Some numerical results on best uniform polynomial approximation of Xα on [0, 1]

  • Amos J. Carpenter
  • Richard S. Varga
Conference paper
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 1550)

Abstract

Let α be a positive number, and let E n (xα; [0, 1]) denote the error of best uniform approximation to xα by polynomials of degree at most n on the interval [0, 1]. Russian mathematician S. N. Bernstein established the existence of a nonnegative constant β(α) such that β(α):=limn→∞(2n)2αE n (xα;[0, 1]) (α>0).

In addition, Bernstein showed that πβ(α)<Γ(2α)|sin(πα)| (α>0), and that Γ(2α)|sin(πα)|(1−1/(2α−1))<πβ(α) (α>1/2), so that the asymptotic behavior of β(α) is thus known when α → ∞.

Still, the problem of trying to determine β(α) more precisely, for all α>0, is intriguing. To this end, we have rigorously determined the numbers lcub;E n (xα;[0, 1])rcub; n=1 40 for thirteen values of α, where these numbers were calculated with a precision of at least 200 significant digits. For each of these thirteen values of α. Richardson’s extrapolation was applied to the products lcub;(2n)2αE n (xα; [0, 1])rcub; n=1 40 to obtain estimates of β(α) to approximately 40 decimal places. Included are graphs of the points (α,β(α)) for the thirteen values of α that we considered.

Keywords

Russian Mathematician Asymptotic Series Decimal Digit Richardson Extrapolation Bernstein Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. N. Bernstein, Sur la meilleure approximation de |x| par des polynômes de degrés donnés, Acta Math. 37 (1913), pp. 1–57.CrossRefMATHGoogle Scholar
  2. [2]
    S. N. Bernstein, Sur la meillcure approximation de |x| p par des polynômes de degrés trés élevés, Bull. Acad. Sci. USSR, Cl. sci. math. nat. 2 (1938), pp. 181–190.MATHGoogle Scholar
  3. [3]
    S. N. Bernstein, Collected Works (Russian), Akad. Nauk SSSR, Moscow, Vol. II, 1954, pp. 262–272.Google Scholar
  4. [4]
    H.-P. Blatt and E. B. Saff, Behavior of zeros of polynomials of near best approximation, J. Approx. Theory 46 (1986), pp. 323–344.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    R. P. Brent, A FORTRAN multiple-precision arithmetic package, ACM Trans. Math. Soft. 4 (1978), pp. 57–70.CrossRefGoogle Scholar
  6. [6]
    C. Brezinski, Algorithms d'Accélération de la Convergence, Éditions Technip, Paris, 1978.MATHGoogle Scholar
  7. [7]
    A. J. Carpenter and R. S. Varga, Some numerical results on best uniform rational approximation of x α on [0, 1], to appear.Google Scholar
  8. [8]
    A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of rational approximation of analytic functions, Mat. Sbornik 134, (176) (1987), pp. 306–352. An English translation appears in Math. USSR Sbornik 62, 2 (1989), pp. 305–348.MATHGoogle Scholar
  9. [9]
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition Prepared by Alan Jeffrey, Academic Press, San Diego, 1979.Google Scholar
  10. [10]
    M. A. Jenkins, Algorithm 493, Zeros of a real polynomial, Collected Algorithms from ACM, 1975, 10 pp.Google Scholar
  11. [11]
    H. L. Loeb, Approximation by generalized rationals, SIAM J. on Numer. Anal. 3 (1966), pp. 34–55.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    E. L. Lusk and R. A. Overbeek, Use of monitors in FORTRAN: A tutotial on the barrier, self-scheduling do-loop, and askfor monitors, Parallel MIMD Computation: The HEP Supercomputer and Its Applications (J. S. Kowalik, ed.), pp. 367–411, The MIT Press, Cambridge, Mass., 1985.Google Scholar
  13. [13]
    G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, New York, 1967.CrossRefMATHGoogle Scholar
  14. [14]
    E. Ya. Remez, Sur le calcul effectiv des polynômes d'approximation de Tchebichef, C.R. Acad. Sci. Paris 199 (1934), pp. 337–340.Google Scholar
  15. [15]
    T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell Publishing Co., Waltham, Mass., 1969.MATHGoogle Scholar
  16. [16]
    H. Stahl, Best uniform rational approximation of |x| on [−1, 1], Mat. Sbornik (to appear).Google Scholar
  17. [17]
    R. S. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics 60, SIAM, Philadelphia, Penn., 1990.CrossRefMATHGoogle Scholar
  18. [18]
    R. S. Varga and A. J. Carpenter, On the Bernstein Conjecture in approximation theory, Constr. Approx. 1 (1985), pp. 333–348. A Russian translation appears in Mat. Sbornik 129, 171 (1986), pp. 535–548.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    R. S. Varga, A. Ruttan, and A. J. Carpenter, Numerical results on best uniform rational approximation of |x| on [−1, +1], Mat. Sbornik (to appear).Google Scholar

Copyright information

© The Euler International Mathematical Institute 1993

Authors and Affiliations

  • Amos J. Carpenter
    • 1
    • 2
  • Richard S. Varga
    • 1
    • 2
  1. 1.Dept. of Math. and Computer ScienceButler UniversityIndianapolis
  2. 2.Institute for Computational MathematicsKent State UniversityKent

Personalised recommendations