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The computation of the dominant eigenpair for symmetric positive semidefinite matrices is fundamental in numerical optimization. This work shifts the paradigm from the classical Rayleigh quotient to an unconstrained difference formulation,…
In this paper we present an algorithm to find the discrete Lagrangian for an autonomous recurrence relation of arbitrary even order $2k$ with $k>1$. The method is based on the existence of a set of differential operators called annihilation…
We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric…
We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ…
In this paper a novel numerical approximation of parametric eigenvalue problems is presented. We motivate our study with the analysis of a POD reduced order model for a simple one dimensional example. In particular, we introduce a new…
The main of this work is to use the unit lower triangular matrices for solving inverse eigenvalue problem of nonnegative matrices and present the easier method to solve this problem.
We prove that the point process of the eigenvalues of real or complex non-Hermitian matrices $X$ with independent, identically distributed entries is hyperuniform: the variance of the number of eigenvalues in a subdomain $\Omega$ of the…
We propose a new iterative algorithm for generating a subset of eigenvalues and eigenvectors of large matrices which generalizes the method of optimal relaxations. We also give convergence criteria for the iterative process, investigate its…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
We analyze the method for calculation of properties of non-relativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors…
A well known numerical task is the inversion of large symmetric tridiagonal Toeplitz matrices, i.e., matrices whose entries equal $a$ on the diagonal and $b$ on the extra diagonals ($a, b\in \mathbb R$). The inverses of such matrices are…
Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these…
This paper analyzes the convergence of fixed-point iterations of the form u = f(u) and the properties of the inverse of the related pentadiagonal matrices, associated with the fourth-order nonlinear beam equation. This nonlinear problem is…
We provide two new methods for computing lower bounds of eigenvalues of symmetric elliptic second-order differential operators with mixed boundary conditions of Dirichlet, Neumann, and Robin type. The methods generalize ideas of Weinstein's…
A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a…
A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert…
The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this…
We prove that few largest (and most important) eigenvalues of random symmetric matrices of various kinds are very strongly concentrated. This strong concentration enables us to compute the means of these eigenvalues with high precision. Our…
We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…