Related papers: Exact Algorithms for Computing Generalized Eigensp…

This paper develops a new class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This approach…

We review our recently developed methods for large-scale electronic structure calculations, both in one-electron theory and many-electron theory. The method are based on the density matrix representation, together with the Wannier state…

In this paper, we describe a new algorithm that approximates the extreme eigenvalue/eigenvector pairs of a symmetric matrix. The proposed algorithm can be viewed as an extension of the Jacobi eigenvalue method for symmetric matrices…

We establish the first globally convergent algorithms for computing the Kreiss constant of a matrix to arbitrary accuracy. We propose three different iterations for continuous-time Kreiss constants and analogues for discrete-time Kreiss…

We review a method providing explicit formulas for the Jack polynomials. Our method is based on the relation of the Jack polynomials to the eigenfunctions of a well-known exactly solvable quantum many-body system of Calogero-Sutherland…

This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of Reed-Solomon codes with a complexity of O(nd).

Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and…

The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the…

Often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigen-function and eigen-value of a non-negative integral kernel. Except in the finite-dimensional case, usually…

Generalized singular values (GSVs) play an essential role in the comparative analysis. In the real world data for comparative analysis, both data matrices are usually numerically low-rank. This paper proposes a randomized algorithm to first…

We present a method of cones for rigorous estimations of eigenvectors, eigenspaces and eigenvalues of a matrix. The key notion is the cone-domination and is inspired by ideas from hyperbolic dynamical systems. We present theorems which…

We consider an effective new method for solving trust-region and norm-regularization problems that arise as subproblems in many optimization applications. We show that the solutions to such subproblems effectively lie in a…

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…

Periodically driven quantum many-body systems play a central role for our understanding of nonequilibrium phenomena. For studies of quantum chaos, thermalization, many-body localization and time crystals, the properties of eigenvectors and…

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…

This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a…

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to…

We present a novel data-driven reformulation of the iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov…

In this paper we consider the problem of computing the stationary distribution of nearly completely decomposable Markov processes, a well-established area in the classical theory of Markov processes with broad applications in the design,…

We investigate the regularizing behavior of an iterative Krylov subspace method for the solution of linear inverse problems in precisions lower than double. Recent works have considered the projection of iterated Tikhonov methods using…