Skip to main content
SpringerLink
Log in

Geometric convergence to e−z by rational functions with real poles

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

In this paper, we show that there exists a sequence of rational functions of the formR n(z)=pn−1(z)/(1+z/n)n,n=1, 2, ..., with degp n−1≦n−1, which converges geometrically toe −z in the uniform norm on [0, +∞), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cody, W. J., Meinardus, G., Varga, R. S.: Chebyshev rational approximations toe −x on [0, +∞) and applications to heat-conduction problems. J. Approximation Theory2, 50–65 (1969)

    Google Scholar 

  2. Ehle, B. L.:A-stable methods and Padé approximation to the exponential. SIAM J. Math. Anal.,4, 671–680 (1973)

    Google Scholar 

  3. Newman, D. J.: Rational approximation toe −x. J. Approximation Theory10, 301–303 (1974)

    Google Scholar 

  4. Saff, E. B., Varga, R. S.: Angular overconvergence for rational functions converging geometrically on [0, +∞), to appear in the Proceedings of the Approximation Theory Conference. Calgary, Alberta, Canada, August 11–13, 1975

  5. Saff, E. B., Varga, R. S., and Ni, W.-C.: Geometric convergence of rational approximations toe −x in infinite sectors. Numer. Math. (to appear).

  6. Schönhage, A.: Zur rationalen Approximierbarkeit vone −x über [0, +∞). J. Approximation Theory7, 395–398 (1973)

    Google Scholar 

  7. Szegö, G.: Orthogonal Polynomials, Colloq. Publication Vol. 23, Amer. Math. Soc., Providence, R. I., 3rd ed., 1967

    Google Scholar 

  8. Underhill, C., Wragg, A.: Convergence properties of Padé approximants to exp (z) and their derivatives. J. Inst. Math. Appl.11, 361–367 (1973)

    Google Scholar 

  9. Walsh, J. L.: Interpolation and Approximation by Rational Functions in the Complex Domain, Colloq. Publication Vol. 20, Amer. Math. Soc., Providence, R. I., 5th ed., 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of South Florida, 33620, Tampa, FL, U. S. A.

    E. B. Saff

  2. Mathematisches Institut, Universität Tübingen, D-7400, Tübingen 1, Bundesrepublik Deutschland

    A. Schönhage

  3. Dept. of Mathematics, Kent State University, 44242, Kent, OH, U. S. A.

    R. S. Varga

Authors
  1. E. B. Saff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Schönhage
    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

Additional information

Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research Council.

Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Saff, E.B., Schönhage, A. & Varga, R.S. Geometric convergence to e−z by rational functions with real poles. Numer. Math. 25, 307–322 (1975). https://doi.org/10.1007/BF01399420

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01399420

Keywords

Search

Navigation