In this paper the problem of the computation of the joint spectral radius of a finite set of matrices isconsidered. We present an algorithm which, under some suitable assumptions, is able to check if a certainproduct in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attemptingto construct a suitable extremal norm for the family, namely a complex polytope norm. As examples fortesting our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel,Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arisingin the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.

An algorithm for finding extremal polytope norms of matrix families

GUGLIELMI, NICOLA;
2008

Abstract

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices isconsidered. We present an algorithm which, under some suitable assumptions, is able to check if a certainproduct in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attemptingto construct a suitable extremal norm for the family, namely a complex polytope norm. As examples fortesting our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel,Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arisingin the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.
Joint spectral radius; Balanced complex polytope; Complex polytope norm
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/1614
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