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From the secant method and using symmetric divided differences, we construct a uniparametric family of secant-type iterative methods with quadratic convergence and better computational efficiency than the secant method. In addition, we experimentally analyze the accessibility of the family from a dynamic point of view and study the local convergence. [ABSTRACT FROM AUTHOR]

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate. [ABSTRACT FROM AUTHOR]

Abstract In 1991, T. Sakurai, T. Torii and H. Sugiura presented a fourth-order iterative algorithm for finding all zeros of a polynomial simultaneously. In this paper, we provide a detailed convergence analysis (local and semilocal) of this method. The new results improve and complement existing results due to Petković et al. (2003) and Petković (2008). Numerical examples are given to show the applicability of our semilocal convergence results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present an inverse iteration algorithm, to find the H-spectral radius and the associated positive eigenvector of a nonnegative weakly irreducible tensor, which always preserve the positivity of approximate eigenvectors. The local quadratical convergence of the proposed algorithm is established based on a basic result of the Newton's method for solving nonlinear equations. Some numerical examples illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract We have very few literature on fixed point iterative methods for solving nonlinear equations. We consider the Stirling method given by Rall (1969). Based on Stirling method, in this paper, we propose a third order Stirling-like method for finding fixed point for nonlinear equations in Banach spaces. We study the local and semilocal convergence of this method for finding the fixed points of nonlinear equations in Banach spaces. The convergence is established under the assumption that the first order Fréchet derivative satisfies the Lipschitz continuity condition. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We consider the numerical examples for local and semilocal convergence case and calculate the existence and uniqueness region of convergence balls even if we fail to apply the results in Rall (1975), Argyros (1995) and Parhi and Gupta (2010)[11,12] due to the F is not contraction on Ω. [ABSTRACT FROM AUTHOR]

Abstract In this paper we establish the construction of a family of free derivative of point to point iterative processes, with quadratic convergence, from two known data in each previous iteration. Besides, we study the accessibility of this family by means of the basins of attraction and the convergence balls. We provide a local convergence analysis for the family of iterative processes free of derivatives, when the operator F is not necessarily Fréchet differentiable. The sufficient convergence conditions are weaker and more flexible than in earlier studies. An application is provided involving mixed Hammerstein nonlinear integral equation with application in real world problems. Finally, we show also a dynamical study and the convergence regions of some members of the family. [ABSTRACT FROM AUTHOR]

This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. These types of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these types of methods (Bermúdez et al., 2012; Petkovic et al., 2013; Traub, 1964). We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it. [ABSTRACT FROM AUTHOR]

The convergence is developed for a large class of Chebyshev-two point-like methods for solving Banach space valued equations. Both the local as well as the semi-local convergence is provided for these methods under general conditions allowing the solution of nondifferentiable equations. Specializations of our methods improve earlier results. Numerical experiments display the superiority of our results. [ABSTRACT FROM AUTHOR]

Solving equations of the form H ( x ) = 0 is usually done by applying iterative methods. The main interest of this paper is to improve the domain of starting points for Steffensen’s method. In general, the accessibility of iterative methods that use divided differences in their algorithms is reduced, since there are difficulties in the choice of starting points to guarantee the convergence of the methods. In particular, by using a decomposition of the operator H and applying a special type of iterative methods, which combine two iterative schemes in the algorithms, we can improve the accessibility of Steffensen’s method. Moreover, we analyze the local convergence of the new iterative method proposed in two cases: when H is differentiable and H is non-differentiable. The dynamical properties show that the method also improves the region of accessibility of Steffensen’s method for non-differentiable operators. So, we present an alternative for the non-applicability of Newton’s method to non-differentiable operators that improves the accessibility of Steffensen’s method. The theoretical results are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

Secant-type iterative methods are usually used to solve nonlinear systems of equations where the operator involved is nondifferentiable or its derivative is costly. From a uniparametric family of secant-type methods, that is reduced to the secant method and Newton's method for some particular values of the parameter involved, with operational cost similar to that of the secant and Newton methods and superlinear convergence, we construct a biparametric family of iterative methods with quadratic convergence, study their local convergence, their efficiency and, based on the dynamics of the methods, their accessibility. [ABSTRACT FROM AUTHOR]

In 2011, Petković, Rančić and Milošević (Petković et al., 2011) introduced and studied a new fourth-order iterative method for finding all zeros of a polynomial simultaneously. They obtained a semilocal convergence theorem for their method with computationally verifiable initial conditions, which is of practical importance. In this paper, we provide new local as well as semilocal convergence results for this method over an algebraically closed normed field. Our semilocal results improve and complement the result of Petković, Rančić and Milošević in several directions. The main advantage of the new semilocal results are: weaker sufficient convergence conditions, computationally verifiable a posteriori error estimates, and computationally verifiable sufficient conditions for all zeros of a polynomial to be simple. Furthermore, several numerical examples are provided to show some practical applications of our semilocal results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of solving constrained systems of nonlinear equations. We propose an algorithm based on a combination of Newton and conditional gradient methods, and establish its local convergence analysis. Our analysis is set up by using a majorant condition technique, allowing us to prove, in a unified way, convergence results for two large families of nonlinear functions. The first one includes functions whose derivative satisfies a Hölder-like condition, and the second one consists of a substantial subclass of analytic functions. Some preliminary numerical experiments are reported. [ABSTRACT FROM AUTHOR]

