728 research outputs found
Perturbation, extraction and refinement of invariant pairs for matrix polynomials
Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under
perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments
with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures
Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems
Invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding (generalized) eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In this work, we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies. Based on the concept of invariant pairs, a theoretically sound and reliable numerical continuation procedure is developed. Particular attention is paid to the situation when the procedure approaches a singularity, that is, when eigenvalues included in the invariant pair collide with other eigenvalues. For the real generic case, it is proven that such a singularity only occurs when two eigenvalues collide on the real axis. It is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair. The viability of our continuation procedure is illustrated by a numerical exampl
How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models
Invariant manifolds are important constructs for the quantitative and
qualitative understanding of nonlinear phenomena in dynamical systems. In
nonlinear damped mechanical systems, for instance, spectral submanifolds have
emerged as useful tools for the computation of forced response curves, backbone
curves, detached resonance curves (isolas) via exact reduced-order models. For
conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and
their reduced dynamics provide a way to identify nonlinear amplitude-frequency
relationships in the form of conservative backbone curves. Despite these
powerful predictions offered by invariant manifolds, their use has largely been
limited to low-dimensional academic examples. This is because several
challenges render their computation unfeasible for realistic engineering
structures described by finite-element models. In this work, we address these
computational challenges and develop methods for computing invariant manifolds
and their reduced dynamics in very high-dimensional nonlinear systems arising
from spatial discretization of the governing partial differential equations. We
illustrate our computational algorithms on finite-element models of mechanical
structures that range from a simple beam containing tens of degrees of freedom
to an aircraft wing containing more than a hundred-thousand degrees of freedom
A block Newton method for nonlinear eigenvalue problems
We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viabilit
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