Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Deflation Techniques for an Implicitly Restarted Arnoldi Iteration
SIAM Journal on Matrix Analysis and Applications
Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils
SIAM Journal on Scientific Computing
A Supernodal Approach to Sparse Partial Pivoting
SIAM Journal on Matrix Analysis and Applications
Spectral methods in MatLab
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL
ACM Transactions on Mathematical Software (TOMS)
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling
SIAM Journal on Matrix Analysis and Applications
An Extension of MATLAB to Continuous Functions and Operators
SIAM Journal on Scientific Computing
Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method
ACM Transactions on Mathematical Software (TOMS)
A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems
Future Generation Computer Systems - Special issue: Selected numerical algorithms
Solving unsymmetric sparse systems of linear equations with PARDISO
Future Generation Computer Systems - Special issue: Selected numerical algorithms
SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations
SIAM Journal on Scientific Computing
Vector Spaces of Linearizations for Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Efficient computation of characteristic roots of delay differential equations using LMS methods
Journal of Computational and Applied Mathematics
Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control)
Applied Numerical Mathematics
Solution operator approximations for characteristic roots of delay differential equations
Applied Numerical Mathematics
The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
A block Newton method for nonlinear eigenvalue problems
Numerische Mathematik
On dominant poles and model reduction of second order time-delay systems
Applied Numerical Mathematics
Krylov-Based Model Order Reduction of Time-delay Systems
SIAM Journal on Matrix Analysis and Applications
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The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP). The DDE can equivalently be expressed with a linear infinite-dimensional operator whose eigenvalues are the solutions to the DEP. We derive a new method by applying the Arnoldi method to the generalized eigenvalue problem associated with a spectral discretization of the operator and by exploiting the structure. The result is a scheme where we expand a subspace not only in the traditional way done in the Arnoldi method. The subspace vectors are also expanded with one block of rows in each iteration. More important, the structure is such that if the Arnoldi method is started in an appropriate way, it has the (somewhat remarkable) property that it is in a sense independent of the number of discretization points. It is mathematically equivalent to an Arnoldi method with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points. We also show an equivalence with the Arnoldi method in an operator setting. It turns out that with an appropriately defined operator over a space equipped with scalar product with respect to which Chebyshev polynomials are orthonormal, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function. The presented method yields the same Hessenberg matrix as the Arnoldi method applied to the operator.