GLOBALLY CONVERGENT INEXACT NEWTON METHODS

Inexact Newton methods for finding a zero of F : R(n) --> R(n) are var iations of Newton's method in which each step only approximately satis fies the linear Newton equation but still reduces the norm of the loca l linear model of F. Here, inexact Newton methods are formulated that incorporate features designed to improve convergence from arbitrary st arting points. For each method, a basic global convergence result is e stablished to the effect that, under reasonable assumptions, if a sequ ence of iterates has a limit point at which F' is invertible, then tha t limit point is a solution and the sequence converges to it. When app ropriate, it is shown that initial inexact Newton steps are taken near the solution, and so the convergence can ultimately be made as fast a s desired, up to the rate of Newton's method, by forcing the initial l inear residuals to be appropriately small. The primary goal is to intr oduce and analyze new inexact Newton methods, but consideration is als o given to ''globalizations'' of (exact) Newton's method that can natu rally be viewed as inexact Newton methods.