We review numerical methods for computing eigenvalues of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-and-conquer algorithms for tridiagonal matrices. The second part of this survey involves the computation of eig...

In this work, we experimentally evaluate and verify model assumptions for our recently proposed algorithm (practical ReProCS) for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt := St + Lt, when Lt lies in a slowly changing low-dimensional subspace. A key application where this problem occurs is in video layering where the goal is to se...

Methods for solving linear systems of equations are at the heart of many computational science applications. Examples include science domains such as astrophysics, biology, chemistry, fusion energy, power system networks, and structural engineering, employing a diverse set of modeling approaches, such as computational fluid dynamics, finite element modeling, and linear programming. In this repo...

For a compact metric space X , consider a linear subspace A of C{X) containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if A separates points, then the closure of A under both minima and maxima is dense in C{X). By the Hahn-Banach Theorem, if A separates probability measures, A is dense in C{X). It is shown that if A separates points from probability mea...

This paper describes a sequential solution to dense non-rigid structure from motion that recovers the camera motion and 3D shape of non-rigid objects by processing a monocular image sequence as the data arrives. We propose to model the time-varying shape with a probabilistic linear subspace of mode shapes obtained from continuum mechanics. To efficiently encode the deformations of dense 3D shap...

this study develops and analyzes preconditioned krylov subspace methods to solve linear systemsarising from discretization of the time-independent space-fractional models. first, we apply shifted grunwald formulas to obtain a stable finite difference approximation to fractional advection-diffusion equations. then, we employee two preconditioned iterative methods, namely, the preconditioned gene...

Given a pair of topological vector spaces $$X,\; Y$$ where X is proper linear subspace Y it examined whether $$Y\setminus X$$ residual in (topological genericity), contains dense except 0 (algebraic genericity) and closed infinite dimensional (spaceability). In the present paper are either sequence or analytic functions on unit disc regarded as via identification function with its Taylor coeffi...

We present a model in which a gauge symmetry of a field theory is intrinsic in the geometry of an extended space time itself. A consequence is that the dimension of our space time is restricted through the BRS cohomology. If the Hilbert space is a dense subspace of the space of all square integrable C functions, the BRS cohomology classes are nontrivial only when the dimension is two or four. p...

In this paper, parallel extensions of a complete symmetric eigensolver, proposed by Yau and Lu in ' l 993' are pre' sented. First, an overview of this invariant subspace decomposition method for dense symmetric matrices is g iven, fo l lowed by numer ical resul ts . Then' works are exposed in progress on distributed-memory implementation. The algorithm's heavy reliance on matrix-matrix multipli...

Abstract. We show that the set of nonnegative equilibrium-like states, namely, like (yd, 0) of the semilinear vibrating string that can be reached from any non-zero initial state (y0, y1) ∈ H 0 (0, 1) × L(0, 1), by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace L(0, 1) × {0} of L(0, 1) × H−1(0, 1). Our main results deal with nonlinear terms wh...