A Jacobi-Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations

SUMMARYThe critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type...

Full description

Saved in:
Bibliographic Details
Published in:Numerical linear algebra with applications 2013-10, Vol.20 (5), p.852-868
Main Authors: Meerbergen, Karl, Schröder, Christian, Voss, Heinrich
Format: Article
Language:English
Subjects:
critical delay
Delay
delay-differential equation
Eigenvalues
Jacobi-Davidson
Linear algebra
Linear systems
Mathematical analysis
Mathematical models
nonlinear eigenvalue problem
Nonlinearity
Solvers
two-parameter eigenvalue problem
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:SUMMARYThe critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.1848