A derivative-free method for the system of nonlinear equations

2006 ◽  
Vol 7 (3) ◽  
pp. 378-384 ◽  
Author(s):  
Pu-yan Nie
Keyword(s):  
Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Derivative Free ◽  
Derivative Free Method

scholarly journals A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 168 ◽  
Author(s):  
Zhifeng Dai ◽  
Huan Zhu
Keyword(s):  
Global Convergence ◽  
Large Scale ◽  
Line Search ◽  
System Of Nonlinear Equations ◽  
Inexact Newton Method ◽  
Monotone Equations ◽  
Smoothing Methods ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Nonlinear Monotone Equations

The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

scholarly journals A New Derivative-Free Method to Solve Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 583
Author(s):  
Beny Neta
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
High Order ◽  
Order Derivative ◽  
Eighth Order ◽  
High Order Derivative ◽  
Derivative Free ◽  
Derivative Free Method

A new high-order derivative-free method for the solution of a nonlinear equation is developed. The novelty is the use of Traub’s method as a first step. The order is proven and demonstrated. It is also shown that the method has much fewer divergent points and runs faster than an optimal eighth-order derivative-free method.

scholarly journals An efficient derivative free iterative method for solving systems of nonlinear equations

2013 ◽  
Vol 7 (2) ◽  
pp. 390-403 ◽  
Author(s):  
Janak Sharma ◽  
Himani Arora
Keyword(s):  
Nonlinear Equations ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
First Order ◽  
Numerical Tests ◽  
Several Variables ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Large Systems ◽  
Theoretical Results

We present a derivative free method of fourth order convergence for solving systems of nonlinear equations. The method consists of two steps of which first step is the well-known Traub's method. First-order divided difference operator for functions of several variables and direct computation by Taylor's expansion are used to prove the local convergence order. Computational efficiency of new method in its general form is discussed and is compared with existing methods of similar nature. It is proved that for large systems the new method is more efficient. Some numerical tests are performed to compare proposed method with existing methods and to confirm the theoretical results.

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