A Class of Steffensen-Type Iterative Methods for Nonlinear Systems (original) (raw)
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Journal of Computational and Theoretical Nanoscience, 2015
We presented multi-step iterative method for solving systems of nonlinear equations, associated with ordinary differential equations (ODEs). In particular we considered the ODEs of the form L x t + f x t = g t : here L • is a linear differential operator and f • is a nonlinear smooth function. The computational efficiency of our proposed iterative method is hidden in frozen Jacobian at a single point. The direct inversion of frozen Jacobian is avoided by using either LU-decomposition or GMRES-type methods. The higher-order Frechet derivatives of systems of nonlinear equations associated with under-discussion ODEs are diagonal matrices, and the computational cost of higher-order Frechet derivatives is same as Jacobian. The use of second-order Frechet-derivative enhances the convergence-order. The inclusion of multi-step increase the convergence-rate and the design of multi-step consist on the product of a matrix polynomial and frozen Jacobian inverse. The convergence-order (CO) obeys the formula CO = 3 m − 1 , where m ≥ 2 is the number of steps per full-cycle of the iterative scheme.
Applied Mathematics and Computation, 2015
We developed multi-step iterative method for computing the numerical solution of nonlinear systems, associated with ordinary differential equations (ODEs) of the form LðxðtÞÞ þ f ðxðtÞÞ ¼ gðtÞ: here LðÁÞ is a linear differential operator and f ðÁÞ is a nonlinear smooth function. The proposed iterative scheme only requires one inversion of Jacobian which is computationally very efficient if either LU-decomposition or GMRES-type methods are employed. The higher-order Frechet derivatives of the nonlinear system stemming from the considered ODEs are diagonal matrices. We used the higher-order Frechet derivatives to enhance the convergence-order of the iterative schemes proposed in this note and indeed the use of a multi-step method dramatically increases the convergence-order. The secondorder Frechet derivative is used in the first step of an iterative technique which produced third-order convergence. In a second step we constructed matrix polynomial to enhance the convergence-order by three. Finally, we freeze the product of a matrix polynomial by the Jacobian inverse to generate the multi-step method. Each additional step will increase the convergence-order by three, with minimal computational effort. The convergence-order (CO) obeys the formula CO ¼ 3m, where m is the number of steps per full-cycle of the considered iterative scheme. Few numerical experiments and conclusive remarks end the paper.
Some Steffensen-type iterative schemes for the approximate solution of nonlinear equations
Miskolc Mathematical Notes, 2021
In this paper, we suggest some new and efficient iterative methods for solving nonlinear equations f (x) = 0. These methods are free from derivatives having high order of convergence. We also give some examples to illustrate the efficiency of these methods. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the classical Steffensen's method. These new methods can be considered as alternative of existing derivative-free methods.
A multi-step frozen Jacobian iterative scheme for solving the system of nonlinear equations associated with IVPs (initial value problems) and BVPs (boundary value problems) is constructed. The multi-step iterative schemes consist of two parts, namely base method and a multi-step part. The proposed iterative scheme uses higher order Fréchet derivatives in the base method part and offers high convergence order (CO) 3m + 1, here s is the number of steps. The increment in the CO per step is three, and we solve three upper and lower triangles systems per step in the multi-step part. A single inversion of the frozen Jacobian is required and in fact, we avoid the direct inversion of the frozen Jacobian by computing the LU factors. The LU-factors are utilized in the multi-step part to solve upper and lower triangular systems repeatedly that makes the iterative scheme computationally efficient. We solve a set of IVPs and BVPs to show the validity, accuracy and efficiency of our proposed iterative scheme.
Journal of Computational and Applied Mathematics, 2018
There are a very small number of high quality derivative free methods with memory for solving a system of nonlinear equations numerically. Motivated and inspired by the fact, we propose a more efficient general class of Steffensen type multipoint methods with memory. This proposed class requires one divided difference and inverse of only one matrix per full iteration. We also demonstrate their applicability and illustrate that these methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations numerically. For quantitative comparison, we have also computed total number of convergent points and convergent percentage along with basins of attractions on a number of test problems to recommend the best quality algorithm.
Some new two step iterative methods for solving nonlinear equations using Steffensen’s method
Journal of Mathematical and Computational Science, 2016
In this paper, we introduce the comparative study of some new two step iterative methods for solving nonlinear equations by using Steffensen's method. Some examples are also discussed. These new methods can be viewed as significant modification and improvement of the Steffensen's method. Numerical comparisons are made with other exiting methods to show the performance of the present methods.
A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems
Mathematics, 2019
In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.
Derivative Free Three-Step Iterative Method to Solve Nonlinear Equations
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH
This article discusses a derivative free three-step iterative method to solve a nonlinear equation using Steffensen method, after approximating the derivative in the method proposed by Abro et al. [Appl. Math. Comput.,55(2019),516-536] by a divided difference method. We show analytically that the method is of order sixth under a condition and for each iteration it requires three function evaluations. Numerical experiments show that the new method is comparable with other discussed method.
New Derivative Free Iterative Method ’ S for Solving Nonlinear Equations Using Steffensen ’ S Method
2016
In this paper, we introduce the comparative study of derivative free new two step iterative method for finding the zeros of the nonlinear equation = 0 without the evaluation of the derivatives . It is established that the new method has convergence order three. The efficiency index of new method is equal to 1.442. The Convergence and error analysis are given. Numerical comparisons are made with other existing methods to show the performance of the presented methods.