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


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.


Chebyshev polynomials Error analysis Roots of polynomials 

Mathematics Subject Classifications (2010)

65G50 65D20 65L70 


  1. 1.
    Bakhvalov, N.S.: The stable calculation of polynomial values. J. Comp. Math. Math. Phys. 11, 1568–1574 (1971)Google Scholar
  2. 2.
    Barrio, R.: Stability of parallel algorithms to evaluate Chebyshev series. Comput. Math. Appl. 41, 1365–1377 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barrio, R.: Rounding error bounds for the Clenshaw and Forsythe algorithms for the evaluation of orthogonal series. J. Comput. Appl. Math. 138, 185–204 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barrio, R.: A unified rounding error bound for polynomial evaluation. Adv. Comput. Math. 19(4), 385–399 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barrio, R., Jiang, H., Serrano, S.: A general condition number for polynomials. SIAM J. Numer. Anal. 51(2), 1280–1294 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berrut, J.-P., Trefethen, L. N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deuflhard, P.: On algorithm for the summation of certain special functions. Computing 17, 37–48 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Driscoll, T.A., Hale, N, Trefethen, L. N. (eds.): Chebfun Guide. Pafnuty Publications, Oxford (2014)Google Scholar
  9. 9.
    Elliott, D.: Error analysis of an algorithm for summing certain finite series. J. Austral. Math. Soc. 8, 213–221 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation (Numerical Mathematics and Scientific Computation). Oxford University Press (2004)Google Scholar
  11. 11.
    Gentleman, W.M.: An error analysis of Goertzel’s (Watt’s) method for computing Fourier coefficients. Comput. J. 12, 160–165 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)zbMATHGoogle Scholar
  13. 13.
    Koepf, W.: Efficient computation of Chebyshev polynomials. In: Wester, M.J. (ed.) Computer Algebra Systems: A Practical Guide, pp. 79–99. Wiley, New York (1999)Google Scholar
  14. 14.
    Paszkowski, S.: Numerical Applications of Chebyshev Polynomials. Warsaw, (in Polish) (1975)Google Scholar
  15. 15.
    Smoktunowicz, A.: Backward stability of Clenshaw’s algorithm. BIT 42(3), 600–610 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Szegö, G.: Orthogonal Polynomials, rev. ed. New York (1959)Google Scholar
  17. 17.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problems. Oxford University Press (1965)Google Scholar
  18. 18.
    Woźniakowski, H.: Numerical stability for solving nonlinear equations. Numer. Math. 27, 373–390 (1977) 600–610Google Scholar

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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