Related papers: Improved power methods for computing eigenvalues o…
A multigrid method is proposed in this paper to solve eigenvalue problems by the finite element method based on the shifted-inverse power iteration technique. With this scheme, solving eigenvalue problem is transformed to a series of…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…
We present a new algorithm for solving an eigenvalue problem for a real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in $O(n^{2})$…
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…
Computing electronic structures of quantum systems is a key task underpinning many applications in photonics, solid-state physics, and quantum technologies. This task is typically performed through iterative algorithms to find the energy…
In this paper, we present a generalized Cuppen's divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We extend the Cuppen's work to the rank two modifications of the form $A =T +\beta_1\bw_1\bw_1^T +…
Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some…
In this paper, we discuss numerical methods for the eigenvalue decomposition of real symmetric matrices. While many existing methods can compute approximate eigenpairs with sufficiently small backward errors, the magnitude of the resulting…
A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter…
Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix…
The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision…
Tenfold improvements in computation speed can be brought to the alternating direction method of multipliers (ADMM) for Semidefinite Programming with virtually no decrease in robustness and provable convergence simply by projecting…
In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or…
Douglas-Rachford splitting and its equivalent dual formulation ADMM are widely used iterative methods in composite optimization problems arising in control and machine learning applications. The performance of these algorithms depends on…
Thanks to its versatility, its simplicity, and its fast convergence, ADMM is among the most widely used approaches for solving a convex problem in distributed form. However, making it running efficiently is an art that requires a fine…
For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the…
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…
The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the…
In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for…
We present an iteration for the computation of simple eigenvalues using a pseudospectrum approach. The most appealing characteristic of the proposed iteration is that it reduces the computation of a single eigenvalue to a small number of…