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Numerical Algorithms

, Volume 69, Issue 4, pp 785–794 | Cite as

The three-term recursion for Chebyshev polynomials is mixed forward-backward stable

  • Alicja SmoktunowiczEmail author
  • Agata Smoktunowicz
  • Ewa Pawelec
Open Access
Original Paper

Abstract

This paper provides an error analysis of the three-term recurrence relation (TTRR) T n+1(x)=2x T n (x)−T n−1(x) for the evaluation of the Chebyshev polynomial of the first kind T N (x) in the interval [−1,1]. We prove that the computed value of T N (x) from this recurrence is very close to the exact value of the Chebyshev polynomial T N of a slightly perturbed value of x. The lower and upper bounds for the function \(C_{N}(x)= |T_{N}(x)| + |x T_{N}^{\prime }(x)|\) are also derived. Numerical examples that illustrate our theoretical results are given.

Keywords

Chebyshev polynomials Error analysis Roots of polynomials 

Mathematics Subject Classifications (2010)

65G50 65D20 65L70 

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Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Alicja Smoktunowicz
    • 1
    Email author
  • Agata Smoktunowicz
    • 2
  • Ewa Pawelec
    • 1
  1. 1.Faculty of Mathematics and Information ScienceUniversity of TechnologyWarsawPoland
  2. 2.School of MathematicsUniversity of EdinburghEdinburgh, ScotlandUK

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