Localization theorems for nonlinear eigenvalue problems

Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function that is analytic on some simply-connected domain \Omega \subset \bbC. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure

On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.Comment: 4 figure

Circular law for non-central random matrices

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let \la_{n,1},...,\la_{n,n} be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution \frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}}) converges weakly as $n\to\infty$ to the uniform law over the unit disc \{z\in\dC;|z|\leq1\}. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and \mathrm{rank}(M)=O(n^\al) with \al<1. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.Comment: accepted in Journal of Theoretical Probabilit

How descriptive are GMRES convergence bounds?

Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES

Error analysis of signal zeros: a projected companion matrix approach

AbstractAn error analysis of so-called signal zeros of polynomials associated with exponentially damped/undamped signals is performed and zero error bounds are derived. The bounds are in terms of the angle between the exact and approximate signal subspace, the signal parameter themselves, the polynomial degree, and the error on the polynomial coeficients. The key idea behind the analysis is to regard signal zeros as eigenvalues of projected companion matrices and then to generate error bounds by exploiting perturbation theorem for eigenvalues. The conclusion drawn from the bounds is that the signal zeros become relatively insensitive to small perturbations on the polynomial coefficients as long as the polynomial degree is large enough and the zeros are extracted as eigenvalues of projected companion matrices. Also, the bounds suggests that signal zero estimates derived from projected companion matrices are more accurate than those obtained from the companion matrices themselves. Illustrative numerical results are provided

Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability as long as the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - SIT^{-1}||_2 \leq \varepsilon$ in at most $O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of exact arithmetic matrix pencil diagonalization.Comment: 58 pages, 8 figures, 2 table

A method to rigorously enclose eigenpairs of complex interval matrices

summary:In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair x=(\lambda,\rv) is found by solving a nonlinear equation of the form $f(x)=0$ via a contraction argument. The set-up of the method relies on the notion of {\em radii polynomials}, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable