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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.
Added: Feb 19, 2016
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.

Added: Feb 20, 2016
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.

Added: Oct 30, 2019
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.

Added: Feb 23, 2016
Article
Nesterov Y., Protasov V. SIAM Journal on Matrix Analysis and Applications. 2013. Vol. 34. No. 3. P. 999-1013.

Research Article

Added: Jan 28, 2016
Article
Protasov V. Y., Logofet D. SIAM Journal on Matrix Analysis and Applications. 2014. Vol. 35. No. 2. P. 749-764.

We study the location of λ2(A), the second positive eigenvalue of a nonnegative matrix A, as the issue of how many positive eigenvalues can be shifted beyond the spectral radius ρ(A) by means of arbitrary changes in elements of one row. The notion of rank-one correction suggests the nearest generalization expanding the changes in one row to any matrix of rank one (still keeping the matrix nonnegative). The main theorem limits the number of those eigenvalues, counting multiplicities, to the increased spectral radius alone. In matrix population models, we treat the projection matrix L = T + F as the rank-one correction of its transition part T by the fertility one F. The matrix T is column substochastic due to its demographic interpretation, hence we conclude that λ2(L) ≤ 1 and specify the rare cases where λ2(L) = 1. The location λ2(L) < 1 ensures that the function R(L) = 1 − det (I − L) has the indicator property, namely, its value is always located on the same side of 1 as is ρ(L). This indicator does not pose any computational problems and helps calibrate L from empirical data.

Added: Feb 19, 2016