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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…

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…

An a posteriori verification method is proposed for the generalized real-symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large-scale electronic state calculations. The proposed method is realized by a…

We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries…

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed.…

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…

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…

Computing many eigenpairs of the Schr{\"o}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a…

Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications.Here we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems.This type…

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…

We propose a novel numerical algorithm for computing the electronic structure related eigenvalue problem of incommensurate systems. Unlike the conventional practice that approximates the system by a large commensurate supercell, our…

In this paper we are concerned to find the eigenvalues and eigenvectors of a real symetric matrix by applying a new numerical method similar to Jacobi method. Our approch consists to use a new orthogonal matrix. The computation of the…

Different possible sources are discussed for enhancement of the calculation time when solving ordinary differential equations systems to forecast space objects' motion. This paper presents an approach for building an integrator of ordinary…

We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.

We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity…

We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the…

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…

We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…

Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…

The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…