TABLE OF CONTENTS Even the best and most robust codes can (and do) fail in practice. In this section we give some guidance that may help you troubleshoot your own solvers or interpret hard-to-understand results from solvers written by others. These are some problems that can arise for all choices of methods. We will also repeat some of these things in subsequent chapters, when we discuss problems that are specific to a method for approximating the Newton direction.
Most nonlinear equation codes, including the ones that accompany this book, are intended to solve problems for which F ? is Lipschitz continuous. The codes will behave unpredictably if your function is not Lipschitz continuously differentiable. If, for example, the code for your function contains
nondifferentiable functions such as the absolute value, a vector norm, or a fractional power;
internal interpolations from tabulated data;
control structures like case or if-then-else that govern the value returned by F; or
calls to other codes,
you may well have a nondifferentiable problem.
If your function is close to a smooth function, the codes may do very well. On the other hand, a nonsmooth nonlinearity can cause any of the failures listed in this section.
The theory, as stated in Theorem 1.4, does not imply that the iteration will converge, only that nonconvergence can be identified easily. So, if the iteration fails to converge to a root, then either the iteration will become...
