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Numerical Algorithms

, Volume 2, Issue 2, pp 171–185 | Cite as

Some numerical results on best uniform rational approximation ofxα on [0,1]

  • Richard S. Varga
  • Amos J. Carpenter
Article

Abstract

Let α be a positive number, and letE n,n (xα;[0,1]) denote the error of best uniform rational approximation from π n,n toxα on the interval [0,1]. We rigorously determined the numbers {E n,n (xα;[0,1])} n =1/30 for six values of α in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of α, Richardson's extrapolation was applied to the products\(\{ e^{\pi \sqrt {4\alpha n} } E_{n,n} (x^\alpha ;[0,1])\} _{n = 1}^{30} \) to obtain estimates of
$$\lambda (\alpha ): = \mathop {\lim }\limits_{n \to \infty } e^{\pi \sqrt {4\alpha n} } E_{n,n} (x^\alpha ;[0,1]) (\alpha > 0).$$

These estimates give rise to two interesting new conjectures in the theory of rational approximation.

Subject classification

AMS (MOS) 41A20 41A50 

Keywords

Rational approximation best approximation Remez algorithm 

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Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1992

Authors and Affiliations

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

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