This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefinite programming subproblem at each iteration. The trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. We prove the global convergence properties of the algorithm under mild assumptions. We also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid Maratos effect. We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear. Finally, some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in C O M P l e i b are given. [ABSTRACT FROM AUTHOR]

We study the local convergence of Chebyshev–Halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation. In Sharma (2015) (see Theorem 1, p. 121) the convergence of the method was shown under hypotheses reaching up to the third derivative. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamics of these methods are also studied. Finally, numerical examples examining dynamical planes are also provided in this study to solve equations in cases where earlier studies cannot apply. [ABSTRACT FROM AUTHOR]

For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H = 1 2 ( A + A T ) is a centrosymmetric matrix and the skew-symmetric part H = 1 2 ( A − A T ) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective. [ABSTRACT FROM AUTHOR]

We present a local convergence analysis for general multi-point-Chebyshev–Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study. [ABSTRACT FROM AUTHOR]

In the analysis of the fluid queues, it is necessary to obtain the nonnegative solution of a nonsymmetric algebraic Riccati matrix equation. Under suitable conditions, this solution can be obtained transforming algebraic Riccati equations into unilateral quadratic matrix equations. In this paper, we use an efficient iterative scheme to approximate a solution of this quadratic matrix equation. We improve the efficiency and the accuracy of Newton's method, widely used in the literature. Moreover, a local convergence result is proved. Finally, we apply this efficient method to approximate the solution of a particular noisy Wiener–Hopf problem and we compare it with Newton's method. Moreover, a predictor–corrector iterative scheme is constructed that improve the accessibility of the aforesaid method. [ABSTRACT FROM AUTHOR]

In this work we are going to use the Kurchatov–Schmidt–Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by x ∗ of operator F. We define F as F : D ⊆ B 1 ⟶ B 2 where B 1 and B 2 stand for Banach spaces, D ⊆ B 1 be a convex set and F be a differentiable mapping according to Fréchet. Under these conditions, for all n = 0 , 1 , 2 , ... and 0 ≤ i ≤ m − 1 using Taylor expansion, KSSLS and KLS, when B 1 = B 2 and high order derivatives and divided differences not appearing in these solvers, the results obtained are the restart of the utilization of these iterative solvers. Moreover, we show under the same set of conditions that the local convergence radii are the same, the uniqueness balls coincide but the error estimates on ‖ x n − x ∗ ‖ differ. It is worth noticing our results improve the corresponding ones (Grau-Sánchez et al., 2011; Kurchatov, 1971 and Shakno, 2009). Finally, we apply our theoretical results to some numerical examples in order to prove the improvement. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a local convergence analysis of inexact Gauss–Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss–Newton like methods and some important, special cases. [Copyright &y& Elsevier]

Abstract: We provide a local convergence analysis for Newton’s method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before . Special cases and numerical examples are also provided in this study. [Copyright &y& Elsevier]

Abstract: A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications. [ABSTRACT FROM AUTHOR]

Abstract: A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study. [Copyright &y& Elsevier]

Abstract: We propose a penalty-function-free non-monotone line search method for nonlinear optimization problems with equality and inequality constraints. This method yields global convergence without using a penalty function or a filter. Each step is required to satisfy a decrease condition for the constraint violation, as well as that for the objective function under some reasonable conditions. The proposed mechanism for accepting steps also combines the non-monotone technique on the decrease condition for the constraint violation, which leads to flexibility and an acceptance behavior comparable with filter based methods. Furthermore, it is shown that the proposed method can avoid the Maratos effect if the search directions are improved by second-order corrections (SOC). So locally superlinear convergence is achieved. We also present some numerical results which confirm the robustness and efficiency of our approach. [Copyright &y& Elsevier]

Abstract: The famous Newton–Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton''s method to a solution of an equation. Here we present a “Kantorovich type” convergence analysis for the Gauss–Newton''s method which improves the result in [W.M. Häußler, A Kantorovich-type convergence analysis for the Gauss–Newton-method, Numer. Math. 48 (1986) 119–125.] and extends the main theorem in [I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004) 315–332]. Furthermore, the radius of convergence ball is also obtained. [Copyright &y& Elsevier]

In this paper, we propose two complementary variants of the projected Levenberg–Marquardt (LM) algorithm for solving convex constrained nonlinear equations. Since the orthogonal projection onto the feasible set may be computationally expensive, we first propose a local LM algorithm in which inexact projections are allowed. The feasible inexact projections used in our algorithm can be easily obtained by means of iterative methods, such as conditional gradient. Local convergence of the proposed algorithm is established by using an error bound condition which is weaker than the standard full-rank assumption. We further present and analyze a global version of this algorithm by means of a nonmonotone line search technique. Numerical experiments are reported to showcase the effectiveness of the proposed algorithms, especially when the projection onto the feasible set is difficult to compute. [ABSTRACT FROM AUTHOR]

This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. We perform a convergence study and an analysis of the efficiency. This analysis gives us the opportunity to select the most efficient method in the family without the necessity of their implementation. The method can be applied to many type of problems, including the discretization of ordinary differential equations, integral equations, integro-differential equations or partial differential equations. Moreover, multi-step iterative methods are computationally attractive. [ABSTRACT FROM AUTHOR]