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

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 The material we present in the final section is part of an original research which is in progress and is still unpublished.

[1]  Jr. A. Willson.  A stability criterion for nonautonomous difference equations with application to the design of a digital FSK oscillator , 1974 .

[2]  L. Villemoes.  Wavelet analysis of refinement equations , 1994 .

[3]  Marino Zennaro,et al.  The Analysis and the Representation of Balanced Complex Polytopes in 2D , 2009, Found. Comput. Math..

[4]  Paul H. Siegel,et al.  On codes that avoid specified differences , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

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

[6]  Vincent D. Blondel,et al.  On the finiteness property for rational matrices , 2007 .

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

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

[9]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

[10]  Vincent D. Blondel,et al.  Overlap-free words and spectra of matrices , 2007, Theor. Comput. Sci..

[11]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[12]  L. Gurvits.  Stability of discrete linear inclusion , 1995 .

[13]  T. Andô,et al.  Simultaneous Contractibility , 1998 .

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

[15]  M. Zennaro,et al.  Finiteness property of pairs of 2× 2 sign-matrices via real extremal polytope norms , 2010 .

[16]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[17]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

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

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

[20]  Victor S. Kozyakin,et al.  On the computational aspects of the theory of joint spectral radius , 2009 .

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

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

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

[24]  I. Daubechies,et al.  Two-scale difference equations II. local regularity, infinite products of matrices and fractals , 1992 .

[25]  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..

[26]  Nikita Sidorov,et al.  An explicit counterexample to the Lagarias-Wang finiteness conjecture , 2010, ArXiv.

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

[28]  S. Dubuc.  Interpolation through an iterative scheme , 1986 .

[29]  U. Reif,et al.  C1-continuity of the generalized four-point scheme , 2009 .

[30]  Jean Berstel,et al.  Growth of repetition-free words -- a review , 2005, Theor. Comput. Sci..

[31]  A. Jadbabaie,et al.  Approximation of the joint spectral radius using sum of squares , 2007, 0712.2887.

[32]  Nicola Guglielmi,et al.  An algorithm for finding extremal polytope norms of matrix families , 2008 .

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

[34]  R. Jungers.  The Joint Spectral Radius: Theory and Applications , 2009 .

[35]  Nicola Guglielmi,et al.  Finding Extremal Complex Polytope Norms for Families of Real Matrices , 2009, SIAM J. Matrix Anal. Appl..

[36]  G. Alexits.  Approximation theory , 1983 .

[37]  R. Shorten,et al.  Quadratic and Copositive Lyapunov Functions and the Stability of Positive Switched Linear Systems , 2007, 2007 American Control Conference.

[38]  J. Vandergraft.  Spectral properties of matrices which have invariant cones , 1968 .

[39]  Nicola Guglielmi,et al.  Exact Computation of Joint Spectral Characteristics of Linear Operators , 2011, Found. Comput. Math..

[40]  Vincent D. Blondel,et al.  On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns , 2006, IEEE Transactions on Information Theory.

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

[42]  Carla Manni,et al.  Convergence analysis of C2 Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas , 2011 .

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

[44]  M. Zennaro,et al.  On the asymptotic regularity of a family of matrices , 2012 .

[45]  V. Protasov.  The generalized joint spectral radius. A geometric approach , 1997 .

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

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

[48]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

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

[50]  J. Mairesse,et al.  Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture , 2001 .

[51]  R. M. Jungers,et al.  Counterexamples to the Complex Polytope Extremality Conjecture , 2009, SIAM J. Matrix Anal. Appl..