An algorithm for finding extremal polytope norms of matrix families

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered. We present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attempting to construct a suitable extremal norm for the family, namely a complex polytope norm. As examples for testing 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 arising in the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.

[1]  G. Gripenberg.  COMPUTING THE JOINT SPECTRAL RADIUS , 1996 .

[2]  N. Guglielmi,et al.  Polytope norms and related algorithms for the computation of the joint spectral radius , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[3]  John N. Tsitsiklis,et al.  The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate , 1997, Math. Control. Signals Syst..

[4]  Vincent D. Blondel,et al.  Computationally Efficient Approximations of the Joint Spectral Radius , 2005, SIAM J. Matrix Anal. Appl..

[5]  M. Maesumi.  Construction of Optimal Norms for Semi-Groups of Matrices , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  G. Ziegler.  Lectures on Polytopes , 1994 .

[7]  F. Wirth.  The generalized spectral radius and extremal norms , 2002 .

[8]  Yang Wang,et al.  Bounded semigroups of matrices , 1992 .

[9]  G. Rota,et al.  A note on the joint spectral radius , 1960 .

[10]  Mau-Hsiang Shih,et al.  Asymptotic Stability and Generalized Gelfand Spectral Radius Formula , 1997 .

[11]  L. Elsner.  The generalized spectral-radius theorem: An analytic-geometric proof , 1995 .

[12]  Nicola Guglielmi,et al.  Stability of one‐leg Θ‐methods for the variable coefficient pantograph equation on the quasi‐geometric mesh , 2003 .

[13]  Nicola Guglielmi,et al.  On the zero-stability of variable stepsize multistep methods: the spectral radius approach , 2001, Numerische Mathematik.

[14]  Fabian R. Wirth,et al.  Complex Polytope Extremality Results for Families of Matrices , 2005, SIAM J. Matrix Anal. Appl..

[15]  Mau-Hsiang Shih,et al.  Simultaneous Schur stability , 1999 .

[16]  I. Daubechies,et al.  Sets of Matrices All Infinite Products of Which Converge , 1992 .

[17]  M. Zennaro,et al.  On the limit products of a family of matrices , 2003 .

[18]  Vincent D. Blondel,et al.  An Elementary Counterexample to the Finiteness Conjecture , 2002, SIAM J. Matrix Anal. Appl..

[19]  M. Zennaro,et al.  Balanced Complex Polytopes and Related Vector and Matrix Norms , 2007 .

[20]  Y. Nesterov,et al.  On the accuracy of the ellipsoid norm approximation of the joint spectral radius , 2005 .

[21]  J. Lagarias,et al.  The finiteness conjecture for the generalized spectral radius of a set of matrices , 1995 .

[22]  V. Protasov.  The Geometric Approach for Computing the Joint Spectral Radius , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[23]  Nicola Guglielmi,et al.  On the asymptotic properties of a family of matrices , 2001 .