Nonlinear Equations - UFRJ

126 [CH. 9: THE PSEUDO-NEWTON OPERATOR
In the projective case s = 1, β scales as ‖X‖ while γ scales as
‖X‖ −1 . α is invariant. This is no more true when s ≥ 2.
We can extend those definitions to projective or multiprojective
space by setting β(f, x) = β(f, X) where X is scaled such that ‖X 1 ‖ =
· · · = ‖X s ‖ = 1. (The same for γ and α).
Lemma 7.9 that was crucial for alpha theory. Now it becomes:
Lemma 9.4. Let X, Y ∈ M and f ∈ F. Assume that u = ‖X −
Y‖γ(f, X) < 1 − √ 2/2. Then,
‖Df(Y) † Df(X)‖ ≤
(1 − u)2
.
ψ(u)
Proof. Expanding Y ↦→ Df(X) † Df(Y) around X, we obtain:
Df(X) † Df(Y) =Df(X) † Df(X)+
+ ∑ 1
k − 1! Df(X)† D k f(X)(Y − X) k−1 .
k≥2
Rearranging terms and taking norms, Lemma 7.6 yields
‖Df(X) † Df(Y) − Df(X) † Df(X)‖ ≤
In particular,
1
(1 − γ‖Y − X‖) 2 − 1.
‖Df(X) † Df(Y) | ker Df(X) ⊥ − Df(X) † Df(X) | ker Df(X) ⊥‖ ≤
1
≤
(1 − γ(f, X)‖Y − X‖) 2 − 1.
Now we have full rank endomorphisms of ker Df(X) ⊥ on the left,
so we can apply Lemma 7.8 to get:
‖Df(Y) −1
| ker Df(X) ⊥ Df(X)‖ ≤
(1 − u)2
. (9.1)
ψ(u)
Because of the minimality property of the pseudo-inverse (see
Lemma 9.2),
‖Df(Y) † Df(X)‖ ≤ ‖Df(Y) −1
| ker Df(X) ⊥ Df(X)‖
so (9.1) proves the Lemma.