BIT Numerical Mathematics

, Volume 32, Issue 1, pp 118–130 | Cite as

Generalized Padé approximations to the exponential function

  • J. C. Butcher
  • F. H. Chipman
Part II Numerical Mathematics


The stability properties of the Padé rational approximations to the exponential function are of importance in determining the linear stability properties of several classes of Runge-Kutta methods. It is well known that the Padé approximationR n,m (z) =N n,m (z)/M n,m (z), whereN n,m (z) is of degreen andM n,m (z) is of degreem, is A-stable if and only if 0 ≤m − n ≤ 2, a result first conjectured by Ehle. In the study of the linear stability properties of the broader class of general linear methods one must generalize these rational approximations. In this paper we introduce a generalization of the Padé approximations to the exponential function and present a method of constructing these approximations for arbitrary order and degree. A generalization of the Ehle inequality is considered and, in the case of the quadratic Padé approximations, evidence is presented that suggests the inequality is both necessary and sufficient for A-stability. However, in the case of the cubic Padé approximations, the inequality is shown to be insufficient for A-stability. A generalization of the restricted Padé approximation, in which the denominator has a singlem-fold zero, is also introduced. A procedure for the construction of these restricted approximations is described, and results are presented on the A-stability of the restricted quadratic Padé approximations. Finally, to demonstrate the connection between a generalized Padé approximation and a general linear method, a specific general linear method is constructed with a stability region corresponding to a given quadratic Padé approximation.


Computational Mathematic Exponential Function General Linear Stability Region Linear Stability 
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]
    Axelsson, O.,A class of A-stable methods, BIT 9 (1969), 185–199.CrossRefGoogle Scholar
  2. [2]
    Burrage, K., and Chipman, F. H.,The stability properties of singly-implicit general linear methods, IMA J. Numer Anal., 5, (1985) 287–295.Google Scholar
  3. [3]
    Butcher, J. C.,The order of numerical methods for ordinary differential equations Math. Comp. 27 (1973), 107–117.Google Scholar
  4. [4]
    Butcher, J. C.,On A-stable implicit Runge-Kutta methods, BIT 17 (1977), 375–378.CrossRefGoogle Scholar
  5. [5]
    Butcher, J. C.,The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, J. Wiley and Sons, Chichester and New York, 1987.Google Scholar
  6. [6]
    Daniel, J. W. and Moore, R. E.,Computation and Theory in Ordinary Differential Equations, W. H. Freeman, San Francisco, 1970.Google Scholar
  7. [7]
    Ehle, B. L.,A-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal. 4 (1973), 671–680.CrossRefGoogle Scholar
  8. [8]
    Wanner, G., Hairer, E. and Nørsett, S. P.,Order stars and stability theorems, BIT 18 (1978), 475–489.CrossRefGoogle Scholar
  9. [9]
    Wright, K.,Some relationships between implicit Runge-Kutta, collocation and Lanczos τ methods, and their stability properties, BIT 10 (1970), 217–227.CrossRefGoogle Scholar

Copyright information

© BIT Foundations 1992

Authors and Affiliations

  • J. C. Butcher
    • 1
    • 2
  • F. H. Chipman
    • 1
    • 2
  1. 1.Dept. of Mathematics and StatisticsThe University of AucklandAucklandNew Zealand
  2. 2.Dept. of MathematicsAcadia UniversityWolfvilleCanada

Personalised recommendations