Related papers: A comparison of eigenvalue condition numbers for m…
In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…
For standard eigenvalue problems, a closed-form expression for the condition numbers of a multiple eigenvalue is known. In particular, they are uniformly 1 in the Hermitian case, and generally take different values in the non-Hermitian…
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices…
The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with…
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the…
We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…
We compute the exact value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite…
An apriori bound for the condition number associated to each of the following problems is given: general linear equation solving, minimum squares, non-symmetric eigenvalue problems, solving univariate polynomials, solving systems of…
This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…
This article study the average conditioning for a random underdetermined polynomial system. The expected value of the moments of the condition number are compared to the moments of the condition number of random matrices. An expression for…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
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…
We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theory fail to predict the observed accuracy of computed solutions. To motivate our ideas, we…
This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate…
In this paper, we establish bounds for the eigenvalues of matrix polynomials. Specifically, we find different generalizations of the Enestrom-Kakeya Theorem for matrix polynomials.
In this article, we study the singular case of an homogeneous generalized discrete time system with given initial conditions. We consider the matrix pencil singular and provide necessary and sufficient conditions for existence and…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…