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination. In ISSAC '92, pages 177--188. ACM, 1992.

Digital Library

[19]

&#201;. Hubert. Notes on triangular sets and triangulation-decomposition algorithms. I. Polynomial systems. In Symbolic and numerical scientific computation (Hagenberg, 2001), volume 2630 of LNCS, pages 1--39. Springer, 2003.

Digital Library

[20]

F. Lemaire, M. Moreno Maza, and Y. Xie. The RegularChains library. In Ilias S. Kotsireas, editor, Maple Conference 2005, pages 355--368, 2005.

[21]

X. Li, M. Moreno Maza, and W. Pan. Computations modulo regular chains. In Proc. ISSAC'09, ACM, 2009.

Digital Library

[22]

X. Li, M. Moreno Maza, R. Rasheed, and &#201;. Schost. The Modpn library: Bringing fast polynomial arithmetic into Maple. In Proc. MICA'08, 2008.

[23]

X. Li, M. Moreno Maza, and &#201;. Schost. Fast arithmetic for triangular sets: From theory to practice. In Proc. ISSAC'07, pages 269--276. ACM, 2007.

Digital Library

[24]

S. McCallum. An improved projection operation for cylindrical algebraic decomposition of 3-dimensional space. J. Symb. Comput., 5(1-2):141--161, 1988.

Digital Library

[25]

S. McCallum. Solving polynomial strict inequalities using cylindrical algebraic decomposition. The Computer Journal, 36(5):432--438, 1993.

[26]

M. Moreno Maza. On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK, 1999. Presented at the MEGA-2000 Conference, Bath, England.

[27]

A. Strzebo'nski. Solving systems of strict polynomial inequalities. J. Symb. Comput., 29(3):471--480, 2000.

Digital Library

[28]

D. M. Wang. Elimination Methods. Springer, New York, 2000.

[29]

W. T. Wu. A zero structure theorem for polynomial equations solving. MM Research Preprints, 1:2--12, 1987.

[30]

B. Xia and L. Yang. An algorithm for isolating the real solutions of semi-algebraic systems. J. Symb. Comput., 34(5):461--477, 2002.

Digital Library

[31]

B. Xia and T. Zhang. Real solution isolation using interval arithmetic. Comput. Math. Appl., 52(6-7):853--860, 2006.

Digital Library

[32]

L. Yang, X. Hou, and B. Xia. A complete algorithm for automated discovering of a class of inequality--type theorems. Science in China, Series F, 44(6):33--49, 2001.

Cited By

Pickering Ldel Río Almajano TEngland MCohen K(2024)Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decompositionJournal of Symbolic Computation10.1016/j.jsc.2023.102276123(102276)Online publication date: Jul-2024
https://doi.org/10.1016/j.jsc.2023.102276
Davenport JNair ASankaran GUncu A(2023)Lazard-style CAD and Equational ConstraintsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597090(218-226)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597090
Bradford RDavenport JEngland MSadeghimanesh AUncu A(2022)The DEWCAD projectACM Communications in Computer Algebra10.1145/3511528.351153855:3(107-111)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3511528.3511538
Show More Cited By

Index Terms

Computing cylindrical algebraic decomposition via triangular decomposition
1. Mathematics of computing
  1. Mathematical software

Recommendations

Triangular decomposition of semi-algebraic systems
ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic ...
Algorithms for computing triangular decompositions of polynomial systems
ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a weakened notion of a polynomial GCD modulo a regular chain, which permits to greatly ...
Quantifier elimination by cylindrical algebraic decomposition based on regular chains
ISSAC '14: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

A quantifier elimination algorithm by cylindrical algebraic decomposition based on regular chains is presented. The main idea is to refine a complex cylindrical tree until the signs of polynomials appearing in the tree are sufficient to distinguish the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 28 July 2009

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ISSAC '09

Sponsor:

ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

62
Total Citations
View Citations
204
Total Downloads

Downloads (Last 12 months)8
Downloads (Last 6 weeks)2

Reflects downloads up to 17 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Pickering Ldel Río Almajano TEngland MCohen K(2024)Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decompositionJournal of Symbolic Computation10.1016/j.jsc.2023.102276123(102276)Online publication date: Jul-2024
https://doi.org/10.1016/j.jsc.2023.102276
Davenport JNair ASankaran GUncu A(2023)Lazard-style CAD and Equational ConstraintsProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597090(218-226)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597090
Bradford RDavenport JEngland MSadeghimanesh AUncu A(2022)The DEWCAD projectACM Communications in Computer Algebra10.1145/3511528.351153855:3(107-111)Online publication date: 12-Jan-2022
https://dl.acm.org/doi/10.1145/3511528.3511538
del Río TEngland M(2022)New Heuristic to Choose a Cylindrical Algebraic Decomposition Variable Ordering Motivated by Complexity AnalysisComputer Algebra in Scientific Computing10.1007/978-3-031-14788-3_17(300-317)Online publication date: 11-Aug-2022
https://doi.org/10.1007/978-3-031-14788-3_17
Kremer GBrandt J(2021)Implementing arithmetic over algebraic numbers A tutorial for Lazard's lifting scheme in CAD2021 23rd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC54541.2021.00013(4-10)Online publication date: Dec-2021
https://doi.org/10.1109/SYNASC54541.2021.00013
England MBradford RDavenport J(2020)Cylindrical algebraic decomposition with equational constraintsJournal of Symbolic Computation10.1016/j.jsc.2019.07.019100:C(38-71)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.1016/j.jsc.2019.07.019
Florescu DEngland M(2020)A Machine Learning Based Software Pipeline to Pick the Variable Ordering for Algorithms with Polynomial InputsMathematical Software – ICMS 202010.1007/978-3-030-52200-1_30(302-311)Online publication date: 8-Jul-2020
https://doi.org/10.1007/978-3-030-52200-1_30
Chen C(2020)Chordality Preserving Incremental Triangular Decomposition and Its ImplementationMathematical Software – ICMS 202010.1007/978-3-030-52200-1_3(27-36)Online publication date: 8-Jul-2020
https://doi.org/10.1007/978-3-030-52200-1_3
Chen CZhu ZChi H(2020)Variable Ordering Selection for Cylindrical Algebraic Decomposition with Artificial Neural NetworksMathematical Software – ICMS 202010.1007/978-3-030-52200-1_28(281-291)Online publication date: 8-Jul-2020
https://doi.org/10.1007/978-3-030-52200-1_28
Florescu DEngland M(2020)Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output CorrectnessMathematical Aspects of Computer and Information Sciences10.1007/978-3-030-43120-4_27(341-356)Online publication date: 18-Mar-2020
https://doi.org/10.1007/978-3-030-43120-4_27
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten