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This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then a real geometric…
In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the…
We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two…
Solving large-scale systems of nonlinear equations/inequalities is a fundamental problem in computing and optimization. In this paper, we propose a generic successive projection (SP) framework for this problem. The SP sequentially projects…
In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the relaxed filtered Krylov subspace method for computing the eigenvalues with the largest real parts and the corresponding eigenvectors of non-symmetric…
Prony's method is a standard tool exploited for solving many imaging and data analysis problems that result in parameter identification in sparse exponential sums $$f(k)=\sum_{j=1}^{T}c_{j}e^{-2\pi i\langle t_{j},k\rangle},\quad k\in…
An algorithm named EigenWave is described to compute eigenvalues and eigenvectors of elliptic boundary value problems. The algorithm, based on the recently developed WaveHoltz scheme, solves a related time-dependent wave equation as part of…
In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and more. The canonical eigenproblem we…
This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the…
We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions…
Many classical Computer Vision problems, such as essential matrix computation and pose estimation from 3D to 2D correspondences, can be tackled by solving a linear least-square problem, which can be done by finding the eigenvector…
We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be…
Consider the problem of registering multiple point sets in some $d$-dimensional space using rotations and translations. Assume that there are sets with common points, and moreover the pairwise correspondences are known for such sets. We…
Inverse source problems arise often in real-world applications, such as localizing unknown groundwater contaminant sources. Being different from Tikhonov regularization, the quasi-boundary value method has been proposed and analyzed as an…
The aim of this paper is to propose an efficient adaptive finite element method for eigenvalue problems based on the multilevel correction scheme and inverse power method. This method involves solving associated boundary value problems on…
This paper develops a new class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This approach…
We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…
Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework…