Regarding the Newton-Kantorovich theorem, suppose that you are given a function , its derivative , its second derivative .

Use the Newton-Kantorovich theorem to find an estimated upper bound on the distance from the current iterate to the nearby solution when the NK conditions hold. (You'll have to assume that the current value of (or a multiple of it to be on the safe side) is an upper bound for all values of nearby, but this usually isn't too far off.)

See the attachments for NK theorem.

(2) App Newton-Kantorovich Theorem. (simplified) Suppose f: R^_IR" is by differentiable and for some K we have ll of C&)-DFC)ll