Abstract
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.
Highlights
IntroductionThe Newton’s method requires the computation of Jacobian matrix, which demands the first-order derivative of the system
Consider the the system of nonlinear equations: F(x) = 0, (1)where F : Rn → Rn is nonlinear map, F is assume to satisfy the following assumptions: Assumption 1. (1) There exists x∗ ∈ Rn such that F(x∗) = 0. (2) F is continuously differentiable mapping in a neighborhood of x∗.The renowned method for finding the solution to (1) is the Newton’s method
A new approach for solving system of nonlinear equations is present, that is based on approximating the Jacobian into a diagonal matrix by means of acceleration parameter
Summary
The Newton’s method requires the computation of Jacobian matrix, which demands the first-order derivative of the system. The computation of some functions derivative are costly in practice, sometimes they are not even available or could not be obtained exactly In this case Newton’s method cannot be applied directly [17, 22, 23]. It is very important to state that the transformation of double step length methods was used in unconstrained optimization problem. They are efficient due to their convergence properties, simple implementation, and low storage requirement [20]. A double step size method for solving unconstrained optimization problem is proposed in [14].
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