Choosing the forcing terms in an inexact Newton method

An inexact Newton method is a generalization of Newton`s method for solving F(x) = 0, F: {Re}{sup n} {r_arrow} {Re}{sup n}, in which each step reduces the norm of the local linear model of F. At the kth iteration, the norm reduction is usefully expressed by the inexact Newton condition where x{sub k} is the current approximate solution and s{sub k} is the step. In many applications, an {eta}{sub k} is first specified, and then an S{sub k} is found for which the inexact Newton condition holds. Thus {eta}{sub k} is often called a {open_quotes}forcing term{close_quotes}. In practice, the choice of the forcing terms is usually critical to the efficiency of the method and can affect robustness as well. Here, the authors outline several promising choices, discuss theoretical support for them, and compare their performance in a Newton iterative (truncated Newton) method applied to several large-scale problems.