The greedy strategy in optimizing the Perron eigenvalue

We address the problems of minimizing and of maximizing the spectral radius overa compact family of non-negative matrices. Those problems being hard in generalcan be efficiently solved for some special families. We consider the so-called prod-uct families, where each matrix is composed of rows chosen independently from givensets. A recently introduced greedy method works very fast. However, it is applicablemostly for strictly positive matrices. For sparse matrices, it often diverges and gives awrong answer. We present the "selective greedy method" thatworks equally well forall non-negative product families, including sparse ones.For this method, we provea quadratic rate of convergence and demonstrate its efficiency in numerical examples.The numerical examples are realised for two cases: finite uncertainty sets and poly-hedral uncertainty sets given by systems of linear inequalities. In dimensions up to 2000, the matrices with minimal/maximal spectral radii in product families are foundwithin a few iterations. Applications to dynamical systemsand to the graph theoryare considere

Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions

We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum maximizing products. A criterion of convergence of the algorithm is proved. As an application we solve two challenging computational open problems. First we find the regularity of the Butterfly subdivision scheme for various parameters $\omega$. In the "most regular" case $\omega = \frac{1}{16}$, we prove that the limit function has H\"older exponent $2$ and its derivative is "almost Lipschitz" with logarithmic factor $2$. Second we compute the H\"older exponent of Daubechies wavelets of high order.Comment: 36 page

Maximal acyclic subgraphs and closest stable matrices

We develop a matrix approach to the Maximal Acyclic Subgraph (MAS) problem by reducing it to finding the closest nilpotent matrix to the matrix of the graph. Using recent results on the closest Schur stable systems and on minimising the spectral radius over special sets of non-negative matrices we obtain an algorithm for finding an approximate solution of MAS. Numerical results for graphs from 50 to 1500 vertices demonstrate its fast convergence and give the rate of approximation in most cases larger than 0.6. The same method gives the precise solution for the following weakened version of MAS: and the minimal $r$ such that the graph can be made acyclic by cutting at most $r$ incoming edges from each vertex. Several modifications, when each vertex is assigned with its own maximal number $r_i$ of cut edges, when some of edges are "untouchable", are also considered. Some applications are discussed

Stability of linear switching systems and Markov-Bernstein inequalities for exponents

We analyse the problem of stability of a continuous time linear switching system (LSS) versus the stability of its Euler discretization. It is well-known that the existence of a positive {\tau} for which the corresponding discrete time system with step size {\tau} is stable implies the stability of LSS. Our main goal is to obtain a converse statement, that is, to estimate the discretization step size {\tau} > 0 up to a given accuracy {\epsilon} > 0. This leads to a method of deciding the stability of continuous time LSS with a guaranteed accuracy. As the first step, we solve this problem for matrices with real spectrum and conjecture that our method stays valid for the general case. Our approach is based on Markov-Bernstein type inequalities for systems of exponents. We obtain universal estimates for sharp constants in those inequalities. Our work provides the first estimate of the computational cost of the stability problem for continuous-time LSS (though restricted to the real-spectrum case)

Tiling of polyhedral sets

A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings. It is shown that in R^d there exist an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to an important particular case when the matrix of affine similarity and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed

On the closest stable/unstable nonnegative matrix and related stability radii

We consider the problem of computing the closest stable/unstable non-negative matrix to a given real matrix. This problem is important in the study of linear dynamical systems, numerical methods, etc. The distance between matrices is measured in the Frobenius norm. The problem is addressed for two types of stability: the Schur stability (the matrix is stable if its spectral radius is smaller than one) and the Hurwitz stability (the matrix is stable if its spectral abscissa is negative). We show that the closest unstable matrix can always be explicitly found. For the closest stable matrix, we present an iterative algorithm which converges to a local minimum with a linear rate. It is shown that the total number of local minima can be exponential in the dimension. Numerical results and the complexity estimates are presented

Self-affine 2-attractors and tiles

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In the general, non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to an affine similarity, by the spectrum of the dilation matrix. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with the free coefficient $\pm 2$. The total number of such polynomials is estimated by the Mahler measure. We present several infinite series of such polynomials. For some of the 2-attractors, their H\"older exponents are found. Some of our results are extended to attractors with an arbitrary number of digits.Comment: 40 pages, 7 figure

Analytic functions in shift-invariant spaces and analytic limits of level dependent subdivision

The structure of exponential subspaces of finitely generated shift-invariant spaces is well understood and the role of such subspaces for the approximation power of refinable function vectors and related multi-wavelets is well studied. In this paper, in the univariate setting, we characterize all analytic subspaces of finitely generated shift-invariant spaces and provide explicit descriptions of elements of such subspaces. Consequently, we depict the analytic functions generated by level dependent (non-stationary) subdivision schemes with masks of unbounded support. And we confirm the belief that the exponential polynomials are indeed the only analytic functions generated by such subdivision schemes with finitely supported masks

Elliptic polytopes and invariant norms of linear operators

We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long matrix products, in the Lyapunov stability, etc.. The main issue in the construction is to decide whether a given ellipse is in the convex hull of others. The computational complexity of this problem is analysed by considering an equivalent optimisation problem. We show that the number of local extrema of that problem may grow exponentially in d. For d=2,3, it admits an explicit solution for an arbitrary number of ellipses; for higher dimensions, several geometric methods for approximate solutions are derived. Those methods are analysed numerically and their efficiency is demonstrated in applications

Convex Optimization methods for computing the Lyapunov Exponent of matrices

We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist