Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis

Juan Xu, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We describe a new algorithm Miranda for isolating the simple zeros of a function f : Rn → Rn within a box B0 ⊆ Rn. The function f and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is “effective” in the sense that our algorithmic description also specifies the numerical precision that is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelidis (MK) test, based on Miranda's Theorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm. We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). This methodology provides a systematic pathway for achieving effective subdivision algorithms in general.

Original languageEnglish (US)
Title of host publicationISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation
PublisherAssociation for Computing Machinery
Pages355-362
Number of pages8
ISBN (Electronic)9781450360845
DOIs
StatePublished - Jul 8 2019
Event44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019 - Beijing, China
Duration: Jul 15 2019Jul 18 2019

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Conference

Conference44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019
CountryChina
CityBeijing
Period7/15/197/18/19

Fingerprint

Subdivision Algorithm
Complexity Analysis
System of equations
Zero
Zero of a function
Analysis of Algorithms
Interval
Algorithm Analysis
Partial derivative
Subdivision
Predicate
Isolation
Pathway
Roots
Synthesis
Sufficient
Polynomial
Methodology
Theorem

Keywords

  • Certified Computation
  • Complexity Analysis
  • Effective Certified Algorithm
  • Miranda Theorem
  • Moore-Kioustelidis Test
  • Root Isolation
  • Subdivision Algorithms
  • System of Real Equations

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Xu, J., & Yap, C. (2019). Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. In ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation (pp. 355-362). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). Association for Computing Machinery. https://doi.org/10.1145/3326229.3326270

Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. / Xu, Juan; Yap, Chee.

ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery, 2019. p. 355-362 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Xu, J & Yap, C 2019, Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. in ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, Association for Computing Machinery, pp. 355-362, 44th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2019, Beijing, China, 7/15/19. https://doi.org/10.1145/3326229.3326270
Xu J, Yap C. Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. In ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery. 2019. p. 355-362. (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). https://doi.org/10.1145/3326229.3326270
Xu, Juan ; Yap, Chee. / Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. ISSAC 2019 - Proceedings of the 2019 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery, 2019. pp. 355-362 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).
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