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Stability of Linear Problems: Joint Spectral Radius of Sets of Matrices

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Current Challenges in Stability Issues for Numerical Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2082))

Abstract

It is well known that the stability analysis of step-by-step numerical methods for differential equations often reduces to the analysis of linear difference equations with variable coefficients. This class of difference equations leads to a family \(\mathcal{F}\) of matrices depending on some parameters and the behaviour of the solutions depends on the convergence properties of the products of the matrices of \(\mathcal{F}\). To date, the techniques mainly used in the literature are confined to the search for a suitable norm and for conditions on the parameters such that the matrices of \(\mathcal{F}\) are contractive in that norm. In general, the resulting conditions are more restrictive than necessary. An alternative and more effective approach is based on the concept of joint spectral radius of the family \(\mathcal{F}\), \(\rho (\mathcal{F})\). It is known that all the products of matrices of \(\mathcal{F}\) asymptotically vanish if and only if \(\rho (\mathcal{F}) < 1\). The aim of this chapter is that to discuss the main theoretical and computational aspects involved in the analysis of the joint spectral radius and in applying this tool to the stability analysis of the discretizations of differential equations as well as to other stability problems. In particular, in the last section, we present some recent heuristic techniques for the search of optimal products in finite families, which constitutes a fundamental step in the algorithms which we discuss. The material we present in the final section is part of an original research which is in progress and is still unpublished.

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Acknowledgements

We thank C.I.M.E. for the excellent support in the organization of the Summer School and INdAM-GNCS for partial funding.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L’Aquila, Via Vetoio - Loc. Coppito, 67100, L’Aquila, Italy

    Nicola Guglielmi

  2. Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34100, Trieste, Italy

    Marino Zennaro

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  1. Nicola Guglielmi
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Correspondence to Nicola Guglielmi .

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Guglielmi, N., Zennaro, M. (2014). Stability of Linear Problems: Joint Spectral Radius of Sets of Matrices. In: Current Challenges in Stability Issues for Numerical Differential Equations. Lecture Notes in Mathematics(), vol 2082. Springer, Cham. https://doi.org/10.1007/978-3-319-01300-8_5

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