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On the A-stability of Padé type approximants of exp(z) whose denominators are \((1-z/n)^ n\) and the associated Runge-Kutta methods. (Spanish. English summary) Zbl 0571.41014

Differential equations and applications, Proc. 7th Congr., Granada/Spain 1984, 67-72 (1985).
[For the entire collection see Zbl 0552.00006.]
The authors consider the Padé approximations of type (n-1/n) and (n/n) for \(e^ z\) with denominators \((1-z/n)^ n\), and develop a necessary condition for A-stability which makes use of Laguerre polynomials. They establish, for example, that for \(n\leq 14\) the only A-stable Padé approximations of type (n-1/n) are those for \(n=1,2,3\) and 4. For Padé approximations of type (n/n), those for \(n=1,2\), and 3 are the only A- stable ones satisfying \(n\leq 30\). B-stability is also discussed. The results here extend results of J. Van Iseghen [Numer. Math. 43, 283-292 (1983; Zbl 0514.65006)]. The present paper contains no proofs. The proofs appear in a Technical Report of the University of Zaragoza.
Reviewer: P.Lappan

MSC:

41A21 Padé approximation
41A20 Approximation by rational functions