Skip to main content
SpringerLink
Log in

Extended numerical computations on the “1/9” conjecture in rational approximation theory

  • Numerical Methods
  • Conference paper
  • First Online:
Rational Approximation and Interpolation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1105))

Abstract

The behavior of the constants λn,n(e −x), denoting the errors of best uniform approximation to e −z on the interval [0,+∞) by real rational functions having numerator and denominator polynomials of degree at most n, has generated much recent interest in the approximation theory literature. Based on high-precision calculations, we present here the table of constants {λn,n (e −x)} 30n =0, rounded to forty significant digits, and we discuss their significance to related conjectures in this area.

Research supported by the Air Force Office of Scientific Research and by the Department of Energy.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.-P. Blatt and D. Braess, “Zur rationalen Approximation von e −x auf [0, ∞)”, J. Approxiation Theory 30(1980), 169–172.

    Article  MathSciNet  MATH  Google Scholar 

  2. Richard Brent, “A FORTRAN multiple-precision arithmetic package”, Assoc. Comput. Mach. Trans. Math. Software 4(1978), 57–70.

    Article  Google Scholar 

  3. C. Brezinski, Algorithmes d'Accélération de la Convergence, Éditions Technip, Paris, 1978.

    MATH  Google Scholar 

  4. W. J. Cody, G. Meinardus, and R. S. Varga, “Chebyshev rational approximation to e −x in [0, +∞) and applications to heat-conduction problems”, J. Approximation Theory 2(1969), 50–65.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, New York, 1967.

    Book  MATH  Google Scholar 

  6. G. Németh, “Notes on generalized Padé approximation”, in Approximation and Function Spaces (Z. Ciesielski, ed.), pp. 484–508, North-Holland Publishing Co., Amsterdam, 1981.

    Google Scholar 

  7. D. J. Newman, “Rational approximation to e −x”, J. Approximation Theory 10(1974), 301–303.

    Article  MathSciNet  MATH  Google Scholar 

  8. H.-U. Opitz and K. Scherer, “On the rational approximation of e −x on [0, ∞)”, Constructive Approximation (to appear).

    Google Scholar 

  9. Q. I. Rahman and G. Schmeisser, “Rational approximation to e −x”, J. Approximation Theory 23(1978), 146–154.

    Article  MathSciNet  MATH  Google Scholar 

  10. Q. I. Rahman and G. Schmeisser, “Rational approximation to e −x. II”, Trans. Amer. Math. Soc. 235(1978), 395–402.

    MathSciNet  MATH  Google Scholar 

  11. E. B. Saff and R. S. Varga, “Some open questions concerning polynomials and rational functions”, in Padé and Rational Approximation (E. B. Saff and R. S. Varga, eds.), pp. 483–488, Academic Press, Inc., New York, 1977.

    Chapter  Google Scholar 

  12. A. Schönhage, “Zur rationalen Approximerbarkeit von e −x über [0, ∞), J. Approximation Theory 7(1973), 395–398.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Schönhage, “Rational approximation to e −x and related L 2-problems”, SIAM J. Numer. Anal. 19(1982), 1067–1082.

    Article  MathSciNet  Google Scholar 

  14. L. N. Trefethen and M. H. Gutknecht, “The Carathéodory-Fejér method for real rational approximation”, SIAM J. Numer. Anal. 20(1983), 420–436.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Computational Mathematies, Kent State University, 44242, Kent, Ohio, USA

    A. J. Carpenter, A. Ruttan & R. S. Varga

Authors
  1. A. J. Carpenter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Ruttan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. S. Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Peter Russell Graves-Morris Edward B. Saff Richard S. Varga

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Carpenter, A.J., Ruttan, A., Varga, R.S. (1984). Extended numerical computations on the “1/9” conjecture in rational approximation theory. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072427

Download citation

  • DOI: https://doi.org/10.1007/BFb0072427

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation