On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously

In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.

Real-world applications depend heavily on the fixed-point solution. In this paper, we have suggested an effective iterative method for fixed points. We have first given the approximate order of convergence for this method using Taylor’s series. The radii of convergence balls for this method can then be calculated using a local convergence theorem that we then present. The semilocal convergence theorem, which determines the starting point’s accuracy, is then presented. We have created some technical lemmas and theorems to serve this purpose. In contrast to an earlier study using the same type of method for nonlinear equations, we have not used the convergence conditions on higher-order Frechet derivatives in our study of convergence. Finally, some numerical examples are provided to support the theoretical findings we made. This highlights the uniqueness of this study.

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.

In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x*, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

In this study we are concerned with the problem of approximating a locally unique solution of an equation on a Banach space. A semilocal convergence theorem is given for the Super-Halley method in Banach spaces. Earlier results have shown that the order of convergence is four for a certain class of operators [4], [5], [8]. These results were not given in affine invariant form, and made use of a real quadratic majorizing polynomial. Here, we provide our results in affine invariant form showing that the order of convergence is at least four. In cases that it is exactly four the rate of convergence is improved. We achieve these results by using a cubic majorizing polynomial. Some numerical examples are given to show that our error bounds are better than earlier ones.

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iterative method for solving large sparse non-Hermitian positive definite system of linear equations. In this paper, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equations, and the HSS method to approximately solve the Newton equation. Local and semilocal convergence theorems are proved under proper conditions. Also, we present the global multi-step modified Newton-HSS methods with a backtracking strategy and analyse its global convergence. Finally, numerical results are given to confirm the feasibility and effectiveness of our method.

In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng’s method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković, and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52).

In this paper, the author studies a Broyden-like method for solving nonlinear equations with nondifferentiable terms, which uses as updating matrices, approximations for Jacobian matrices of differentiable terms. Local and semilocal convergence theorems are proved. The results generalize those of Broyden, Dennis and More.

In the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to its computable initial condition. The obtained theorems improve and complement all existing such kind of convergence results about this method. At the end of the paper, we provide three numerical examples to show the applicability of our semilocal theorem to some physics problems. Within the examples, we propose a new algorithm for the experimental study of the dynamics of the simultaneous methods and compare the convergence and dynamical behaviors of the modified and the classical Weierstrass methods.

In this paper, we establish a general theorem for iteration functions in a cone normed space over \({{\mathbb {R}}}^n\). Using this theorem together with a general convergence theorem of Proinov (J Complex 33:118–144, 2016), we obtain a local convergence theorem with a priori and a posteriori error estimates as well as a theorem under computationally verifiable initial conditions for the Schroder’s iterative method considered as a method for simultaneous computation of polynomial zeros of unknown multiplicity. Numerical examples which demonstrate the convergence properties of the proposed method are also provided.

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.