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Advantages and disadvantages of Newton's method

The results that I have derived above show that Newton's method produces excellent local convergence. However, there is no reason to expect that the algorithm will behave well when x(0) is chosen far from x*. Indeed, the algorithm may not even be defined; when an iterate x(k) is encountered with the property that J(x(k)) or $\nabla^2f(x^{(k)})$ is singular, then Newton's method does not define x(k+1). Moreover, in the case of a minimization problem, the sequence may converge to a stationary point of f that is not a local minimizer, such as a local maximizer or saddle point.

For these reasons, it is necessary to enhance Newton's method to obtain global convergence (that is, convergence to a solution from a starting point that may be far away). Whatever techniques are used to ensure global convergence should not ruin the excellent local convergence exhibited by Newton's method.

Mark S. Gockenbach