Related papers: New Power Method for Solving Eigenvalue Problems

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive…

This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem…

We investigate the automatic differentiation of dominant eigensolver where only a small proportion of eigenvalues and corresponding eigenvectors are obtained. Backpropagation through the dominant eigensolver involves solving certain…

Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we…

We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(\lambda)\mathbf{u}= 0$ on compact continua $\Omega \subset \mathbb{C}$. The method combines two key components: (1) a rational minimax…

In this paper, we first establish the convergence criteria of the residual iteration method for solving quadratic eigenvalue problem- s. We analyze the impact of shift point and the subspace expansion on the convergence of this method. In…

We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as…

Computing effective eigenvalues for neutron transport often requires a fine numerical resolution. The main challenge of such computations is the high memory effort of classical solvers, which limits the accuracy of chosen discretizations.…

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the…

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix…

We present a new straightforward principal component analysis (PCA) method based on the diagonalization of the weighted variance-covariance matrix through two spectral decomposition methods: power iteration and Rayleigh quotient iteration.…

Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m>>n collocation points. We show how eigenvalue problems can be solved in this…

We propose a new iterative algorithm for generating a subset of eigenvalues and eigenvectors of large matrices which generalizes the method of optimal relaxations. We also give convergence criteria for the iterative process, investigate its…

In recent years, the power system research community has seen an explosion of novel methods for formulating and solving power network optimization problems. These emerging methods range from new power flow approximations, which go beyond…

The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a…

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order…

We propose a new concept of a relatively inexact stochastic subgradient and present novel first-order methods that can use such objects to approximately solve convex optimization problems in relative scale. An important example where…

We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when…

We revisit the problem of robust principal component analysis with features acting as prior side information. To this aim, a novel, elegant, non-convex optimization approach is proposed to decompose a given observation matrix into a…

Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This…