En enhanced matrix-free method via double step length approach for solving systems of nonlinear equations

Abstract

<jats:p>A variant method for solving system of nonlinear equations is presented. This method use the special form of iteration with two step length parameters, we suggest a derivative-free method without computing the Jacobian via acceleration parameter as well as inexact line search procedure. The proposed method is proven to be globally convergent under mild condition. The preliminary numerical comparison reported in this paper using a large scale benchmark test problems show that the proposed method is practically quite effective.</jats:p>

A Derivative-Free Decent Method Via Acceleration Parameter for Solving Systems of Nonlinear Equations

“…In this section, we compared the performance of our method with an improved derivative free method via double direction approach for solving systems of nonlinear equations (IDFDD) [5]. For the both algorithms the following parameters are set ω1 = ω2 = 10 −4 , r = 0.2 and = 1 ( +1) 2 .…”

“…The methods were tested on some Benchmark test problems with different initial points. problem 1 and 3 below are from [5] while problem 2 is from [11].…”

“…Furthermore, the search direction is generally required to satisfy the descent condition ∇ ( ) < 0. The derivative-free direction can be obtained in several ways [4,5,7,9,12]. An iterative method that generates a sequence { } satisfying (3) or (5) is called a norm descent method.…”

Efficient matrix-free direction method with line search for solving large-scale system of nonlinear equations

“…More applications of the problem (1) in economic equilibrium analysis, chemical equilibrium systems, compressive sensing, and control theory can be found in [14,17,21] and in the references therein. Some iterative methods for solving these problems include Newton and quasi-Newton methods [3,12,15,18], the Gauss-Newton methods [7,22], the Levenberg-Marquardt methods [16,19,23], the derivative-free methods [9,13,25,29], the subspace methods [24], and the tensor methods [26]. The most popular schemes for solving (1) are based on successive linearization [3], where the search direction d k is obtained by solving the following linear equation:…”

“…However a double direction method for solving unconstrained optimization problem was presented by Petrovic and Stanimirovic [8]. In [9] Halilu and Waziri incorporated the work in [8] to solve the system of nonlinear equations, and approximated the Jacobian matrix with diagonal matrix via acceleration parameter. The global convergence of the scheme [9] is established under mild conditions.…”