TABLE OF CONTENTS Recall from section 1.4 that an inexact Newton method approximates the Newton direction with a vector d such that
The parameter ? is called the forcing term.
Newton iterative methods realize the inexact Newton condition (3.1) by applying a linear iterative method to the equation for the Newton step and terminating that iteration when (3.1) holds. We sometimes refer to this linear iteration as an inner iteration. Similarly, the nonlinear iteration (the while loop in Algorithm nsolg) is often called the outer iteration.
The Newton Krylov methods, as the name suggests, use Krylov subspace-based linear solvers. The methods differ in storage requirements, cost in evaluations of F, and robustness. Our code, nsoli.m, includes three Krylov linear solvers: GMRES [64], BiCGSTAB [77], and TFQMR [31]. Following convention, we will refer to the nonlinear methods as Newton-GMRES, Newton-BiCGSTAB, and Newton-TFQMR.
Krylov iterative methods approximate the solution of a linear system Ad = b with a sum of the form
where r 0 = b ? Ad 0 and d 0 is the initial iterate. If the goal is to approximate a Newton step, as it is here, the most sensible initial iterate is d 0 = 0, because we have no a priori knowledge of the direction, but, at least in the local phase of the iteration, expect it to be small.
We express this in compact form as d k
