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Tighter bounds of errors of numerical roots

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Abstract

LetP(z) be a monic univariate polynomial overC, of degreen and having roots ζ1,..., ζ n . Given approximate rootsz 1,...,z n, with ζ i ≃z i (i = 1,...,n), we derive a very tight upper bound of ¦ζ i −z i¦, by assuming that ζ i has no close root. The bound formula has a similarity with Smale’s and Smith’s formulas. We also derive a lower bound of ¦ζ i −z i¦ and a lower bound of min{ζ j −z i¦ ¦j ≠i}.

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba-shi, 305, Ibaraki, Japan

    Tateaki Sasaki

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  1. Tateaki Sasaki
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Correspondence to Tateaki Sasaki.

Additional information

Work supported in part by Japanese Ministry of Education, Science and Culture under Grants 15300002.

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Sasaki, T. Tighter bounds of errors of numerical roots. Japan J. Indust. Appl. Math. 24, 219–226 (2007). https://doi.org/10.1007/BF03167534

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  • DOI: https://doi.org/10.1007/BF03167534

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