Abstract
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.
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A1244.
Visiting Professor from NTH, Trondheim, Norway.
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References
Burrage, K., "A special family of Runga-Kutta methods for solving stiff differential equations", BIT 18 (1978), pp. 22–41.
Euler, L., Opera Omnia, Series Prima, Vol. 11, Leipzig and Berlin, 1913.
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.
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.
Lawson, J.D., "Generalized Runga-Kutta processes for stable systems with large Lipschitz constants", SIAM J. Numer. Anal. 4 (1967), pp. 372–380.
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.
_____, "One-step methods of Hermite type for numerical integration of stiff systems", BIT 14 (1974), pp. 63–77.
_____, "Restricted Padé-approximations to the exponential function", SIAM J. Numer. Anal. 15 (1978), pp. 1008–1029.
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.
Swayne, D.A., "Computation of Rational Functions with Matrix Argument with Application to Initial-Value Problems", Ph.D. Thesis, University of Waterloo, 1975.
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.
Wanner, G., Hairer, E., and Norsett, S.P., "Order stars and stability theorems", BIT 18 (1978), pp. 475–489.
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Norsett, S.P., Trickett, S.R. (1984). Exponential fitting of restricted rational approximations to the exponential function. 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/BFb0072433
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DOI: https://doi.org/10.1007/BFb0072433
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