An algorithm to compute sets enclosingM-numerical ranges with applications in numerical analysis

Abstract

One of the possible generalizations of the classical numerical range studied in the literature is the so-called M-numerical range. In this paper we present an algorithm which can be used to construct a polygon V enclosing the M-numerical range of a given square matrix A. The algorithm can be executed such that V encloses the M-numerical range of A tightly.The concept of M-numerical ranges can be used to investigate stability properties of numerical methods for approximating the solution to linear initial value problems. To illustrate our algorithm, we consider a numerical method for a given initial-boundary value problem. A modified version of our algorithm will be used to obtain sufficient conditions for the stability of the numerical method under consideration.