Related papers: Structured eigenvalue problems in electronic struc…

We analyze the systematic errors made when using the generalized eigenvalue problem to extract energies and matrix elements in lattice gauge theory. Effective theories such as HQET are also discussed. Numerical results are shown for the…

The eigenvector-dependent nonlinear eigenvalue problem (NEPv) $A(P)V=V\Lambda$, where the columns of $V\in\mathbb{C}^{n\times k}$ are orthonormal, $P=VV^{\mathrm{H}}$, $A(P)$ is Hermitian, and $\Lambda=V^{\mathrm{H}}A(P)V$, arises in many…

Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a…

Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and…

We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured…

In this research paper, structured bi-matrix variate, matrix quadratic equations are considered. Some lemmas related to determining the eigenvalues of unknown matrices are proved. Also, a method of determining the diagonalizabe unknown…

We present a closed-form solution to Wahba's problem in the quaternion domain for the special case of two vector observations. Existing approaches, including Davenport's $q$-method, QUEST, Horn's method, and ESOQ algorithms, recover the…

This paper is to give a new understanding and applications of the subspace projection method for selfadjoint eigenvalue problems. A new error estimate in the energy norm, which is induced by the stiff matrix, of the subspace projection…

Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an…

Sensitivity of an eigenvalue $\lambda_i$ to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as $\kappa_i = \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where $\left<L_i\right|$ and…

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not…

We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some…

Under suitable scaling, the structure of self-gravitating polytropes is described by the standard Lane-Emden equation (LEE), which is characterised by the polytropic index $n$. Here we use the known exact solutions of the LEE at $n=0$ and…

We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by…

In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential…

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…

We study the eigenvalue problem for some special class of anti-triangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices.…

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation…

The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank…

We introduce a basis set consisting of three-dimensional Deslauriers--Dubuc wavelets and solve numerically the Schr\"odinger equations of H and He atoms and molecules $\mathrm{H}_2$, $\mathrm{H}_2^+$, and $\mathrm{LiH}$ with HF and DFT…