Rational Approximation and Interpolation pp 466-476 | Cite as

# Exponential fitting of restricted rational approximations to the exponential function

Numerical Methods

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## Abstract

Let R_{n/m}(z, γ)=P_{n}(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 z_{1}. It is shown that there are exactly min(n+1, m) different positive values of γ with this property.

## Keywords

Rational Approximation Laguerre Polynomial Exponential Fitting Real Polis Pade Approximation
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## References

- [1]Burrage, K., "A special family of Runga-Kutta methods for solving stiff differential equations", BIT 18 (1978), pp. 22–41.MathSciNetCrossRefzbMATHGoogle Scholar
- [2]Euler, L., Opera Omnia, Series Prima, Vol. 11, Leipzig and Berlin, 1913.Google Scholar
- [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]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]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]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]_____, "One-step methods of Hermite type for numerical integration of stiff systems", BIT 14 (1974), pp. 63–77.MathSciNetCrossRefzbMATHGoogle Scholar
- [8]_____, "Restricted Padé-approximations to the exponential function", SIAM J. Numer. Anal. 15 (1978), pp. 1008–1029.MathSciNetCrossRefzbMATHGoogle Scholar
- [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]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]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]Wanner, G., Hairer, E., and Norsett, S.P., "Order stars and stability theorems", BIT 18 (1978), pp. 475–489.MathSciNetCrossRefzbMATHGoogle Scholar

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