Abstract
In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.
Highlights
IntroductionFrom the technique in (9), it is easy to see that the Jacobian matrix must be computed at every iteration, which will increase the computing difficulty, especially for the large-scale problems or when the matrix is expensive to calculate
Consider the systems of nonlinear equations: F (x) = 0, (1)where F : Rn → Rn is nonlinear map
We introduce the derivative-free line search proposed by Li and Fukushima [10] in order to compute our step lengths αk and βk
Summary
From the technique in (9), it is easy to see that the Jacobian matrix must be computed at every iteration, which will increase the computing difficulty, especially for the large-scale problems or when the matrix is expensive to calculate Considering these points, a new backtracking inexact technique is presented by Yuan∗ and Lu [12] in order to obtain the step length αk:. To globalize a quasi-Newton method, Li and Fukushima [10] proposed an approximately norm descent line search technique and established global and superlinear convergence of a Gauss-Newton based BFGS method for solving symmetric nonlinear equations. The proposed method has a norm descent property without computing the Jacobian matrix with less number of iterations and CPU time that is globally convergent.
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