Abstract
This paper clears up to the following three conjectures:
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1.
The conjecture of Ehle [1] on theA-acceptability of Padé approximations toe z, which is true;
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2.
The conjecture of Nørsett [5] on the zeros of the “E-polynomial”, which is false;
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3.
The conjecture of Daniel and Moore [2] on the highest attainable order of certainA-stable multistep methods, which is true, generalizing the well-known Theorem of Dahlquist.
We further give necessary as well as sufficient conditions forA-stable (acceptable) rational approximations, bounds for the highest order of “restricted” Padé approximations and prove the non-existence ofA-acceptable restricted Padé approximations of order greater than 6.
The method of proof, just looking at “order stars” and counting their “fingers”, is very natural and geometric and never uses very complicated formulas.
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References
B. L. Ehle,On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems, Thesis 1969, CSRR2010, Department of Computer Science, University of Waterloo, Ontario, Canada, conjecture 3.1. Also see a paper of similar title in SIAM J. Math. Anal. 4 (1973), 671–680.
J. W. Daniel and R. E. Moore,Computation and theory in ordinary differential equations, Freeman and Co., 1970, page 80.
Y. Genin,An algebraic approach to A-stable linear multistep-multiderivative integration formulas, BIT 14 (1974), 382–406.
R. Jeltsch,Note on A-stability of multistep-multiderivative methods, BIT 16 (1976), 74–78.
S. P. Nørsett,C-polynomials for rational approximations to the exponential function, Numer. Math. 25 (1975), 39–56.
S. P. Nørsett and A. Wolfbrandt,Attainable order of rational approximations to the exponential function with only real poles, BIT 17 (1977), 200–208.
W. B. Rubin,A-stability and composite multistep methods, Ph. D. Thesis, EE Dept., Syracuse University, New York 1973.
H. J. Stetter,Analysis of discretization methods in ordinary differential equations, Springer-Verlag 1973.
S. P. Nørsett,Restricted Padé approximations to the exponential function, to appear in SIAM J. Numer. Anal. 1978.
id., similar title, Math. and Computation, No. 474, Trondheim.
S. P. Nørsett,One step methods of Hermite type for numerical integration of stiff systems, BIT 14 (1974), 63–77.
M. Crouzeix and F. Ruamps,On rational approximations to the exponential, R.A.I.R.O. Analyse Numérique 11, no 3 (1977), 241–243.
G. Szegö,Orthogonal polynomials, AMS 1939.
Behnke-Sommer,Theorie der analytischen Funktionen, Springer 1962.
A. Wolfbrandt,A note on a recent result of rational approximations to the exponential function, BIT 17 (1977), 367–368.
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Wanner, G., Hairer, E. & Nørsett, S.P. Order stars and stability theorems. BIT 18, 475–489 (1978). https://doi.org/10.1007/BF01932026
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DOI: https://doi.org/10.1007/BF01932026