Related papers: Verified eigenvalue and eigenvector computations u…
We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolver, can reduce a given eigenproblem into a generalized…
Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting…
The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of…
Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for…
This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that…
We revisit a classical problem in numerical linear algebra: given an $k$-dimensional subspace $\mathcal{Q}$ that approximates the leading eigenspace of an $n\times n$ positive semi-definite matrix $A$, the goal is to extract high-accuracy…
This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed…
Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair…
Under the hypothesis that the deviations of the desired eigenvectors of the matrix $A$ from the underlying subspace tend to zero, the Ritz vectors may not converge and have poor or little accuracy. This phenomenon is not unusual and…
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace $\X$, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$…
We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms…
This paper introduces a method for computing eigenvalues and eigenvectors of a generalized Hermitian, matrix eigenvalue problem. The work is focused on large scale eigenvalue problems, where the application of a direct inverse is out of…
The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The…
Recently, a non-classical eigenvalue solver, called RIM, was proposed to compute (all) eigenvalues in a region on the complex plane. Without solving any eigenvalue problem, it tests if a region contains eigenvalues using an approximate…
The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator $A$. If the trial subspace is $A$-invariant, the Ritz…
From characterizing the speed of a thermal system's response to computing natural modes of vibration, eigenvalue analysis is ubiquitous in engineering. In spite of this, eigenvalue problems have received relatively little treatment compared…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
We present a new algorithm for solving an eigenvalue problem for a real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in $O(n^{2})$…
A new algorithm, denoted by RSRR, is presented for solving large-scale nonlinear eigenvalue problems (NEPs) with a focus on improving the robustness and reliability of the solution, which is a challenging task in computational science and…
We address the problem of computing the eigenvalue backward error of the Rosenbrock system matrix under various types of block perturbations. We establish computable formulas for these backward errors using a class of minimization problems…