TABLE OF CONTENTS While one cannot know the error without knowing the solution, in most cases the norm of F ( x) can be used as a reliable indicator of the rate of decay in e as the iteration progresses [42]. Based on this heuristic, we terminate the iteration in our codes when
The relative ? r and absolute ? a error tolerances are both important. Using only the relative reduction in the nonlinear residual as a basis for termination (i.e., setting ? a = 0) is a poor idea because an initial iterate that is near the solution may make (1.12) impossible to satisfy with ? a = 0.
One way to quantify the utility of termination when F ( x) is small is to compare a relative reduction in the norm of the error with a relative reduction in the norm of the nonlinear residual. If the standard assumptions hold and x 0 and x are sufficiently near the root, then
where
is the condition number of F ?( x*) relative to the norm . From (1.13) we conclude that, if the Jacobian is well conditioned (i.e., ?( F ?( x*)) is not very large), then (1.12) is a useful termination criterion. This is analogous to the linear case, where a small residual implies a small error if the matrix is well conditioned.
Another approach, which is supported...
