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Regular version of the site
Of all publications in the section: 6
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Article
Protasov V. Y. SIAM Journal on Matrix Analysis and Applications. 2013. Vol. 34. No. 3. P. 1174-1188.
Article
Protasov V. Y., Guglielmi N. SIAM Journal on Matrix Analysis and Applications. 2016. Vol. 37. No. 1. P. 18-52.

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 Holder exponent 2 and its derivative is “almost Lipschitz” with logarithmic factor 2. Second we compute the Holder exponent of Daubechies wavelets of high order.

Article
Guglielmi N., Protasov V. Y. SIAM Journal on Matrix Analysis and Applications. 2018. Vol. 39. No. 4. P. 1642-1669.

We consider the problem of computing the closest stable/unstable nonnegative matrix

to a given real matrix. 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. The problem

of computing the closest stable matrix to a nonnegative matrix is a hard problem even if the stable

matrix is not constrained to be nonnegative. Adding the nonnegativity constraint makes the problem

even more dicult. 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.

Article
Nesterov Y., Protasov V. Y. SIAM Journal on Matrix Analysis and Applications. 2013. Vol. 34. No. 3. P. 999-1013.

We suggest a new approach to finding the maximal and the minimal spectral radii of linear operators from a given compact family of operators, which share a common invariant cone (e.g., family of nonnegative matrices). In the case of families with the so-called product structure, this leads to efficient algorithms for optimizing the spectral radius and for finding the joint and lower spectral radii of the family. Applications to the theory of difference equations and to problems of optimizing the spectral radius of graphs are considered.