An experimental study of approximation algorithms for the joint spectral radius

We describe several approximation algorithms for the joint spectral radius and compare their performance on a large number of test cases. The joint spectral radius of a set Σ of $n \times n$ matrices is the maximal asymptotic growth rate that can be obtained by forming products of matrices from Σ. This quantity is NP-hard to compute and appears in many areas, including in system theory, combinatorics and information theory. A dozen algorithms have been proposed this last decade for approximating the joint spectral radius but little is known about their practical efficiency. We overview these approximation algorithms and classify them in three categories: approximation obtained by examining long products, by building a specific matrix norm, and by using optimization-based techniques. All these algorithms are now implemented in a (freely available) MATLAB toolbox that was released in 2011. This toolbox allows us to present a comparison of the approximations obtained on a large number of test cases as well as on sets of matrices taken from the literature. Finally, in our comparison we include a method, available in the toolbox, that combines different existing algorithms and that is the toolbox’s default method. This default method was able to find optimal products for all test cases of dimension less than four.

[1]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[2]  Donald Ervin Knuth,et al.  The Art of Computer Programming, Volume II: Seminumerical Algorithms , 1970 .

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

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

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

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

[7]  M. Maesumi.  An efficient lower bound for the generalized spectral radius , 1996 .

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

[9]  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..

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

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

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

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

[14]  V. Kozyakin.  Proof of a Counterexample to the Finiteness Conjecture in the Spirit of the Theory of Dynamical Systems , 2005 .

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

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

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

[18]  V. Kozyakin.  Structure of extremal trajectories of discrete linear systems and the finiteness conjecture , 2007 .

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

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

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

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

[23]  V. Kozyakin.  On accuracy of approximation of the spectral radius by the Gelfand formula , 2008, 0810.2856.

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

[25]  Vincent D. Blondel,et al.  Joint Spectral Characteristics of Matrices: A Conic Programming Approach , 2010, SIAM J. Matrix Anal. Appl..

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

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

[28]  V. Kozyakin.  ITERATIVE BUILDING OF BARABANOV NORMS AND COMPUTATION OF THE JOINT SPECTRAL RADIUS FOR MATRIX SETS , 2008, 0810.2154.

[29]  V. Kozyakin.  A relaxation scheme for computation of the joint spectral radius of matrix sets , 2008, 0810.4230.

[30]  V. Blondel,et al.  Approximating the joint spectral radius using a genetic algorithm framework , 2011 .

[31]  Christos Georgakis,et al.  18th IFAC World Congress: Milano, Italy, August 28 to September 2, 2011 , 2013 .