The examination of nonlinear stability and solvability of the algebraic equations for the implicit Taylor series method (Q1294500)

Implicit Taylor series methods -- also called Obreschkoff or HOP methods -- have received much attention during the last years (mainly due to efficient automatic differentiation techniques). This article shows that, even when the method is A-stable, it may not be suited for the solution of stiff problems. More precisely, linear problems \(y'=g(x)y\) with smooth \(g(x)\) satisfying \(g(x)\leq 0\) are constructed in such a way that the numerical solution of the method does not exist for all stepsizes \(h>0\).