Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.Comment: Submitted to Linear Algebra App

Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples

A family of simultaneous zero-finding methods

AbstractApplying Hansen-Patrick's formula for solving the single equation f(z) = 0 to a suitable function appearing in the classical Weierstrass' method, two one-parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived. It is shown that all the methods of these families have fourth-order of convergence. Some computational aspects of the proposed methods and numerical examples are given

Higher-order iterative methods for approximating zeros of analytic functions

AbstractIterative methods with extremely rapid convergence in floating-point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions (inside a simple smooth closed contour in the complex plane) are presented. The R-order of convergence of the basic total-step and single-step methods, as well as their improvements which use Newton's and Halley's corrections, is given. Some hybrid algorithms that combine the efficiency of ordinary floating-point iterative methods with the accuracy control provided by interval arithmetic are also considered

Some improved inclusion methods for polynomial roots with Weierstrass' corrections

AbstractOne decade ago, the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in [1]. Following Nourein's idea [2], some modifications of this method with the increased convergence are proposed. The acceleration of convergence is attained by using Weierstrass' corrections without additional calculations, which provides a high computational efficiency of the modified methods. It is proved that their R-orders of convergence are asymptotically greater than 3.5. The presented interval methods are realized in circular complex arithmetic

A new higher-order family of inclusion zero-finding methods

AbstractStarting from a suitable fixed point relation, a new one-parameter family of iterative methods for the simultaneous inclusion of complex zeros in circular complex arithmetic is constructed. It is proved that the order of convergence of this family is four. The convergence analysis is performed under computationally verifiable initial conditions. An approach for the construction of accelerated methods with negligible number of additional operations is discussed. To demonstrate convergence properties of the proposed family of methods, two numerical examples results are given

The Root and Bell’s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial, Part I: Correction methods

AbstractIn this paper we consider the error propagation of the Root and Bell’s disk iteration methods enhanced by incorporating a correction term and a choice of a disk inversion formula in the methods, for the simultaneous computation of the zeros of a polynomial. The asymptotic error propagation is proved to be the same in both methods. This result is important considering the fact that these methods are in popular usage in the simultaneous computation of the zeros of a polynomial. The proof of the results herein follows the ideas of [M.S. Petkovic, C. Carstensen, Some improved inclusion methods for polynomial roots with Weierstrass corrections, Comput. Math. Appl. 25 (3) (1993) 59–67]. When the refinement process of correction is efficient, it is this mode of correction we have desired to propose

Um Novo Método Simultâneo de Sexta Ordem Tipo Ehrlich para Zeros Polinomiais Complexos

This paper presents a new iterative method for the simultaneous determination of simple polynomial zeros. The proposed method is obtained from the combination of the third-order Ehrlich iteration with an iterative correction derived from Li's fourth-order method for solving nonlinear equations. The combined method developed has order of convergence six. Some examples are presented to illustrate the convergence and efficiency of the proposed Ehrlich-type method with Li correction for the simultaneous approximation of polynomial zeros.Este artigo apresenta um novo método iterativo para a determinação simultânea de zeros polinomiais simples. O~método proposto é obtido a partir da combinação da iteração de Ehrlich de terceira ordem com uma correção iterativa derivada do método de Li de quarta ordem para a resolução de equações não lineares. O método combinado desenvolvido tem ordem de convergência seis. Alguns exemplos são apresentados para ilustrar a convergência e eficiência do método tipo Ehrlich com correção de Li proposto para a aproximação simultânea de zeros polinomiais