Generalized Padé approximations to the exponential function

Abstract

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.