Two-scale difference equations II. local regularity, infinite products of matrices and fractals
SIAM Journal on Mathematical Analysis
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
An Elementary Counterexample to the Finiteness Conjecture
SIAM Journal on Matrix Analysis and Applications
Computationally Efficient Approximations of the Joint Spectral Radius
SIAM Journal on Matrix Analysis and Applications
Structure of extremal trajectories of discrete linear systems and the finiteness conjecture
Automation and Remote Control
Overlap-free words and spectra of matrices
Theoretical Computer Science
Finding Extremal Complex Polytope Norms for Families of Real Matrices
SIAM Journal on Matrix Analysis and Applications
Joint Spectral Characteristics of Matrices: A Conic Programming Approach
SIAM Journal on Matrix Analysis and Applications
On codes that avoid specified differences
IEEE Transactions on Information Theory
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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.