System theory for numerical analysis

System theory for numerical analysis has recently become a focus of research. In this paper we regard dynamics of Newton’s method as a nonlinear feedback system and derive convergence conditions, based on the internal model principle and systems of Lur’e type. We then focus our attention on the analysis of the region of absolute stability of Runge-Kutta type methods. We derive a linear matrix inequality condition which characterizes a relationship between Runge-Kutta coefficients and the corresponding stability region. We also propose a new optimization procedure for designing a Runge-Kutta method based on this characterization. Copyright c ©2005 IFAC