Ljiljana Cvetković

The distance to instability provides a natural robustness measure for stable continuous-time dynamical systems, but its direct computation can become computationally expensive for large non-normal matrices. In this paper, we develop a structure-aware framework for obtaining certified lower bounds for the distance to instability by exploiting block decompositions of the system matrix. Building on an existing block comparison result, we formulate a recursive two-block procedure in which local stability margins can be supplied by different certified estimates. A new theoretical ingredient is a Frobenius-resolvent lower estimate which, for the class of substochastic network blocks considered here, leads to an explicit DTI bound depending only on the block dimension and the model parameters, thereby avoiding singular-value and resolvent computations for such blocks. The resulting framework admits adaptive and hybrid implementations: individual blocks may be treated by direct computation, explicit estimates, or further partitioning according to the quality and computational cost of the available local information. Particular attention is given to hierarchical and nearly triangular interaction patterns, for which directional coupling can substantially simplify the admissibility condition. Numerical experiments show that finer partitions do not necessarily produce sharper bounds and that carefully selected coarse, possibly highly unbalanced partitions can provide a favorable balance between accuracy and computational cost. A large hierarchical socio-economic network example illustrates the ability of the explicit local estimate to replace direct DTI computation for a high-dimensional subsystem while retaining a substantial portion of the corresponding block bound.

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.

{P1,P2}-Nekrasov matrices represent a generalization of Nekrasov matrices via permutations. In this paper, we obtained an error bound for linear complementarity problems for fP1; P2g-Nekrasov matrices. Numerical examples are given to illustrate that new error bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of {P1,P2}-Nekrasov matrices in the block case, considering two different types of block generalizations. Numerical examples show that new norm bounds for the block case can give tighter results compared to already known bounds for the point-wise case.

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

In this paper we present a new algorithm for the computation of the minimal Gersgorin set that can be considered an extension of the results from [5]. While the general approach to calculation of the boundary of the minimal Gersgorin set is kept, the core numerical calculation is changed. Namely, the problem is formulated in such a way that the eigenvalue computations are replaced by LU decompositions, allowing the algorithm to be used for larger matrices more efficiently. To illustrate the benefits, we compare both algorithms on several test matrices.