19 papers · ranked by Valyu relevance
Haoze He, Daniel Kressner, Bor Plestenjak
It is well known that a family of ${n}\times{n}$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to…
Brendan Gavin, Agnieszka Międlar, Eric Polizzi
The linear FEAST algorithm is a method for solving linear eigenvalue problems. It uses complex contour integration to calculate the eigenvectors whose eigenvalues that are located inside some user-defined region in the complex plane. This makes it possible to parallelize the process of solving eigenvalue problems by…
Benyamin Ghojogh, Fakhri Karray, Mark Crowley
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also…
Thomas J. Anastasio, Andrea K. Barreiro, Jared C. Bronski
We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved…
Patrick Dumond, Natalie Baddour
An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for…
Marta M. Betcke, Heinrich Voss
In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which…
Henrik Eisenmann, Yuji Nakatsukasa
We present a new approach to compute selected eigenvalues and eigenvectors of the twoparameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems…
Pavel Osinenko, Grigory Devadze, Stefan Streif
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or Jordan normal forms. Unfortunately, numerical algorithms computing eigenvectors are…
P. Cheema, M. M. Alamdari, G. A. Vio
Least Eigenvalue of an Unknown Mass Matrix Authors: ['P. Cheema' 'M. M. Alamdari' 'G. A. Vio'] In the field of structural engineering analysis, a common requirement is to calculate the modal frequencies of a structure that has undergone an update, either naturally (such as from material degradation), or due to manmade…
Juán Tolosa
Starting from a mistake done by a student, we discover an unexpected method of finding both eigenvectors for a 2×2 matrix with distinct eigenvalues in a single computation. We discuss a connection with the Cayley-Hamilton theorem, and show the corresponding generalization for a 3 × 3 matrix. The arguments should be…
John W. Pearson, Jennifer Pestana, David J. Silvester
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying…
Eric Hermes, Khachik Sargsyan, Habib Najm, Judit Zádor
We present a new algorithm for the optimization of molecular structures to saddle points on the potential energy surface using a redundant internal coordinate system. This algorithm automates the procedure of defining the internal coordinate system, including the handling of linear bending angles, e.g. through the…
Lewi Stone
In his theoretical work of the 70’s, Robert May introduced a Random Matrix Theory (RMT) approach for studying the stability of large complex biological systems. Unlike the established paradigm, May demonstrated that complexity leads to instability in generic models of biological networks. The RMT approach has since…
David Thompson, Johan Gielis
Our understanding of quantum phenomena often begins with simple particle-in-a-box style problems, the solutions of which introduce the student to foundational quantum concepts such as degeneracy and quantization. Simple model geometries of confinement afford analytic solutions, which are readily derivable, easily…
Sucheta Ghosh, Shankar Prasad Bhattacharyya
The quantum states of hydrogen atom in one dimension can be obtained by a careful application of the well-known Frobenius method. The exercise is highly educative and brings to focus the subtle aspects of quantum mechanics. The allowed states turn out to be only of odd parity and non-degenerate, having energy given by…
Arkajit Mandal, Michael Taylor, Pengfei Huo
We provide a simple and intuitive theory to explain how coupling a molecule to an optical cavity can modify ground-state chemical reactivity by exploiting intrinsic quantum behaviors of light-matter interactions. Using the recently developed Polarized Fock States representation, we demonstrate that the change of the…
Junya Watanabe
Quantification of the magnitude of trait covariation plays a pivotal role in the study of phenotypic evolution, for which statistics based on dispersion of eigenvalues of a covariance or correlation matrix—eigenvalue dispersion indices—are commonly used. This study remedies major issues over the use of these…
John Herbert, Aniket Mandal
A fundamental tenet of quantum mechanics is that properties should be independent of representation. In self-consistent field methods such as density functional theory, this manifests as a requirement that properties be invariant with respect to unitary transformations of the occupied molecular orbitals and…
Authors not listed
This article is the second in a two-part tutorial review on electronic spin-dependent dynamics. In Part I, we presented the fundamental theory within the adiabatic Born– Huang framework that describes the interaction between nuclear motion and the elec- tronic (spin and spatial) degrees of freedom. In particular, we…