Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

Collocation Methods for Exploring Perturbations in Linear Stability Analysis

“…It can be explained using pseudospectra: the -pseudospectrum of the Jacobian matrix, defined for M = I in (1.3), is the set of eigenvalues of J + E for E ≤ . (A generalization to forms of M considered in the present study is discussed in [11].) Transient growth is exhibited when some elements of this set protrude into the right-half of the complex plane [23].…”

“…Our aim in this study is to develop and explore a simple procedure to study the sensitivity of the eigenvalues of (1.3) when the dynamical system comes from models of incompressible flow. As observed in [11], it is not practical to compute pseudospectra for the large-scale systems that arise in this setting. This difficulty is addressed in [11] by projecting such systems into invariant subspaces of (shifted versions of) J −1 M , which have smaller dimension and for which computation of pseudospectra is feasible.…”

“…As observed in [11], it is not practical to compute pseudospectra for the large-scale systems that arise in this setting. This difficulty is addressed in [11] by projecting such systems into invariant subspaces of (shifted versions of) J −1 M , which have smaller dimension and for which computation of pseudospectra is feasible. It is shown in [11] that these pseudospectra estimates provide interior bounds on pseudospectra of (1.3) as well as insight into transient growth of solutions.…”