Related papers: Counting eigenvalues in domains of the complex fie…
We describe algorithms for computing eigenpairs (eigenvalue--eigenvector) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not…
Using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator. This relation can be used in practice to compute eigenvalue invariants of the covariance…
In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used…
In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting…
Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. In this paper, we consider the following problem: given two parametric real symmetric matrices and an…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted…
We present a method of cones for rigorous estimations of eigenvectors, eigenspaces and eigenvalues of a matrix. The key notion is the cone-domination and is inspired by ideas from hyperbolic dynamical systems. We present theorems which…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…
We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such…
Convergence properties of binary stationary subdivision schemes for curves have been analyzed using the techniques of z-transforms and eigenanalysis. Eigenanalysis provides a way to determine derivative continuity at specific points based…
We consider path-connected sets of matrices and the induced paths between eigenvalues. We discuss the equivalence relation generated by these paths, and how it relates to the presence of higher multiplicity eigenvalues realized by the set.…
The polynomial eigenvalue problem arises in many applications and has received a great deal of attention over the last decade. The use of root-finding methods to solve the polynomial eigenvalue problem dates back to the work of…
The Google matrix is a positive, column-stochastic matrix that is used to compute the pagerank of all the web pages on the Internet: the eigenvector corresponding to the eigenvalue 1 is the pagerank vector. Due to its huge dimension, of the…
Contour integral algorithms seek to compute a small number of eigenvalues located within a bounded region of the complex plane. These methods can be applied to both linear and nonlinear matrix eigenvalue problems. In the latter case, the…
The bootstrap is a technique recently developed to get energy eigenvalues of bound states and correlation functions. There are three crucial steps - recursive equations, positivity constraints, search space. We calculate recursive equations…
Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the…
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…
For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These…