Exponential fitting of restricted rational approximations to the exponential function

  • Syvert P. Norsett
  • Stewart R. Trickett
Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)


Let Rn/m(z, γ)=Pn(z; γ)/(1-γz)m be a restricted rational approximation to exp(z), zεℂ, of order n for all real γ. In this paper we discuss how γ can be used to obtain fitting at a real non-positive point z1. It is shown that there are exactly min(n+1, m) different positive values of γ with this property.


Rational Approximation Laguerre Polynomial Exponential Fitting Real Polis Pade Approximation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Burrage, K., "A special family of Runga-Kutta methods for solving stiff differential equations", BIT 18 (1978), pp. 22–41.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Euler, L., Opera Omnia, Series Prima, Vol. 11, Leipzig and Berlin, 1913.Google Scholar
  3. [3]
    Iserles, A., "Generalized order star theory,", Padé Approximation and its Applications, Amsterdam 1980 (ed. M.G. De Bruin and H. van Rossum), LNiM 888, Springer-Verlag, Berlin, 1981, pp. 228–238.CrossRefGoogle Scholar
  4. [4]
    Lau, T., "A Class of Approximations to the Exponential Function for the Numerical Solution of Stiff Differential Equations", Ph.D. Thesis, University of Waterloo, 1974.Google Scholar
  5. [5]
    Lawson, J.D., "Generalized Runga-Kutta processes for stable systems with large Lipschitz constants", SIAM J. Numer. Anal. 4 (1967), pp. 372–380.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Norsett, S.P., "An A-stable modification of the Adams-Bashforth methods", Conf. on the Numerical Solution of Differential Equations (ed. A. Dold and B. Eckmann), LNiM 109, Springer-Verlag, Berlin, 1969, pp. 214–219.CrossRefGoogle Scholar
  7. [7]
    _____, "One-step methods of Hermite type for numerical integration of stiff systems", BIT 14 (1974), pp. 63–77.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    _____, "Restricted Padé-approximations to the exponential function", SIAM J. Numer. Anal. 15 (1978), pp. 1008–1029.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Norsett, S.P., and Wolfbrandt, A., "Attainable order of rational approximations to the exponential function with only real poles", BIT 17 (1977), pp. 200–208.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Swayne, D.A., "Computation of Rational Functions with Matrix Argument with Application to Initial-Value Problems", Ph.D. Thesis, University of Waterloo, 1975.Google Scholar
  11. [11]
    Trickett, S.R., "Rational Approximations to the Exponential Function for the Numerical Solution of the Heat Conduction Problem", Master's Thesis, University of Waterloo, 1984.Google Scholar
  12. [12]
    Wanner, G., Hairer, E., and Norsett, S.P., "Order stars and stability theorems", BIT 18 (1978), pp. 475–489.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Syvert P. Norsett
    • 1
  • Stewart R. Trickett
    • 2
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations