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- A family of Ehrlich-type accelerated methods with King’s correction for the simultaneous approximation of polynomial complex zerosPublication . Machado, Roselaine Neves; Lopes, Luiz GuerreiroIn this paper, we present a new family of accelerated iterative methods for the simultaneous approximation of simple complex zeros of a polynomial. These simultaneous methods are constructed on the basis of the third order Ehrlich iteration, accelerated by using the so-called Gauss–Seidel approach, and combined with a correction based on King’s family of optimal fourth order iterative methods for solving nonlinear equations. Using King’s correction, the R-order of convergence of the basic accelerated method is increased from at least 3 to at least 6. A numerical example is provided to illustrate the convergence and effectiveness of the proposed family of combined accelerated methods for the simultaneous approximation of simple polynomial zeros.
- Ehrlich-type methods with king’s correction for the simultaneous approximation of polynomial complex zerosPublication . Neves Machado, Roselaine; Lopes, Luiz GuerreiroThere are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King’s family of iterative methods for nonlinear equations. The use of King’s correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros. the advantage of being inherently parallel and avoid the pro cess of deflation, although these iterative methods usually need good initial approximations for all the zeros in order to con verge. In this work, a new family of numerical methods for the si multaneous approximation of simple complex polynomial ze ros is presented. The proposed simultaneous methods are con structed on the basis of the well-known third order Ehrlich– Aberth iteration [1, 6], combined with an iterative correction from the optimal fourth order two-step King’s method for solv ing nonlinear equations [10]