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Firstly, we define an (r , r , ⋯ , r) -slice tensor, and then solve an (r , r , ⋯ , r) -slice tensor equation by direct method. Secondly, we propose a new splitting of the coefficient tensor to develop a slice tensor splitting method, which is different from the tensor splitting methods in [1,2]. At each iteration of this method, an (r , r , ⋯ , r) -slice tensor equation is solved. Finally, we prove the local convergence of the proposed method under appropriate conditions, and give numerical experiments to illustrate the effectiveness of this method. [ABSTRACT FROM AUTHOR]

Levenberg–Marquardt methods (LMMs) are the most typical algorithms for solving nonlinear equations F (x) = 0 , where F : R n → R m is a continuously differentiable function. They sequentially solve subproblems represented as squared residual of the Newton equations with the L 2 regularization to determine the search direction. However, since the subproblems of the LMMs are usually reduced to linear equations with n variables, it takes much time to solve them when m ≪ n. In this paper, we propose a new LMM which generalizes the L 2 regularization of the subproblems of the ordinary LMMs. By virtue of the generalization, we can choose a suitable regularization term for each given problem. Moreover, we show that a sequence generated by the proposed method converges globally and quadratically under some reasonable assumptions. Finally, we conduct numerical experiments to confirm that the proposed method performs better than the existing LMMs for some problems that satisfy m ≪ n. [ABSTRACT FROM AUTHOR]

In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N , the N th method of this family has the order of convergence 2 N + 1. In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family. [ABSTRACT FROM AUTHOR]

Abstract This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used. [ABSTRACT FROM AUTHOR]

In this paper we revise the proofs of the results obtained in “Convergence radius of Osada’s method under Hölder continuous condition” [4], because the remainder of the Taylor’s expansion used for the obtainment of the local convergence radius is not correct. So we perform the complete study in order to modify the equation for getting the local convergence radius, the uniqueness radius and the error bounds. Moreover a dynamical study for the third order Osada’s method is also developed. [ABSTRACT FROM AUTHOR]

In this paper, we present two new derivative-free families of Chebyshev–Halley type methods for solving nonlinear equations numerically. Both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence. Furthermore, accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. The self-accelerating parameter is estimated from the current and previous iteration. This self-accelerating parameter is calculated using Newton’s interpolation polynomial of third and fourth degrees. Consequently, the R -orders of convergence are increased from 4 to 6 and 8 to 12, respectively, without any additional functional evaluation. The results require high-order derivatives reaching up to the eighth derivative. That is why we also present an alternative approach using only the first or at most the fourth derivative. We also obtain the radius of convergence and computable error bounds on the distances involved. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we consider a concrete variety of real life problems coming from different disciplines e.g., Kepler’s equation of motion, Planck’s radiation law problem, fractional conversion in a chemical reactor, the trajectory of an electron in the air gap between two parallel plates, Van der Waal’s equation which explains the behavior of a real gas by introducing in the ideal gas equations, in order to check the applicability and effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

By means of a nice idea, a Newton–Kurchatov type iterative process is constructed for solving nonlinear equations in Banach spaces. We analyze the local convergence of this iterative process. This study have an important and novel feature, since it is applicable to non-differentiable operators. So far, most of the local convergence results considered by other authors may apply only to differentiable operators due to the conditions that are required on the solution of the nonlinear equation. [ABSTRACT FROM AUTHOR]

In this article, we propose a quasi-Newton algorithm to solve a matrix polynomial equation, which can be seen as a generalization of the algorithm of the same type to solve the matrix quadratic equation proposed in Macías et al. (2016). The proposed algorithm reduces the computational cost of the Newton–Schur method traditionally used to solve this type of equations. We show that this algorithm is local and even quadratically convergent. Finally, we present numerical experiments that ratify the theoretical results developed. [ABSTRACT FROM AUTHOR]

The principle aim of this manuscript is to present a new highly efficient and optimal eighth-order family of iterative methods to solve nonlinear equations in the case of simple roots. The derivation of this scheme is based on weight function and rational approximation approaches. The proposed family requires only four functional evaluations (viz. f ( x n ) f ′( x n ) f ( y n ) and f ( z n )) per iteration. Therefore, the proposed family is optimal in the sense of Kung–Traub hypotheses. In addition, we given a theorem which describing the order of convergence of the proposed family. Moreover, we present a local convergence analysis using hypotheses only on the first-order derivative, since in our preceding theorem we used hypotheses on higher-order derivatives that do not appear in these methods. In this way, we expand the applicability of these methods even further. Furthermore, a variety of nonlinear equations is considered for the numerical experiments. It is observed from the numerical experiments that our proposed methods perform better than the existing optimal methods of same order, when the accuracy is checked in multi precision digits. [ABSTRACT FROM AUTHOR]

In this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in Banach spaces that allows us to improve the convergence domains. Specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition. For some values of the parameter, these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We have considered a number of numerical examples including a nonlinear Hammerstein equation and computed the radii of the convergence balls. It is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods. The global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials. [ABSTRACT FROM AUTHOR]

