The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. Here we present a Lanczoslike algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a strategy for its numerical implementation is also outlined and will be subject of future investigation.

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Identifying important components in a network is one of the major goals of network analysis. Popular and effective measures of importance of a node or a set of nodes are defined in terms of suitable entries of functions of matrices f (A). These kinds of measures are particularly relevant as they are able to capture the global structure of connections involving a node. However, computing the entries of f (A) requires a significant computational effort. In this work we address the problem of estimating the changes in the entries of f (A) with respect to changes in the edge structure. Intuition suggests that, if the topology or the overall weight of the connections in the new graph G are not significantly distorted, relevant components in G maintain their leading role in G. We propose several bounds giving mathematical reasoning to such intuition and showing, in particular, that the magnitude of the variation of the entry f (A) k decays exponentially with the shortest-path distance in G that separates either k or from the set of nodes touched by the edges that are perturbed. Moreover, we propose a simple method that exploits the computation of f (A) to simultaneously compute the all-pairs shortest-path distances of G, with essentially no additional cost. The proposed bounds are particularly relevant when the nodes whose edge connection tends to change more often or tends to be more often affected by noise have marginal role in the graph and are distant from the most central nodes.

We derive a priori residual-type bounds for the Arnoldi approximation of a matrix function and a strategy for setting the iteration accuracies in the inexact Arnoldi approximation of matrix functions. Such results are based on the decay behavior of the entries of functions of banded matrices. Specifically, we will use a priori decay bounds for the entries of functions of banded non-Hermitian matrices by using Faber polynomial series. Numerical experiments illustrate the quality of the results.

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the method's efficacy and properties in the scalar case.

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The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.