Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poores...

The use of localized basis sets is essential in linear-scaling electronic structure calculations, and since such basis sets are mostly non-orthogonal, it is necessary to solve the generalized eigenvalue problem Hx = "Sx. In this work, an iterative method for nd-ing the lowest few eigenvalues and corresponding eigenvectors for the generalized eigenvalue problem based on the conjugate gradient me...

In this article we report on our e orts to test and expand the current state-of-the-art in eigenvalue solvers applied to the eld of nanotechnology. We singled out the nonlinear conjugate gradients method as the blackbone of our e orts for their previous success in predicting the electronic properties of large nanostructures and made a library of three di erent solvers (two recent and one new) t...

This research focuses on nding a large number of eigenvalues and eigenvectors of a sparse symmetric or Hermitian matrix, for example, nding 1000 eigenpairs of a 100,000 100,000 matrix. These eigenvalue problems are challenging because the matrix size is too large for traditional QR based algorithms and the number of desired eigenpairs is too large for most common sparse eigenvalue algorithms. I...

Rigidity of formations in multi-robot systems is important for formation control, localization, and sensor fusion. This work proposes a rigidity maintenance gradient controller for a multi-agent robot team. To develop such a controller, we first provide an alternative characterization of the rigidity matrix and use that to introduce the novel concept of the rigidity eigenvalue. We provide a nec...

This report describes an algorithm for the efficient computation of several extreme eigenvalues and corresponding eigenvectors of a large-scale standard or generalized real symmetric or complex Hermitian eigenvalue problem. The main features are: (i) a new conjugate gradient scheme specifically designed for eigenvalue computation; (ii) the use of the preconditioning as a cheaper alternative to ...

1. Basic gradient estimate; different variations of the gradient estimates; 2. The theorem of Brascamp-Lieb, Barkey-Émery Riemannian geometry, relation of eigenvalue gap with respect to the first Neumann eigenvalue; the Friedlander-Solomayak theorem, 3. The definition of the Laplacian on L space, theorem of Sturm, 4. Theorem of Wang and its possible generalizations. 1 Gradient estimate of the f...

This paper establishes converses to the well-known result: for any vector ũ such that the sine of the angle sin θ(u, ũ) = O( ), we have ρ(ũ) def = ũ∗Aũ ũ∗ũ = λ+O( ), where λ is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and “∗” denotes complex conjugate transpose. It shows that if ρ(ũ) is close to A’s largest eigenvalue, then ũ is close to the corresponding ei...

We numerically analyze the possibility of turning off postsmoothing (relaxation) in geometric multigrid when used as a preconditioner in conjugate gradient linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening Multigrid (SMG) method is provided by the hypre parallel software package. We solve linear systems using two variants (standard and flexible) of the preconditio...