In this paper, we present new error bounds for the Lanczos method and the shift-andinvert Lanczos method for computing e−τAv for a large sparse symmetric positive semidefinite matrix A. Compared with the existing error analysis for these methods, our bounds relate the convergence to the condition numbers of the matrix that generates the Krylov subspace. In particular, we show that the Lanczos m...

Using the work by Bampi and Caviglia, we write the Weyl-Lanczos equations as an exterior differential system. Using Janet-Riquier theory, we compute the Cartan characters for all spacetimes with a diagonal metric and for the plane wave spacetime since all spacetimes have a plane wave limit. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet-Riq...

While discussing the convergence of the Lanczos method, Trefethen and Bau observed a relationship with electric charge distributions, and claimed that the Lanczos iteration tends to converge to eigenvalues in regions of \too little charge" for an equilibrium distribution. Recently, Kuijlaars found a theoretical justiication for this phenomenon by considering the Lanczos method applied to a suit...

This paper considers the problem of computing charge densities in a density functional theory (DFT) framework. In contrast to traditional, diagonalization-based, methods, we utilize a technique which exploits a Lanczos basis, without explicit reference to individual eigenvectors. The key ingredient of this new approach is a partial reorthogonalization strategy whose goal is to ensure a good lev...

We study quasistationary physical process for black holes within the context of Lanczos-Lovelock gravity. We show that the Wald entropy of the stationary black holes in Lanczos-Lovelock gravity monotonically increases for quasistationary physical processes in which the horizon is perturbed by the accretion of positive energy matter and the black hole ultimately settles down to a stationary stat...

Recent years have witnessed the popularity of using rank minimization as a regularizer for various signal processing and machine learning problems. As rank minimization problems are often converted to nuclear norm minimization (NNM) problems, they have to be solved iteratively and each iteration requires computing a singular value decomposition (SVD). Therefore, their solution suffers from the ...

Minimum mode following algorithms are widely used for saddle point searching in chemical and material systems. Common to these algorithms is a component to find the minimum curvature mode of the second derivative, or Hessian matrix. Several methods, including Lanczos, dimer, Rayleigh-Ritz minimization, shifted power iteration, and locally optimal block preconditioned conjugate gradient, have be...

The key problem of the theory of exterior differential systems (EDS) is to decide whether or not a system is in involution. The special case of EDSs generated by one-forms (Pfaffian systems) can be adequately illustrated by a 2-dimensional example. In 4 dimensions two such problems arise in a natural way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known from the work of Ba...