Related papers: Iterative Methods for Computing Eigenvalues and Ei…
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…
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…
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 propose an efficient algorithm for computing a common eigenvector of a finite set of square matrices. As an immediate consequence we obtain an algorithm for determining whether the matrices admit a simultaneous triangulation, and, if so,…
We develop the first stochastic incremental method for calculating the Moore-Penrose pseudoinverse of a real matrix. By leveraging three alternative characterizations of pseudoinverse matrices, we design three methods for calculating the…
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…
The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…
The power method is a basic method for computing the dominant eigenpair of a matrix. In this paper, we propose a structure-preserving power-like method for computing the dominant conjugate pair of purely imaginary eigenvalues and the…
We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of…
We use a unified framework to summarize sixteen randomized iterative methods including Kaczmarz method, coordinate descent method, etc. Some new iterative schemes are given as well. Some relationships with \textsc{mg} and \textsc{ddm} are…
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient…
We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear…
Some iterative techniques are defined to solve reversible inverse problems and a common formulation is explained. Numerical improvements are suggested and tests validate the methods.
Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods…
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…
In this paper, we present a protocol for computing the principal eigenvector of a collection of data matrices belonging to multiple semi-honest parties with privacy constraints. Our proposed protocol is based on secure multi-party…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift…
We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with…
We test iterative algorithms, MR, QMR$\gamma_5$ and BiCG$\gamma_5$, to compare their efficiency in matrix inversion with multi-quarks (shifted matrices) within one iteration process. Our results on the 8^3 x 12 and 16^3 x 24 show that MR…