TABLE OF CONTENTS Algorithm nsolg is a general formulation of an inexact Newton Armijo iteration. The methods in Chapters 2 and 3 are special cases of nsolg. There is a lot of freedom in Algorithm nsolg. The essential input arguments are the initial iterate x, the function F, and the relative and absolute termination tolerances ? a and ? r. If nsolg terminates successfully, x will be the approximate solution on output.
Within the algorithm, the computation of the Newton direction d can be done with direct or iterative linear solvers, using either the Jacobian F ?( x) or an approximation of it. If you use a direct solver, then the forcing term ? is determined implicitly; you do not need to provide one. For example, if you solve the equation for the Newton step with a direct method, then ? = 0 in exact arithmetic. If you use an approximate Jacobian and solve with a direct method, then ? is proportional to the error in the Jacobian. Knowing about ? helps you understand and apply the theory, but is not necessary in practice if you use direct solvers.
If you use an iterative linear solver, then usually (1.10) is the termination criterion for that linear solver. You'll need to make a decision about the forcing term in that case (or accept the defaults from a code like nsoli.m, which we describe in...
