A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

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

Algebraic methods for dynamic systems

Algebraic methods for application to dynamic control system

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

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

Iteration functions re-visited

Two classes of Iteration Functions (IFs) are derived in this paper. The first (one-point IFs) was originally derived by Joseph Traub using a different approach to ours (simultaneous IFs). The second is new and is demonstrably shown to be more informationally efficient than the first. These IFs apply to polynomials with arbitrary complex coefficients and zeros, which can also be multiple