On efficient matrix-free method via quasi-Newton approach for solving system of nonlinear equations

Abstract

In this paper, a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis of the new method is discussed based on some standard condition. Preliminary numerical results on some test problems show that the method is promising.

A projection method for solving monotone nonlinear equations with application

“…Numerous problems, including the sub-problems in the generalized proximal algorithms with Breman distances, some ℓ 1− norm regularization problems in image restoration problems, compressive sensing problems, and variational inequality problems [4][5][6][7] can be transformed into the form of (1). Numerous approaches, including Levenberg-Marquardt, Newton, and quasi-Newton methods, have been presented for finding a solution to (1) (see [8][9][10][11][12][13][14][15][16][17]). However, the Newton method stands among the best for finding a solution to (1).…”

“…The conjugate gradient method is one of the most developed algorithms for solving large-scale problems due to its appealing practical characteristics, such as easy computation, low memory requirement, sufficient descent property, and strong global convergence [4][5][6][15][16][17][18][19].…”