iterative method

In our research, a new third-order iterative method has been introduced. This method involves the creation of a new quadrature formula by averaging Simpson's 1/3 rd and Trapezoidal rules. The newly developed quadrature formula is then used to establish the new iterative scheme, which modifies the Newton-Raphson method. It has been demonstrated that the new iterative technique exhibits a convergence order of 3. Finally, examples have been provided to illustrate the effectiveness of the new process. The results indicate that the new approach finds the root of the nonlinear equation in fewer iterations compared to other methods, suggesting the potential superiority of our newly developed scheme.

‎ In this paper, an iterative method of successive approximations based on the trapezoidal quadrature rule to solve two-dimensional Fredholm integral equations of second kind (2DFIE) is proposed. The error estimation of the proposed method is presented. The benefit of the method is that we do not have to solve a system of algebraic equations. Finally, a numerical example verify the theoretical results and show the accuracy of the method.

In this paper, we provide a quadrature-based iterative approach to solve three-dimensional nonlinear fuzzy Volterra integral equations of the second kind. The error estimation as well as convergence analysis of the proposed method are also provided. Finally, numerical experiments validate the theoretical findings. The suggested method's advantages include   precision, accuracy and ease of use.

This paper introduces two families of modified Householder’s method (HM) that are optimal in line with Kung-Traub conjecture given in [4]. The modification techniques employed involved approximation of the function derivatives in the HM with divided difference operator, a polynomial function approximation and the modified Wu function approximation in [17]. These informed the formation of two families of methods that that are optimal and do not or require function derivative evaluation. The both families do not breakdown when f(·) ≈ 0 as in the case with the HM and many existing modified HM. From the convergence investigation carried out on the methods, the sequence of approximations produced by the methods, converged to solution of nonlinear equation with order four. The implementation of the methods was illustrated and numerical results obtained were compared with that of some recently developed methods.

We propose an algorithm that estimates the real roots of differentiable functions on closed intervals. Then, we extend this algorithm to real differentiable functions that are dominated by a polynomial. For each starting point, our method converges to the nearest root to the right or left hand side of that point. Our algorithm can look for missed roots as well and theoretically it misses no root. Furthermore, we do not find the roots by randomly chosen initial guesses. The iterated sequences in our algorithms converge linearly. Therefore, the rate of convergence can be accelerated considerably to make it comparable to Newton-Raphson and other high-speed methods. We have illustrated our algorithms with some concrete examples. Finally, the pseudo-codes of the related algorithms are presented at the end of this manuscript.

This study is aimed at performing a comprehensive numerical evalua-tion of the iterative solution techniques without memory for solving non-linear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes are be evaluated, in which the ability of the methods to solve different types of nonlinear equations is be studied. First, 26 different iterative methods with the best performance are reviewed. These methods are selected based on performing more than 46000 analyses on 166 different available nonlinear solvers. For the easier application of the techniques, consistent mathematical notation is employed to present reviewed approaches. After presenting the diverse methodologies suggested for solving nonlinear equations, the performances of the reviewed methods are evaluated by solving 28 different nonlinear equations. The utilized test functions, which are selected from the re-viewed research works, are solved by all schemes and by assuming different initial guesses. To select the initial guesses, endpoints of five neighboring intervals with different sizes around the root of test functions are used. Therefore, each problem is solved by ten different starting points. In order to calculate novel computational efficiency indices and rank them accu-rately, the results of the obtained solutions are used. These data include the number of iterations, number of function evaluations, and convergence times. In addition, the successful runs for each process are used to rank the evaluated schemes. Although, in general, the choice of the method de-pends on the problem in practice, but in practical applications, especially in engineering, changing the solution method for different problems is not feasible all the time, and accordingly, the findings of the present study can be used as a guide to specify the fastest and most appropriate solution technique for solving nonlinear problems.

The Haar wavelet collocation with iteration technique is applied for solving a class of time-fractional physical equations. The approximate solutions obtained by two dimensional Haar wavelet with iteration technique are compared with those obtained by analytical methods such as Adomian decomposition method (ADM) and variational iteration method (VIM). The results show that the present scheme is effective and appropriate for obtaining the numerical solution of the timefractional Modified Camassa-Holm equation and Time fractional Modified Degasperis-Procesi equation.

The current research develops a derivative-free family without memory methods. The proposed method consisting of two steps and one parameter for solving nonlinear equations is brought forward.\,The basin of attraction of the proposed methods has investigated using different weight functions.\,Numerical examples are experimented with to check the performance of the proposed schemes. Furthermore, the theoretical order of convergence is confirmed on the experiment work.

In this manuscript, for approximation of solutions to equations that are nonlinear, a new class of two-point iterative structure that is based on a weight function involving two converging power series, is developed. For any method constructed from the developed class of methods, it requires three separate functions evaluation in a complete iteration cycle that is of order four convergence. Also, some well-known existing methods are typical members of the new class of methods. The numerical test on some concrete methods derived from the class of methods indicates that they are effective and competitive when employed in solving a nonlinear equation.

