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This paper introduces a novel method for eigenvalue computation using a distributed cooperative neural network framework. Unlike traditional techniques that face scalability challenges in large systems, our decentralized algorithm enables…
Various Neural Networks employ time-consuming matrix operations like matrix inversion. Many such matrix operations are faster to compute given the Singular Value Decomposition (SVD). Previous work allows using the SVD in Neural Networks…
Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and…
The quantum mechanical expression relating two commuting operators is reformulated such that the power method (also called method of moments) for iteratively calculating eigenvalues and eigenvectors becomes applicable. The new iterative…
In this paper we present an efficient algorithm to compute the eigen decomposition of a matrix that is a weighted sum of the self outer products of vectors such as a covariance matrix of data. A well known algorithm to compute the eigen…
This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that…
The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total…
Pairwise comparison matrices are frequently applied in multi-criteria decision making. A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and…
The matrix element method utilizes ab initio calculations of probability densities as powerful discriminants for processes of interest in experimental particle physics. The method has already been used successfully at previous and current…
Approximate computing has shown to provide new ways to improve performance and power consumption of error-resilient applications. While many of these applications can be found in image processing, data classification or machine learning, we…
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…
Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of…
In recent years, a new kind of accelerated hardware has gained popularity in the Artificial Intelligence (AI) and Machine Learning (ML) communities which enables extremely high-performance tensor contractions in reduced precision for deep…
In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding…
In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct…
Many problems in physics, chemistry and other fields are perturbative in nature, i.e. differ only slightly from related problems with known solutions. Prominent among these is the eigenvalue perturbation problem, wherein one seeks the…
The numerical solution of eigenvalue problems is essential in various application areas of scientific and engineering domains. In many problem classes, the practical interest is only a small subset of eigenvalues so it is unnecessary to…
Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…