相关论文: Computing and deflating eigenvalues while solving …
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on…
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 give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in…
Connections of the conjugate gradient (CG) method with other methods in computational mathematics are surveyed, including the connections with the conjugate direction method, the subspace optimization method and the quasi-Newton method BFGS…
The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using…
The need to compute small con-eigenvalues and the associated con-eigenvectors of positive-definite Cauchy matrices naturally arises when constructing rational approximations with a (near) optimally small $L^{\infty}$ error. Specifically,…
We examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The five methods examined here range from the simple power iteration method to the more complicated QR iteration method. The…
We propose a new method for computing the eigenvalue decomposition of a dense real normal matrix $A$ through the decomposition of its skew-symmetric part. The method relies on algorithms that are known to be efficiently implemented, such as…
We present a quantum algorithm that analyzes time series data simulated by a quantum differential equation solver. The proposed algorithm is a quantum version of the dynamic mode decomposition algorithm used in diverse fields such as fluid…
The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However,…
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value…
We examine and compare several iterative methods for solving large-scale eigenvalue problems arising from nuclear structure calculations. In particular, we discuss the possibility of using block Lanczos method, a Chebyshev filtering based…
Estimation of actual errors from the residue in iterative solutions is necessary for efficient solution of large problems when their condition number is much larger than one. Such estimators for conjugate gradient algorithms used to solve…
We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient…
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 consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients, or solving a system of coupled differential equations in the Mori…
Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a…
A new inverse iteration algorithm that can be used to compute all the eigenvectors of a real symmetric tri-diagonal matrix on parallel computers is developed. The modified Gram-Schmidt orthogonalization is used in the classical inverse…
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…
Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters…