Solving a system of equations using Newton's method

Question

The following paper 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! :)

Answer 1

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.