Localization and continuation of nonlinear eigenvalues

Abstract: 

Nonlinear eigenvalue problems are ubiquitous in the stability analysis of nonlinear systems, such as vibrating systems or systems with delay. Numerical discretizations then lead to large and sparse parameterized nonlinear eigenvalue problems

A(s,λ)v=0,v∈Cm,A(s,λ)v=0,v∈Cm,

where the matrix family A(s,λ)∈Cm×mA(s,λ)∈Cm×m depends smoothly on the real parameter s∈Rs∈R and analytically on the eigenvalue parameter λ∈Cλ∈C. We aim at an algorithm that detects a small swarm of eigenvalues λλ within a prescribed complex domain and that continues the swarm with respect to the parameter ss.

 A new computational procedure is presented that determines the eigenvalues (and the corresponding eigenvectors) in the interior of a smooth contour of the complex plane. The method builds on Cauchy's integral formula and on a theorem of Keldysh. Then we discuss a continuation method that pursues the swarm of eigenvalues with the parameter and that deflates and inflates the swarm when collisions with outside eigenvalues occur.