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The Nearest Real Polynomial with a Real Multiple Zero in a Given Real Interval

  • Hiroshi Sekigawa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

Abstract

Given f ∈ ℝ[x] and a closed real interval I, we provide a rigorous method for finding a nearest polynomial with a real multiple zero in I, that is, \(\tilde{f}\in\mathbb{R}[x]\) such that \(\tilde{f}\) has a multiple zero in I and \(\|f - \tilde{f}\|_\infty\), the infinity norm of the vector of coefficients of Open image in new window , is minimal. First, we prove that if a nearest polynomial Open image in new window exists, there is a nearest polynomial \(\tilde{g}\in\mathbb{R}[x]\) such that the absolute value of every coefficient of \(f-\tilde{g}\) is \(\|f - \tilde{f}\|_\infty\) with at most one exceptional coefficient. Using this property, we construct h ∈ ℝ[x] such that a zero of h is a real multiple zero α ∈ I of \(\tilde{g}\). Furthermore, we give a rational function whose value at α is \(\|f - \tilde{f}\|_\infty\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hiroshi Sekigawa
    • 1
  1. 1.NTT Communication Science Laboratories, Nippon Telegraph and Telephone CorporationAtsugi-shiJapan

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