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Mathematics 2018, 6(9), 166; https://doi.org/10.3390/math6090166

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

1
College of Science, Civil Aviation University of China, Tianjin 300300, China
2
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Author to whom correspondence should be addressed.
Received: 12 July 2018 / Revised: 24 August 2018 / Accepted: 3 September 2018 / Published: 11 September 2018
(This article belongs to the Special Issue Computer Algebra in Scientific Computing)
Full-Text   |   PDF [293 KB, uploaded 11 September 2018]

Abstract

In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a simple real zero of the related square system with the interval methods, we assert that the certified zero is a local minimum of sum of squares of the input polynomials. If the value of sum of squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of isolated zeros of polynomial systems and their multiplicity structures. View Full-Text
Keywords: over-determined polynomial system; isolated zeros; minimum point; sum of squares; interval methods over-determined polynomial system; isolated zeros; minimum point; sum of squares; interval methods
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Dou , X.; Cheng , J.-S. A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems. Mathematics 2018, 6, 166.

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