COMPLETE ALGEBRAIC CHARACTERIZATION OF A-STABLE RUNGE-KUTTA METHODS

Important stability concepts for Runge-Kutta methods are I-, A-, and B -stability. For these properties there exist very similar algebraic ch aracterizations. The characterization of B-stability is known for S-ir reducible methods. In this paper, an algebraic characterization of I-s tability and A-stability related to the coefficients of the method is deduced without any assumption on the Runge-Kutta methods. The corresp onding linear dynamic system and its transfer function is considered. The positive real lemma characterizes the passivity of the system or e quivalently the positive realness of the transfer function by the Lyap unov equation. Dropping the assumption of controllability and observab ility a generalization is possible using the Kalman canonical decompos ition. Interpreting the modified stability function of a Runge-Kutta m ethod as the transfer function, the positive real lemma yields a compl ete algebraic characterization of A-stable Runge-Kutta methods.