A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems

In this paper,we propose a fifth-order scheme for solving systems of nonlinear equations.The convergence analysis of the proposed technique is discussed.The proposed method is generalized and extended to be of any odd order of the form 2n1.The scheme is composed of three steps,of which the first two steps are based on the two-step Homeier’s method with cubic convergence,and the last is a Newton step with an appropriate approximation for the derivative.Every iteration of the presented method requires the evaluation of two functions,two Fréchet derivatives,and three matrix inversions.A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed.The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order.Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order.As an application,we apply the new scheme to some real-life problems,including the mixed Hammerstein integral equation and Burgers’equation.Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ

The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind q-Bernstein operators related to B′ezier basis functions with shape parameterλ∈[−1,1].Firstly,we compute some basic results such as moments and central moments,and derive the Korovkin type approximation theorem for these operators.Next,we estimate the order of convergence in terms of the usual modulus of continuity,for the functions belong to Lipschitz-type class and Peetre’s K-functional,respectively.Lastly,with the aid of Maple software,we present the comparison of the convergence of these newly defined operators to the certain function with some graphical illustrations and error estimation table.

Modified Formula for Cotes Rule with Fifths Derivatives of Endpoint

This paper presents a Modified Formula for Cotes rule with fifths derivatives of endpoint and its truncation error. It also displays an analysis on convergence order of compound formula. Though compound modified formula for Cotes rule with endpoint derivatives just calculates a newly-added fifths derivative of the two endpoints for each time compared with compound Cotes formula calculation, there are 2 more ranks of the convergence order in this modified formula. Examples of numerical calculation have validated theoretical analysis.

On Polar Decomposition of Tensors with Einstein Product and a Novel Iterative Parametric Method

This study aims to investigate the polar decomposition of tensors with the Einstein product for thefirst time.The polar decomposition of tensors can be computed using the singular value decomposition of the tensors with the Einstein product.In the following,some iterative methods forfinding the polar decomposi-tion of matrices have been developed into iterative methods to compute the polar decomposition of tensors.Then,we propose a novel parametric iterative method tofind the polar decomposition of tensors.Under the obtained conditions,we prove that the proposed parametric method has the order of convergence four.In every iteration of the proposed method,only four Einstein products are required,while other iterative methods need to calculate multiple Einstein products and one tensor inversion in each iteration.Thus,the new method is superior in terms of efficiency index.Finally,the numerical comparisons performed among several well-known methods,show that the proposed method is remarkably efficient and accurate.

HIGH ORDER ACCURATE QUINTIC NONPOLYNOMIAL SPLINE FINITE DIFFERENCE APPROXIMATIONS FOR THE NUMERICAL SOLUTION OF NON-LINEAR TWO POINT BOUNDARY VALUE PROBLEMS

We develop a new sixth-order accurate numerical scheme for the solution of two point boundary value problems.The scheme uses nonpolynomial spline basis and high order finite difference approximations.With the help of spline functions,we derive consistency conditions and it is used to derive high order discretizations of the first derivative.The resulting difference schemes are solved by the standard Newton’s method and have very small computing time.The new method is analyzed for its convergence and the efficiency of the proposed scheme is illustrated by convection-diffusion problem and nonlinear Lotka–Volterra equation.The order of convergence and maximum absolute errors are computed to present the utility of the new scheme.

Regularization with Closed Linear Operators

In this paper, the regularization with closed linear operators is used to solve an operator equation of the first kind. When all initial data are known approximately, we choose the regular parameter by using the general Arcangeli's criterion to give the convergence and the asymptotic orders of convergence of the regular solution.

Optimal Eighth Order Convergent Iteration Scheme Based on Lagrange Interpolation

In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. The scheme requires three evaluations of the function and one evaluation of the first derivative per iteration. Numerical examples are included to confirm the theoretical results and to show the competitive performance of the proposed iteration scheme.

COMPACT DIFFERENCE SCHEMES FOR THE DIFFUSION AND SCHRODINGER EQUATIONS. APPROXIMATION, STABILITY, CONVERGENCE, EFFECTIVENESS, MONOTONY

Various compact difference schemes （both old and new, explicit and implicit, one-level and two-level）, which approximate the diffusion equation and SchrSdinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.

A FAMILY OF HIGH-ODER PARALLEL ROOTFINDERS FOR POLYNOMIALS

Presents a family of parallel iterations for finding all zeros of a polynomial without evaluation of derivatives. Construction of iterations; Convergence of the iterations; Details on the numerical examples.

A Class of Iterative Formulae for Solving Equations

Using the forms of Newton iterative function, the iterative function of Newton’s method to handle the problem of multiple roots and the Halley iterative function, we give a class of iterative formulae for solving equations in one variable in this paper and show that their convergence order is at least quadratic. At last we employ our methods to solve some non-linear equations and compare them with Newton’s method and Halley’s method. Numerical results show that our iteration schemes are convergent if we choose two suitable parametric functions λ（x） and μ（x）. Therefore, our iteration schemes are feasible and effective.