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

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.

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Correspondence to Alicja Smoktunowicz.

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The research of Agata Smoktunowicz was funded by ERC grant 320974.

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Smoktunowicz, A., Smoktunowicz, A. & Pawelec, E. The three-term recursion for Chebyshev polynomials is mixed forward-backward stable. Numer Algor 69, 785–794 (2015). https://doi.org/10.1007/s11075-014-9925-x

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Keywords

  • Chebyshev polynomials
  • Error analysis
  • Roots of polynomials

Mathematics Subject Classifications (2010)

  • 65G50
  • 65D20
  • 65L70