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Computational Methods in Applied Mathematics

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Volume 19, Issue 1


Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials

Alwin Stegeman
  • Group Science, Engineering and Technology, KU Leuven – Kulak, E. Sabbelaan 53, 8500 Kortrijk; and Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium
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/ Lieven De LathauwerORCID iD: https://orcid.org/0000-0001-5562-5014
Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/cmam-2018-0025


The problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.

Keywords: Rayleigh Quotient Iteration; Univariate Polynomials; Common Roots; Noisy Polynomials, Numerical Polynomial Algebra

MSC 2010: 13P15; 15A18; 65F15


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About the article

Received: 2017-08-31

Revised: 2018-03-06

Accepted: 2018-05-02

Published Online: 2018-07-12

Published in Print: 2019-01-01

Funding Source: Fonds Wetenschappelijk Onderzoek

Award identifier / Grant number: G0F6718N

Research supported by (1) Research Council KU Leuven: C1 project c16/15/059-nD and (2) FWO: EOS project G0F6718N (SELMA).

Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 147–163, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0025.

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