The \(A\)-stability of methods with Padé and generalized Padé stability functions
type: Item
instance of: scholarly article
title: The \(A\)-stability of methods with Padé and generalized Padé stability functions
zbMATH Open document ID: 1016.65057
DOI: 10.1023/A:1021155922184
published in: Numerical Algorithms
publication date: 2003-03-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1879358
zbMATH Keywords: generalized Padé approximations to the exponential
zbMATH Keywords: stability
zbMATH Keywords: order stars
zbMATH Keywords: order arrows
author: J. C. Butcher
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: a:1021155922184
OpenAlex ID: W2165320992
Recommended article: Q3345103
Recommended article: Order and stability of generalized Padé approximations
Recommended article: On the generalisation of Padé approximation approach for the construction of \(p\)-stable hybrid linear multistep methods
Recommended article: On the Stability and Instability of Padé Approximants
Recommended article: Q4309822
Recommended article: Q3528093
Recommended article: Generalized stability of discretization and its applications to numerical solutions of non-linear partial differential equations
Recommended article: Q4696085
Recommended article: Q3762781
Recommended article: Q4843437
description: scientific article; zbMATH DE number 1879358