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


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.


  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Barrio, R.: A unified rounding error bound for polynomial evaluation. Adv. Comput. Math. 19(4), 385–399 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Barrio, R., Jiang, H., Serrano, S.: A general condition number for polynomials. SIAM J. Numer. Anal. 51(2), 1280–1294 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Berrut, J.-P., Trefethen, L. N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Deuflhard, P.: On algorithm for the summation of certain special functions. Computing 17, 37–48 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Driscoll, T.A., Hale, N, Trefethen, L. N. (eds.): Chebfun Guide. Pafnuty Publications, Oxford (2014)

  9. 9.

    Elliott, D.: Error analysis of an algorithm for summing certain finite series. J. Austral. Math. Soc. 8, 213–221 (1968)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation (Numerical Mathematics and Scientific Computation). Oxford University Press (2004)

  11. 11.

    Gentleman, W.M.: An error analysis of Goertzel’s (Watt’s) method for computing Fourier coefficients. Comput. J. 12, 160–165 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)

    Google 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)

  14. 14.

    Paszkowski, S.: Numerical Applications of Chebyshev Polynomials. Warsaw, (in Polish) (1975)

  15. 15.

    Smoktunowicz, A.: Backward stability of Clenshaw’s algorithm. BIT 42(3), 600–610 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Szegö, G.: Orthogonal Polynomials, rev. ed. New York (1959)

  17. 17.

    Wilkinson, J.H.: The Algebraic Eigenvalue Problems. Oxford University Press (1965)

  18. 18.

    Woźniakowski, H.: Numerical stability for solving nonlinear equations. Numer. Math. 27, 373–390 (1977) 600–610

Download references

Author information



Corresponding author

Correspondence to Alicja Smoktunowicz.

Additional information

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

Rights and permissions

Open Access This 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Smoktunowicz, A., Smoktunowicz, A. & Pawelec, E. The three-term recursion for Chebyshev polynomials is mixed forward-backward stable. Numer Algor 69, 785–794 (2015).

Download citation


  • Chebyshev polynomials
  • Error analysis
  • Roots of polynomials

Mathematics Subject Classifications (2010)

  • 65G50
  • 65D20
  • 65L70