The semilocal and local convergence in Banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the Fréchet derivative satisfies the Hölder condition. The Hölder condition generalizes the Lipschtiz condition. The importance of our work lies in the fact that many examples are available which fail to satisfy the Lipschtiz condition but satisfy the Hölder condition. The existence and uniqueness theorem is established with error bounds for the solution. The convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained. These examples include nonlinear Hammerstein and Fredholm integral equations and a boundary value problem. It is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the local convergence of a uniparametric family of secant-like methods for solving nonlinear operators in Banach spaces. This new study has an important and novel feature, since it is applicable to non-differential operators. So far, the results of local convergence usually considered can be only applied to differentiable operators. [ABSTRACT FROM AUTHOR]

Kyurkchiev and Andreev (1985) constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer N ≥ 1, the N th method of this family has the order of convergence N + 1 . Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates. [ABSTRACT FROM AUTHOR]

A trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper. In the current iteration, the trial step providing sufficient descent is generated by solving a corresponding trust-region subproblem. Then, the step size is decided by backtracking line search together with filter technique to obtain the next iteration point. The advantage of this method is that resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be avoided and hence the expensive computation can be lessened. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. Second order correction steps are introduced in the proposed algorithm to overcome Maratos effect. Convergence analysis shows that fast local convergence can be achieved under some mild assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

We present a new semilocal convergence analysis for Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies such as (Amat and Busquier, 2003; Amat et al., in press; Argyros and Hilout, 2012; Argyros et al., 2014; Argyros and Magreñán, 2014, 2015; Dennis, 1971; Ezquerro et al., 2000; Ortega and Rheinboldt, 1970; Potra and Pták, 1984; Schmidt, 1978). Numerical examples are also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]

We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

We present a unified convergence analysis of Inexact Newton-like methods in order to approximate a locally unique solution of a nonlinear operator equation containing a nondifferentiable term in a Banach space setting. The convergence conditions are more general and the error analysis more precise than in earlier studies such as (Argyros, 2007; Cătinaş, 2005; Cătinaş, 1994; Chen and Yamamoto, 1989; Dennis, 1968; Hernández and Romero, 2005; Potra and Pták, 1984; Rheinboldt, 1977). Special cases of our results can be used to find zeros of derivatives. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

A new predictor–corrector iterative procedure, that combines Newton’s method as predictor scheme and a fifth-order iterative method as a corrector, is designed for solving nonlinear equations in Banach spaces. We analyze the local order of convergence and the regions of accessibility of the new method comparing it with Newton’s method, both theoretical and numerically. [ABSTRACT FROM AUTHOR]

Abstract: A local convergence analysis for an efficient Secant-type method (STM) for solving nonlinear equations is given using both the Lipschitz continuous and center-Lipshchitz continuous divided differences of order one. An estimate of the radius of the convergence ball of (STM) is provided, the error estimate matching its convergence order is established. Numerical examples validating the theoretical results are also provided in the concluding section of this study. [Copyright &y& Elsevier]

Abstract: Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples. [Copyright &y& Elsevier]

Abstract: In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem. [Copyright &y& Elsevier]

Abstract: We are concerned with the problem of approximating a locally unique solution of a generalized equation using a Newton-like method in a Banach space setting. Using some ideas introduced by us in [I.K. Argyros, A unifying local–semilocal convergence analysis and applications for Newton-like methods, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, USA, 2005] for nonlinear equations, we provide a local analysis leading to the super–linear convergence of the method. Our approach has the following advantages (under the same computational cost) over earlier work [M.H. Geoffroy, A. Piétrus, Local convergence of some iterative methods for solving generalized equations, J. Math. Anal. Appl. 290 (2004) 497–505]: finer error on the distances involved, and a larger radius of convergence leading to a wider choice of initial guesses and fewer computations to arrive at a desired error tolerance. [Copyright &y& Elsevier]

Abstract: In this note, we are concerned with the further study for a significant improvement on Newton’s iterative method proposed by Wu [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78]. We present a natural extension and development of the improvement on Newton’s method for system of nonlinear equation. The convergence is presented and the numerical results are given to show the efficiency of the extended method for system of nonlinear equations. [Copyright &y& Elsevier]

Abstract: This paper presents the new convergence analysis for the Newton-like method proposed by [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78.] Compared with the original version of the convergence analysis, the restriction imposed on f″(x) is removed thoroughly. In order to guarantee the quadratic convergence of the Newton-like method, it is suffices to suppose that f′(x ∗)≠0 and f′(x) is local Lipschitz near x ∗, where x ∗ is a solution of nonlinear equation f(x)=0. Moreover, some comments are given with examples for the Newton-like method, in comparison with Newton’s method. It can be concluded that the new algorithm is more feasible, effective than Newton’s method. In particularly, when it happens that, the derivative of the function f(x) at an iterate is singular or almost singular, the Newton-like method is vast superior to the classical Newton method. The numerical results of the paper strongly support the conclusion. [Copyright &y& Elsevier]

This paper aims to introduce an algorithm for solving large-scale regularized least-squares problems subject to quadratic inequality constraints. The algorithm recasts the least-squares problem in terms of a parameterized eigenvalue problem. Only two of the smallest eigenpairs of the parameterized problem need to be computed to find the optimal solution from the parameterized eigenvector. Safeguards are introduced to adjust the parameter and a two-point interpolating scheme is developed for updating it. A local convergence theory for this algorithm is presented. It is shown that this algorithm is superlinearly convergent. [Copyright &y& Elsevier]