6. Inexact Newton Methods
C. T. Kelley · Society for Industrial and Applied Mathematics eBooks · 1995
Theorem 5.4.1 describes how errors in the derivative/function affect the progress in the Newton iteration. Another way to look at this is to ask how an approximate solution of the linear equation for the Newton step affects the iteration. This was the view taken in [55] where inexact Newton methods in which the step satisfies‖F′(xc)s+F(xc)‖≤ηc‖F(xc)‖(6.1)are considered. Any approximate step is accepted provided that the relative residual of the linear equation is small. This is quite useful because conditions like (6.1) are precisely the small linear residual termination conditions for iterative solution of the linear system for the Newton step. Such methods are not new. See [145] and [175], for example, for discussion of these ideas in the context of the classical stationary iterative methods. In most of this chapter we will focus on the approximate solution of the equation for the Newton step by GMRES, but the other Krylov subspace methods discussed in Chapters 2 and 3 and elsewhere can also be used. We follow [69] and refer to the term ηc on the right hand side of (6.1) as the forcing term.