The main objective of this paper is to introduce and study a new type of iterative method to approximate a common solution of split variational inclusion problem and a finite family of fixed point problems in real Hilbert spaces. Furthermore, we show that the sequence generated by the proposed iterative method converges strongly to a common solution to these problems. The method and results presented in this paper extend and unify some recent known results in this field. Finally, a numerical example is used to demonstrate the convergence analysis of the sequences generated by the iterative method.

In the present paper, we propose a numerical method based on the combination of the fixed point method and quadrature formula for solving two-dimensional nonlinear Fredholm integral equations of the second kind. Using uniform and partial modulus of continuity, the error estimation is given. Also, the numerical stability with respect to the choice of the first iteration is proved. Moreover, the accuracy of the method and the correctness of the theoretical results are shown by some examples.

In this research, we investigate the fuzzy integral equations related to traffic flow. Using the Banach fixed point theorem, we prove the existence and uniqueness of the solution for such equations. Using the Picard iterative method, we obtain the upper bound for an accurate and approximate solution. Finally, we obtain an error estimation between the exact solution and the solution of the iterative method. Example shows the applicabilityof our results.

In this paper, a third order convergent method for finding the Moore-Penrose inverse of a matrix is presented and analysed. Then, we develop the method to find Drazin inversion. This method is very robust to find the Moore-Penrose and Drazin inverse of a matrix. Finally, numerical examples show that the efficiency of the proposed method is superior over other proposed methods.

In this paper, a new class of absolute value equations is studied as follows:Ax-B|x|-b=o, ( B≠I, σ_"max" (|B|)<σ_"min" (A) ), This new class of absolute value equations, the single value absolute matrix B is less than the single value matrix A and the matrix B is not exclusively the identity matrix..Therfore the power of choice is wider than other methods of the absolute value equations and all matrices are arbitrary and this new class of absolute value equation is the NP hard problem..We solve this new class using a generalized Newton method and also convergence and numerical stability. Also, by testing the numerical examples of the efficiency and effectiveness of the solution method for the new class, it has been studied with other works that have been done including Lotfi and Zainali and Mangasarain and Khaksars method.Eceptthis new class and Lotfi and Zainali method are quadratic convergence, the rest methods are linear convergence.

In this work, solving non-linear two-dimensional Hammerstein integral equations is considered by an iterative method of successive approximation. This method is an efficient approach based on a combination of the quadrature formula and the successive approximations method. Also, the convergence analysis and the numerical stability of the suggested method are studied. Finally, to survey the accuracy of the present method, some numerical experiments are given.

In this chapter, we investigate the global attractivity of the recursive sequence ${mathcal{U}_n} subset mathcal{P}(N)$ defined by[mathcal{U}_{n+k} = mathcal{Q} + frac{1}{k} sum_{j=0}^{k-1} mathcal{A}^* psi(mathcal{U}_{n+j}) mathcal{A}, n=1,2,3ldots,]where $mathcal{P}(N)$ is the set of $N times N$ Hermitian positive definite matrices, $k$ is a positive integer,$mathcal{Q}$ is an $N times N$ Hermitian positive semidefinite matrix, $mathcal{A}$ is an $N times N$ nonsingular matrix, $mathcal{A}^*$ is the conjugate transpose of $mathcal{A}$ and $psi : mathcal{P}(N) to mathcal{P}(N)$ is a continuous. For this, we first introduce $mathcal{FG}$-Prev{s}i'c contraction condition for $f: mathcal{X}^k to mathcal{X}$ in metric spaces and study the convergence of the sequence ${x_n}$ defined by[x_{n+k} = f(x_n, x_{n+1}, ldots, x_{n+k-1}), n = 1, 2, ldots]with the initial values $x_1,ldots, x_k in mathcal{X}$. We furnish our results with some examples throughout the chapter. Finally, we apply these results to obtain matrix difference equations followed by numerical experiments.

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Recently, Zhang et al. [Applied Mathematics Letters 104 (2020) 106287] proposed a preconditioner to improve the convergence speed of three types of Jacobi iterative methods for solving multi-linear systems. In this paper, we consider the Jacobi-type method which works better than the other two ones and apply a new preconditioner. The convergence of proposed preconditioned iterative method is studied. It is shown that the new approach is superior to the recently examined one in the literature. Numerical experiments illustrate the validity of theoretical results and the efficiency of the proposed preconditioner.

In this paper, a numerical iterative algorithm based on combination of the successive approximations method and the quadrature formula for solving two-dimensional nonlinear Volterra integral equations is proposed. This algorithm uses a trapezoidal quadrature rule for Lipschitzian functions applied at each iterative step. The convergence analysis and error estimate of the method are proved. Finally, two numerical examples are presented to show the accuracy of the proposed method.