Related papers: Forward Error-Oriented Iterative Refinement for Ei…
In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians. In earlier works such estimates were computed by solving global optimization problems, which could be…
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…
Computation of the large sparse matrix exponential has been an important topic in many fields, such as network and finite-element analysis. The existing scaling and squaring algorithm (SSA) is not suitable for the computation of the large…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
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…
This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Pad\'e methods shown to be…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an ill-conditioned linear system $Ax=b$, we show that, if the inverse of the…
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…
Motivated by applications such as sparse PCA, in this paper we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
In this paper, the author present a reliable symbolic computational algorithm for inverting a general comrade matrix by using parallel computing along with recursion. The computational cost of our algorithm is O(n^2). The algorithm is…
This paper proposes an efficient method for computing partial eigenvalues of large sparse matrices what can be called the inexact inverse power method (IIPM). It is similar to the inexact Rayleigh quotient method and inexact Jacobi-Davidson…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
In this paper, a new type of multi-level correction scheme is proposed for solving eigenvalue problems by finite element method. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which…
Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…
In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or…
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are…
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…
Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally…