The \(A\)-stability of methods with Padé and generalized Padé stability functions
This paper concerns some questions on Padé and generalized Padé approximations to the exponential function connected to the stability of numerical methods. Firstly, an alternative proof to Ehle's result that the diagonal and the first two subdiagonals are the only Padé table approximations that are \(A\)-stable is given. Next the author comes back to generalized Padé approximations, which in the quadratic case are given by polynomials in two variables \( Q(w,z)= P(z) w^2 + Q(z) w + R(z)\) where \( P, Q, R \) are polynomials with degrees \(k,l,m\), respectively, and satisfy \( Q(e^z,z) = C z^{p+1} + \ldots \) and the order \(p\) is \( p=k+l+m+1\). An extension of Ehle's results to generalized Padé approximations (Butcher-Chipman) is the conjecture that \( 2 k \leq p+2\). Here the author, by introducing the so called order arrows that are lines of steepest ascent and descent from the origin of \( R(z) e^{-z}\) for a Padé approximation \(R(z)\), proves the above conjecture in some particular cases.
generalized Padé approximations to the exponential
scientific article; zbMATH DE number 1879358