Vagnoni, Cristina (2008) Algorithms for the computation of the joint spectral radius. [Ph.D. thesis]
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The asymptotic behaviour of the solutions of a discrete linear dynamical system is related to the spectral radius R of its associated family F. In particular, a system is stable if R <= 1 and there exists an extremal norm for the family F.
This kind of systems is important for a large number of applications. In particular, we mention the stability analysis of numerical methods for ordinary differential equations.
In the last decades some algorithms have been proposed in order to find real extremal norms of polytope type in the case of finite families (see for example [BT79] and [GZ05]). However, recently it has been observed that it is more useful to consider complex polytope norms, which are norms whose unit ball is a balanced complex polytope (see for example [GWZ05] and [MiSa06]).
In this work, using the theory developed in [GZ07], we extend the algorithm for the construction (i.e. for the geometric representation) of the unit ball of real polytope norms to the complex space. In order to succeed in our purpose, we first needed to get a deeper theoretical knowledge of the balanced complex polytopes. However, due to the extreme increase in complexity of the geometry of such objects with the dimension n of the space, we have confined ourselves to face the two-dimensional case.
In particular, we have given original theoretical results on the geometry of two-dimensional balanced complex polytopes in order to present the first two efficient algorithms, one for the construction of a balanced complex polytope in the two-dimensional space and one for the computation of the complex polytope norm of a two-dimensional vector starting from the knowledge of the boundary of the corresponding unit ball.
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