PREVIOUS chapters have taken a largely variational approach to deriving numericalalgorithms. That is, we define an objective function or energy E(~x), possibly with constraints, and pose our algorithms as approaches to a corresponding minimization or maximization problem. A sampling of problems that we solved this way is listed below:

The formulation of numerical problems in variational language is a powerful and general technique. To make it applicable to a larger class of nonlinear problems, we will design algorithms that can perform minimization or maximization in the absence of a special form for the energy E.