Abstract Time evolution and scattering simulation in phenomenological models are of great interest for testing validating the potential near-term quantum computers to simulate field theories. Here, we one-particle propagation two-particle one-dimensional transverse Ising model 3 4 spatial sites with periodic boundary conditions on a computer. We use Lanczos algorithm obtain all energy levels co...

Some integer factorization algorithms require several vectors in the null space of a sparse m x n matrix over the field GF(2). We modify the Lanczos algorithm to produce a sequence of orthogonal subspaces of GF(2)", each having dimension almost N, where N is the computer word size, by applying the given matrix and its transpose to N binary vectors at once. The resulting algorithm takes about n ...

The first half of this thesis develops an eigenspace approach to computing the basisindependent isotypic decomposition of a vector in a representation of a finite group. The approach takes advantage of well-chosen diagonalizable linear transformations to compute isotypic projections through a series of eigenspace projections, and at its heart is an efficient eigenspace projection method built a...

We introduce a new implementation of time-dependent density-functional theory which allows the entire spectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a single standard ground-state calculation. This method is particularly well suited for large systems and/or large basis sets, such as plane waves or real-space grids. By using a superoperator...

For the solutions of linear systems of equations with un-symmetric coeecient matrices, we has proposed an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a couple two-term procedure that generates Lanczos vectors sc...

Pseudospectra of rectangular matrices vary continuously with the matrix entries, a feature that eigenvalues of these matrices do not have. Some properties of eigenvalues and pseudospectra of rectangular matrices are explored, and an efficient algorithm for the computation of pseudospectra is proposed. Applications are given in (square) eigenvalue computation (Lanczos iteration), square pseudosp...

In 7] and 8], the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n n large symmetric positive deenite matrix were analyzed. In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue. We analyze polynomial algorithms using Krylov information for two error criteria: the randomized error and the randomized ...

The s-step Lanczos method is an attractive alternative to the classical Lanczos method as it enables an O(s) reduction in data movement over a fixed number of iterations. This can significantly improve performance on modern computers. In order for s-step methods to be widely adopted, it is important to better understand their error properties. Although the s-step Lanczos method is equivalent to...

We propose a novel approach called the Local Lanczos Spectral Approximation (LLSA) for identifying all latent members of a local community from very few seed members. To reduce the computation complexity, we first apply a fast heat kernel diffusing to sample a comparatively small subgraph covering almost all possible community members around the seeds. Then starting from a normalized indicator ...