Backward error measures for roots of polynomials

Abstract

We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.

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Funding

Sascha Timme was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) Graduiertenkolleg Facets of Complexity (GRK 2434). Marc Van Barel was partially supported by the Research Council KU Leuven, C1-project (Numerical Linear Algebra and Polynomial Computations), and by the Fund for Scientific Research Flanders (Belgium), G.0828.14N (Multivariate polynomial and rational interpolation and approximation), and EOS Project No 30468160.

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Correspondence to Marc Van Barel.

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Telen, S., Timme, S. & Van Barel, M. Backward error measures for roots of polynomials. Numer Algor 87, 19–39 (2021). https://doi.org/10.1007/s11075-020-00956-z

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Keywords

  • Backward error
  • Roots of polynomials
  • Tropical backward error
  • Element-wise mixed backward error
  • Tropical roots.