IN the previous two chapters, we developed general algorithms for minimizing a functionf(~x) with or without constraints on ~x. In doing so, we relaxed our viewpoint from numerical linear algebra that we must find an exact solution to a system of equations and instead designed iterative methods that successively produce better approximations of the minimizer. Even if we never find the position ~x∗ of a local minimum exactly, such methods generate ~xk with smaller and smaller f(~xk), in many cases getting arbitrarily close to the desired optimum.