Solving a system of equations using Newton's method

The following paper http://benisrael.net/Newton-MP.pdf provides a way to solve a system of equations using Newton's method. (The theorem begins at the end of page 2)

I can't understand the assumptions made in the proof :

Basically, what I want to know is , how does the author get to this assumption? :

$$\frac{M}{2}*||T_{u}|| + N < K < 1$$ where M is the Lipschitz constant, $T_{u}$ is the pseudo-inverse of the jacobian at u, and N comes from (11).

Based on these assumptions , the rest of the proof is clear to me. But the basic assumptions aren't. Any help would be greatly appreciated! :)

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Constants:

• $\alpha$ tells something about the initial residual (or the first Newton step)
• $N$ is the Lipschitz constant of $u \mapsto T_u$
• $K$ yields an uniform upper bound on the pseudo-inverses

Assumptions:

• $h=\alpha K<1$: the initial residual is small compared to the bounds of the inverses
• $\frac12 M \|T_u\|+N<K<1$ is most likely the right form of an inequality as later used in the proof ;)

These assumptions seem pretty standard for Newton-Kantorovich results.

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Thanks for answering, but I need some more help! " $\frac12 M \|T_u\|+N<K<1$ is most likely the right form of an inequality as later used in the proof ;)" Can you be more clear about what this assumption exactly means, and when is this valid? Even providing some reference article would do! :) – Ojas Jun 26 '14 at 5:54
The inequality means that the norm of the pseudoinverses is small compared to other quantities. If you have a concrete problem then you can try to check whether this assumption is valid. That is the problem with proofs like this: there are assumptions that cannot be checked. – daw Jun 26 '14 at 8:06
I always thought that if you have a good enough initial guess for the problem, then Newton's method should converge to a solution. Does this mean that the norm of the pseudo-inverse is small near the solution? – Ojas Jun 26 '14 at 8:54
You need a good initial guess of course. However, here 'good' does not mean close to a given solution, but something else. Note that the theorem does not assume the existence of a solution in the given ball, rather the existence comes as an extra conclusion. That is the difference between the 'standard' Newton method proof and this so-called Newton-Kantorovich theorem. – daw Jun 26 '14 at 8:58
I understand that. But can we reason in the opposite direction? Existence of a solution implies $||T_u||$ is small? – Ojas Jun 26 '14 at 9:04