The Inexact Newton Method with Semi Automatic Differentiation

Nonlinear optimization plays an important role in science computation and engineering analysis. Newton like method is popular for solving the nonlinear optimization problem. an inexact Newton algorithm is proposed recently, in which the preconditioned conjugate gradient method is applied to solve the Newton equations. Later, the algorithm is improved by efficiently using automatic differentiation. In practical application, large-scale systems of nonlinear equations typically exhibit either sparsity or other special structures in their Jacobian matrices. In this paper, we propose the structure inexact Newton algorithm (SINA), The algorithm utilized Semi-AD techniques can improve the algorithm efficiency by avoiding the unnecessary computation. Based on the efficiency coefficient defined by Brent, a theoretical efficiency ratio of SINA to the old algorithm is introduced. It has-been shown that SINA is much more efficient than the old one. Furthermore, this theoretical conclusion is supported by numerical experiments.