相关论文: Spectrum through pseudospectrum
Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix…
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the…
We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral…
In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…
We study the pseudospectrum of the non-selfadjoint Zakharov-Shabat system in the semiclassical regime. The pseudospectrum may be defined as the union of the spectra of perturbations of the Zakharov-Shabat system, thus it is relevant to the…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade. Here we present a new algorithm based on slicing the spectrum that takes advantage of the…
Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be…
We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate…
A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary value problems for many trial values of the spectral parameter. We present numerical results…
Let $A$ be a $n\times n$ complex Hermitian matrix and let $\lambda(A)=(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising…
A hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These…
We generalize a recently proposed approach for the construction of pseudo-Hermitian Hamiltonians with real spectra. Present technique is based on a simple and straightforward similarity transformation of the coordinate and momentum.
Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region…
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…
The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However,…
In this paper, we describe a new algorithm that approximates the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi eigenvalue method for symmetric matrices…
We generalize $\epsilon$-pseudospectra and the associated computational algorithms to the generalized eigenvalue problem. Rank one perturbations are used to determine the $\epsilon$-pseudospectra.
Given a sparse Hermitian matrix $A$ and a real number $\mu$, we construct a set of sparse vectors, each approximately spanned only by eigenvectors of $A$ corresponding to eigenvalues near $\mu$. This set of vectors spans the column space of…
This work is concerned with variational analysis of so-called spectral functions and spectral sets of matrices that only depend on eigenvalues of the matrix. Based on our previous work [H. T. B\`ui, M. N. B\`ui, and C. Clason, Convex…