On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations

We present derivative free methods with memory with increasing order of convergence for solving systems of nonlinear equations. These methods relied on the basic family of fourth order methods without memory proposed by Sharma et al. (Appl. Math. Comput. 235, 383–393, 2014). The order of convergence of new family is increased from 4 of the basic family to \(2+\sqrt {5} \approx 4.24\) by suitable variation of a free self-corrected parameter in each iterative step. In a particular case of the family even higher order of convergence \(2+\sqrt {6} \approx 4.45\) is achieved. It is shown that the new methods are more efficient in general. The presented numerical tests confirm the theoretical results.     

On some one-point hybrid iterative methods

When we choose an iterative process for solving a nonlinear equation, the region of accessibility of the iterative process is certainly useful. We know that the higher the order of convergence of the iterative process, the smaller the region of accessibility. In this paper, we present a simple modification of the classic third-order iterative processes, so as to consider, for each of them, the same region of accessibility as that of the Newton method, that is to say a method of order of convergence two.     

On some versions of incomplete block-matrix factorization iterative methods

Various forms of preconditioning matrices for iterative acceleration methods are discussed. The preconditioning is based on two versions of incomplete block-matrix factorization.     

Some optimal iterative methods and their with memory variants

Based on the fourth-order method of Liu et al. [10], eighth-order three-step iterative methods without memory, which are totally free from derivative calculation and reach the optimal efficiency index are presented. The extension of one of the methods for multiple zeros without the knowledge of multiplicity is presented. Further accelerations will be provided through the concept of with memory iteration methods. Moreover, it is shown by way of illustration that the novel methods are useful on a series of relevant numerical problems when high precision computing is required.     

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.     

Derivative free iterative methods with memory of arbitrary high convergence order

In this article we present three derivative free iterative methods with memory to solve nonlinear equations. With the process developed, we can obtain n-step derivative free iterative methods with memory of arbitrary high order. Numerical examples are provided to show that the new methods have an equal or superior performance, on smooth and nonsmooth equations, compared to classical iterative methods as Steffensen’s and Newton’s methods and other derivative free methods with and without memory with high order of convergence.     

On the partial synchronization of iterative methods

The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms have a number of advantages. There is no need for a global time in a parallel system (no need for synchronization), and all computational resources are efficiently loaded (the minimum processor idle time). The method of partial synchronization of iterations for systems of equations was proposed by the authors earlier. In this article, this method is generalized to include the case of nonlinear equations of the form x = F(x), where x is an unknown column vector of length n, and F is an operator from ?ⁿ into ?ⁿ. We consider operators that do not satisfy conditions that are sufficient for the convergence of asynchronous iterations, with simple iterations still converging. In this case, one can specify such an incidence of the operator and such properties of the parallel system that asynchronous iterations fail to converge. Partial synchronization is one of the effective ways to solve this problem. An algorithm is proposed that guarantees the convergence of asynchronous iterations and the Monte Carlo method for the above class of operators. The rate of convergence of the algorithm is estimated. The results can prove useful for solving high-dimensional problems on multiprocessor computational systems.     

Improvements of the efficiency of some three-step iterative like-Newton methods

An improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative. The third and fourth order of known two-step like Newton methods have been improved and the efficiency has also been increased.     

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.     

Point estimation and some applications to iterative methods

We consider one of the crucial problems in solving polynomial equations concerning the construction of such initial conditions which provide a safe convergence of simultaneous zero-finding methods. In the first part we deal with the localization of polynomial zeros using disks in the complex plane. These disks are used for the construction of initial inclusion disks which, under suitable conditions, provide the convergence of the Gargantini-Henrici interval method. They also play a key role in the convergence analysis of the fourth order Ehrlich-Aberth method with Newton's correction for the simultaneous approximation of all zeros of a polynomial. For this method we state the initial condition which enables the safe convergence. The initial condition is computationally verifiable since it depends only on initial approximations, which is of practical importance.     

Superattracting cycles for some Newton type iterative methods

The paper is devoted to the analysis of certain dynamical properties of a family of iterative Newton type methods used to find roots of non-linear equations. We present a procedure for constructing polynomials in such a way that superattracting cycles of any prescribed length occur when these iterative methods are applied. This paper completes the study begun in Amat, Bermúclez, Busquier, et al., (2009).     

Introducing memory to a family of multi-step multidimensional iterative methods with weight function

In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen’s method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we freeze the divided differences used from the second step and use a weight function on already evaluated operators. Therefore, we define a family of multi-step methods with convergence order $2m$, where $m$ is the number of steps, free of derivatives, with several parameters and with dynamic behaviour, in some cases, similar to Steffensen’s method. In addition, we study how to increase the convergence order of the defined family by introducing memory in two different ways: using the usual divided differences and the Kurchatov divided differences. We perform some numerical experiments to see the behaviour of the proposed family and suggest different weight functions to visualize with dynamical planes in some cases the dynamical behaviour.     

On iterative methods for equilibrium problems

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings. We prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions, which is the optimality condition for some minimization problem. We also give some comments on the results of Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455–469]. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area.     

On iterative methods for solving equations with covering mappings

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element x _i+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, x _i) = y _i, where y _i can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine x _i+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) ? φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.     

On the problem of starting points for iterative methods

Numerical Algorithms - We propose an algorithm to find a starting point for iterative methods. Numerical experiments show empirically that the algorithm provides starting points for different...     

Note on the estimation of the order of convergence of some iterative methods

In this paper we obtain upper and lower bounds for the unique positive roots of certain sequences of polynomials. The results are then applied to the determination of theR-order of iterative numerical processes.