Abstract
An algorithm for solving large-scale systems of nonlinear equations based on the transformation of the Newton method with the line search into a derivative-free descent method is introduced. Main idea used in the algorithm construction is to approximate the Jacobian by an appropriate diagonal matrix. Furthermore, the step length is calculated using inexact line search procedure. Under appropriate conditions, the proposed method is proved to be globally convergent under mild conditions. The numerical results presented show the efficiency of the proposed method.
Highlights
Consider the systems of nonlinear equations: F(x) = 0, (1)where F: Rn → Rn is nonlinear map.Among various methods for solving nonlinear equations (1), Newton’s method is quite welcome due to its nice properties such as the rapid convergence rate, the decreasing of the function value sequence [10]
The iterative formula of a Newton method is given by xk+1 = xk + sk, sk = αkdk, k = 0,1, ⋯, where, αk is a step length to be computed by a line search technique [2, 3], xk+1 represents a new iterative point, xk is the previous iteration, while dk is the search direction to be calculated by solving the following linear system of equations, Fo(xk)dk = −F(xk), (2)
We are going to establish the following global convergence theorem to show that under some suitable conditions, there exist an accumulation point of {xk} which is a solution of problem (1)
Summary
At each iteration, Newton’s method needs the computation of the derivative Fo as well as the solution of some system of linear equations. The iterative formula of a Newton method is given by xk+1 = xk + sk, sk = αkdk , k = 0,1, ⋯, where, αk is a step length to be computed by a line search technique [2, 3], xk+1 represents a new iterative point, xk is the previous iteration, while dk is the search direction to be calculated by solving the following linear system of equations, Fo(xk)dk = −F(xk), (2) Newton method with line search is norm descent.
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