Counting eigenvalues in domains of the complex field

The main objective of this paper is to study the sensitivity of eigenvalues in their computational domain under perturbations, and to provide a solid intuition with some numerical example as well as to represent them in graphically. The sensitivity of eigenvalues, estimated by the condition number of the matrix of eigenvectors has been discussed with some numerical example. Here, we have also demonstrated, other approaches imposing some structures on the complex eigenvalues, how this structure affects the perturbed eigenvalues as well as what kind of paths do they follow in the complex plane.

2020

A quadrature rule of a measure µ on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against µ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.

Symmetry

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

Journal of Applied Mathematics and Physics

In computing the desired complex eigenpair of a matrix, we show that by adding Ruhe's normalization to the matrix pencil, we obtain a square nonlinear system of equations. In this work, we show that the corresponding Jacobian is non-singular at the root and that with an appropriately chosen initial guesses, Ruhe's normalization with a fixed complex vector not only converges quadratically but also faster than the earlier Algorithms for the numerical computation of the complex eigenpair of a matrix. The mathematical tools used in this work are Newton and Gauss-Newton's methods.

Journal of Computational and Applied Mathematics, 2003

We consider the quadrature method developed by Kravanja et al. (BIT 39 (4) (1999) 646) for computing all the zeros of an analytic function that lie inside the unit circle. A new perturbation result for generalized eigenvalue problems allows us to obtain a detailed upper bound for the error between the zeros and their approximations. To the best of our knowledge, it is the ÿrst time that such an error estimate is presented for any quadrature method for computing zeros of analytic functions. Numerical experiments illustrate our results.

Applied Mathematics and Computation, 2005

In this study, the collocation method of the weight residual methods are investigated for the approximate computation of higher Sturm-Liouville eigenvalues, where trial solution is accepted as the Chebyshev series. The obtained approximate eigenvalues are compared with the previous computational results [ANZIAM

Numerical Linear Algebra with Applications, 2016

Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an exact count is not necessary and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure.

We consider the quadrature method developed by Kravanja, Sakurai and Van Barel (BIT 39 (1999), nr. 4, 646-682) for computing all the zeros of an analytic function that lie inside the unit circle. A new perturbation result for generalized eigenvalue problems allows us to obtain a detailed upper bound for the error between the zeros and their approximations. To the best of our knowledge, it is the first time that such an error estimate is presented for any quadrature method for computing zeros of analytic functions. Numerical experiments illustrate our results.

IEEE Transactions on Magnetics, 2009

Infinite series involving associated Legendre functions of general degree and order are used to describe the scalar potential in the vicinity of a cubic corner. The method of analysis closely follows methods used to describe the magnetic field in the gap of a ring head for magnetic recording and the degree of the Legendre functions can be approximated when the determinant of the derived matrix vanishes. We solve with both Dirichlet and Neumann boundary conditions, but study symmetric and antisymmetric solutions separately. Other geometries considered are the two-dimensional straight edge and the quarter-plane. We also discuss the degeneracy of higher order eigenvalues.

Applied Mathematics and Computation, 2010

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 numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c