The Joint Spectral Radius and its approximation

Series: 
Applied and Computational Mathematics Seminar
Monday, November 12, 2012 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
GT Math
,  
Organizer: 
Given F, finite set of square matrices of dimension n, it is possible to define the Joint Spectral Radius or simply JSR as a generalization of the well known spectral radius of a matrix.      The JSR evaluation proves to be useful for instance in the analysis of the asymptotic behavior of solutions of linear difference equations with variable coefficients, in the construction of compactly supported wavelets of and many others contexts.     This quantity proves, however, to be hard to compute in general. Gripenberg in 1996 proposed an algorithm for the computation of lower and upper bounds to the JSR based on a four member inequality and a branch and bound technique.    In this talk we describe a generalization of Gripenberg's method based on ellipsoidal norms that achieve a tighter upper bound, speeding up the approximation of the JSR.  We show the performance of this new algorithm compared with Gripenberg's  one.  This talk is based on joint work with V.Y.Protasov.