The gradient method is based on the first-order term in the Taylor expansion for . By taking a second-order term as well and solving the quadratic minimization problem iteratively, Newton's method for functional minimization is obtained. Essentially,Newton's method requires the error surface to be close toquadratic, and its effectiveness is directly tied to the accuracy of this assumption. For most problems, the error surface can be well approximated by a quadratic form near the solution. For this reason, Newton's method tends to give very rapid (``quadratic'') convergence in the last stages of iteration.
Newton's method is derived as follows: The Taylor expansion of about gives
for some , where . It is now necessary to assume that . Differentiating with respect to , where is presumed to be minimum, this becomes
Solving for yields
Applying Eq. (7) iteratively, we obtain the following.
where is given as an initial condition.
When , the answer is obtained after the first iteration. In particular, when the error surface is a quadratic form in , Newton's method produces in one iteration, i.e., for every .
For Newton's method, there is the following general result on the existence of a critical point (i.e. a point at which the gradient vanishes) within a sphere of a Banach space.
Theorem. (Kantorovich) Let be a point in for which exists, and set
Let denote the sphere . Set . If there exists a number such that
for in , and such that , then for some in , and the Newton sequence Eq. (7) converges to it. Furthermore, the rate of convergence is quadratic, satisfying
Proof. See Goldstein [Goldstein 1967], p. 143.