15A18 Eigenvalues, singular values, and eigenvectors
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- eigenvalues (4)
- matrix polynomial (3)
- nonlinear eigenvalue problem (3)
- $\mu$-values (2)
- Hamiltonian matrix (2)
- Smith form (2)
- backward error (2)
- condition numbers (2)
- matrix pencil (2)
- palindromic eigenvalue problem (2)
- palindromic matrix polynomial (2)
- perturbation theory (2)
- perturbations (2)
- robust control (2)
- singular value decomposition (2)
- spectral value sets (2)
- structured linearization (2)
- $CUR$ decomposition (1)
- (even eigenvalue problem (1)
- Gramians (1)
- H-even matrix (1)
- H-odd matrix (1)
- Helmholtz equation (1)
- Hermitian matrix polynomial (1)
- Hessenberg matrix (1)
- Implicitely restarted Krylov method (1)
- Jacobi algorithm (1)
- Jordan form (1)
- Jordan structure (1)
- Markov chains (1)
- Maxwell equation (1)
- Nonlinear eigenvalue problem (1)
- Nonlinear eigenvalue problems (1)
- Perron-Frobenius theorem (1)
- Polynomial eigenvalue problem (1)
- QR algorithm (1)
- Schur form (1)
- Tucker decomposition (1)
- URV decomposition (1)
- Weyl function (1)
- acoustic field computation (1)
- alternating matrix polynomial (1)
- anti-triangular form (1)
- approximation (1)
- automotive industry (1)
- balanced truncation (1)
- block-Arnoldi method (1)
- bulge chasing (1)
- bulge exchange (1)
- canonical form (1)
- complex skew-symmetric matrix polynomial (1)
- complex symmetric linear system (1)
- complex symmetric matrix polynomial (1)
- compound matrix (1)
- conformation dynamics (1)
- convergence theory) (1)
- descriptor system (1)
- differential-algebraic equation (1)
- differentiation index (1)
- dispersive metallic photonic crystals (1)
- distance to singularity (1)
- eigenvalue backward error (1)
- elementary divisor (1)
- elementary divisors (1)
- even matrix polynomial (1)
- even matrix polynomials (1)
- even pencil (1)
- even/odd matrix polynomial (1)
- frequency response problem (1)
- gyroscopic system (1)
- homogeneous polynomial (1)
- implicit QR algorithm (1)
- invariant factor (1)
- invariant polynomial (1)
- invariant polynomials (1)
- invariant subspace (1)
- least squares (1)
- linear systems (1)
- low rank perturbation (1)
- matrix pencils (1)
- model reduction (1)
- non-equivalence deflation (1)
- odd matrix polynomial (1)
- odd matrix polynomials (1)
- palindromic (1)
- palindromic QR-algorithm (1)
- palindromic matrix pencil (1)
- palindromic matrix polynomials (1)
- palindromic/even eigenvalue problem (1)
- passive system (1)
- passivity (1)
- perturbation analysis (1)
- pole condition (1)
- polynomial (1)
- projector chain (1)
- pseudospectra (1)
- purely imaginary (1)
- rank $k$ (1)
- resonance problems (1)
- second-order systems (1)
- singular pencil (1)
- singular values (1)
- skew QR factorization (1)
- skew QRQ$^T$ factorization (1)
- skew Takagi factorization (1)
- skew-Hamiltonian matrix (1)
- skew-Hermitian matrix polynomial (1)
- skew-symmetric/symmetric pencil (1)
- spurious solutions (1)
- structure preserving method (1)
- structured backward error (1)
- structured condition number (1)
- structured deflation method (1)
- structured eigenvalue backward error (1)
- structured perturbation (1)
- structured perturbations (1)
- symplectic matrix (1)
- tractability index (1)
- transparent boundary condition (1)
- unitary congruence (1)
The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster near the zero eigenvalues. For
the solution of the eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which uses a hybrid method that combines the Jacobi-Davidson, the shift-invert residual Arnoldi and nonlinear Arnoldi methods to compute the clustered eigenvalues. Numerical results illustrate that the method is robust even for the case of computing many eigenvalues in very large problems.
Structured eigenvalue backward errors of matrix pencils and polynomials with palindromic structures
(2014)
We derive formulas for the backward error of an approximate eigenvalue of a *-palindromic
matrix polynomial with respect to *-palindromic perturbations. Such formulas are also obtained
for complex T-palindromic pencils and quadratic
polynomials. When the T-palindromic polynomial is real, then we derive the backward error
of a real number considered as an approximate eigenvalue of the matrix polynomial with
respect to real T-palindromic perturbations.
In all cases the corresponding minimal structure preserving perturbations are obtained as well.
The results are illustrated by numerical experiments. These show that there is
significant difference between the backward errors with respect to structure
preserving and arbitrary perturbations in many cases.
For regular matrix pencils the distance in norm to the nearest singular pencil
under low rank perturbation is studied. Characterizations of this distance are derived via the Weyl function of the perturbation. Special attention is paid to the Hermitian pencil case.
Estimates for the distance of a given pencil to the set of singular pencils are obtained.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
New perturbation results for the behavior of eigenvalues and Jordan forms of real and complex matrices
under generic rank one perturbations are discussed. Several results that are available in the complex
case are proved as well for the real case and the assumptions on the genericity are weakened.
Rank one perturbations that lead to maximal algebraic multiplicities of the ``new" eigenvalues are also
discussed.
We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency
response problems that arise in the analysis, simulation and optimization of acoustic fields.
We report about the cooperation with the company SFE in Berlin. We present the challenges
in the current industrial problems and the state-of-the-art of current methods. The difficulties
that arise with current off-the-shelf methods are discussed and several industrial examples are presented. It is documented that industrial cooperation is by no means a one-way street
of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.
We discuss the perturbation analysis for
eigenvalues and eigenvectors of structured homogeneous matrix polynomials with
Hermitian, skew-Hermitian, H-even and H-odd structure.
We construct minimal structured perturbations (structured backward errors) such that an
approximate eigenpair is an exact eigenpair of an appropriate perturbed structured matrix
polynomial. We present various comparisons with unstructured backward
errors and previous error bounds derived for the non-homogeneous case
and show that our bounds present a significant improvement.
Many applications give rise to matrix polynomials whose coefficients have
a kind of reversal symmetry, a structure we call palindromic.
Several properties of scalar palindromic polynomials are derived,
and together with properties of compound matrices, used to
establish the Smith form of regular and singular T-palindromic matrix polynomials,
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials are characterized,
and necessary conditions for the
existence of structured linearizations established.
In the odd degree case, a constructive procedure for building
palindromic linearizations shows that the necessary conditions are sufficient as well.
The Smith form for *-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of
characteristic two are presented.
In this work we propose a general framework for the structured perturbation
analysis of several classes of structured matrix polynomials in homogeneous
form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate eigenpair is an exact eigenpair of an appropriately perturbed matrix polynomial. This work extends previous work for the non-homogeneous case (we include infinite eigenvalues) and we show that the structured backward errors improve the known unstructured backward errors.
A new implicitly-restarted Krylov subspace method
for real symmetric/skew-symmetric generalized eigenvalue problems
is presented. The new method improves and generalizes the SHIRA method
to the case where the skew symmetric matrix is singular.
It computes a few eigenvalues and eigenvectors of the matrix pencil
close to a given target point. Several applications from control theory are
presented and the properties of the new method are illustrated by benchmark
examples.