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Mathematics of Computation

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Multiple zeros of nonlinear systems


Authors: Barry H. Dayton, Tien-Yien Li and Zhonggang Zeng
Journal: Math. Comp. 80 (2011), 2143-2168
MSC (2010): Primary 65H10; Secondary 68W30
DOI: https://doi.org/10.1090/S0025-5718-2011-02462-2
Published electronically: February 3, 2011
MathSciNet review: 2813352
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Abstract: As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity.

Furthermore, this paper elaborates and proves some fundamental properties of the multiplicity, including local finiteness, consistency, perturbation invariance, and depth-deflatability.

As a justification of this formulation, the multiplicity is proved to be consistent with the multiplicity defined in algebraic geometry for the special case of polynomial systems.

The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.


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Additional Information

Barry H. Dayton
Affiliation: Department of Mathematics, Northeastern Illinois University, Chicago, Illinois 60625
Email: bhdayton@neiu.edu

Tien-Yien Li
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: li@math.msu.edu

Zhonggang Zeng
Affiliation: Department of Mathematics, Northeastern Illinois University, Chicago, Illinois 60625
Email: zzeng@neiu.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02462-2
Received by editor(s): September 26, 2009
Received by editor(s) in revised form: June 24, 2010
Published electronically: February 3, 2011
Additional Notes: The second author’s research was supported in part by NSF under Grant DMS-0811172.
The third author’s research was supported in part by NSF under Grant DMS-0715127.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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