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Application of Quantifier Elimination to Solotareff’s Approximation Problem

  • George E. Collins
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 121)

Abstract

Solotareff’s approximation problem is that of obtaining the best uniform approximation on the interval [−1,+1] of a real polynomial of degree n by one of degree n−2 or less. Without loss of generality we take the approximated polynomial to be x n + rx n 1,r ≥ 0. We treat r as a parameter and seek to compute the coefficients of the best approximations, for small fixed values of n, as piecewise algebraic functions of r. We succeed only for n ≤ 4, but also find that we can easily compute the coefficients for n = 5 for fixed values of r. The results serve to display the capabilities and limitations of our quantifier elimination program qepcad. We also prove that the coefficients of the best approximation, for any n, are continuous functions of r.

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References

  1. [1]
    N. I. Achieser. Theory of Approximation. Frederick Ungar Publishing Co., New York, 1956.Google Scholar
  2. [2]
    G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Lecture Notes In Computer Science, Vol. 33, pages 134–183, Springer-Verlag, New York, 1975.Google Scholar
  3. [3]
    G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12 (3): 299–328, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
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    G. E. Collins. Quantifier elimination by cylindrical algebraic decomposition - twenty years of progress. In Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer- Verlag, Vienna, 1995.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1996

Authors and Affiliations

  • George E. Collins
    • 1
  1. 1.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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