Stefano Maset

Stefano Maset

The paper \cite{M0} studied, for a \emph{complex} linear ordinary differential equation $y^\prime(t)=Ay(t)$, the long-time propagation to the solution $y(t)$ of a perturbation of the initial value. By measuring the perturbations with relative errors, this paper introduced a directional pointwise condition number, defined for a specific initial value and for a specific direction of perturbation of this initial value, and a pointwise condition number, defined for a specific initial value and the worst-case scenario for the direction of perturbation. The asymptotic (long-time) behaviors of these two condition numbers were determined. The present paper analyzes such asymptotic behaviors in depth, for a \emph{real} linear ordinary differential equation in a generic case.

Stefano Maset

We are interested in the relative conditioning of the problem $y_0\mapsto \mathrm{e}^{tA}y_0$, i.e., the relative conditioning of the action of the matrix exponential $\mathrm{e}% ^{tA}$ on a vector with respect to perturbations of this vector. The present paper is a qualitative study of the long-time behavior of this conditioning. In other words, we are interested in studying the propagation to the solution $y(t)$ of perturbations of the initial value for a linear ordinary differential equation $y^\prime(t)=Ay(t)$, by measuring these perturbations with relative errors. We introduce three condition numbers: the first considers a specific initial value and a specific direction of perturbation; the second considers a specific initial value and the worst case by varying the direction of perturbation; and the third considers the worst case by varying both the initial value and the direction of perturbation. The long-time behaviors of these three condition numbers are studied.

Stefano Maset

We investigate the propagation of initial value perturbations along the solution of a linear ordinary differential equation \( y'(t) = Ay(t) \). This propagation is analyzed using the relative error rather than the absolute error. Our focus is on the long-term behavior of this relative error, which differs significantly from that of the absolute error. The present paper is a practical sequel to the theoretical papers \cite{M1,M2} on the long-time behavior of the relative error: it includes applicative examples and important issues not addressed in \cite{M1,M2}. In addition, the present paper shows that understanding the long-term behavior provides insights into the growth of the relative error over all times, not just at large times. Therefore, it represents a crucial and fundamental aspect of the conditioning of linear ordinary differential equations, with applications in, for example, non-normal dynamics.

Dimitri Breda, Stefano Maset, Rossana Vermiglio

This paper deals with the approximation of the spectrum of linear and nonautonomous delay differential equations through the reduction of the relevant evolution semigroup from infinite to finite dimension. The focus is placed on classic collocation, even though the requirements that a numerical scheme has to fulfill in order to allow for a correct approximation of the spectral elements are recalled. This choice, motivated by the analyticity of the underlying eigenfunctions, allows for a convergence of infinite order, as rigorously demonstrated through a priori error bounds when Chebyshev nodes are adopted. Fundamental applications such as determination of asymptotic stability of equilibria (autonomous case) and limit cycles (periodic case) follow at once.