Nonlinear Equations - UFRJ

128 [CH. 9: THE PSEUDO-NEWTON OPERATOR
In the multi-projective setting, we define
∑
d proj (X, Y) = √ s d proj (X i , Y i ) 2 .
i=1
Again, this is scaling invariant and we have
d proj (x, y) ≤ dRiem(x, y)
Definition 9.6 (Approximate zero of the first kind). Let f ∈ F 1 ×
· · · × F n , and z ∈ M/H with f(z) = 0. An approximate zero of the
first kind associated to z is a point X 0 ∈ M, such that
1. The sequence (X) i defined inductively by X i+1 = N pseu (f, X i )
is well-defined.
2.
d proj (X i , Z) ≤ 2 −2i +1 d proj (X 0 , Z).
Theorem 9.7 (Smale). Let f ∈ F 1 × · · · × F s and let Z be a nondegenerate
zero of f, scaled such that ‖Z 1 ‖ = · · · = ‖Z s ‖ = 1. Let
X 0 be scaled such that d proj (X 0 , Z) = ‖X 0 − Z‖. If
‖X 0 − Z‖ ≤ 3 − √ 7
2γ(f, Z) ,
then X 0 is an approximate zero of the first kind associated to Z.
This is an improvement of Corollary 1 in [33]. The improvement
is made possible because we do not rescale X 1 , X 2 , . . . .
Proof of Theorem 7.5. Set γ = γ(f, Z), u 0 = ‖X 0 − Z‖γ, and let h γ ,
(u i ) be as in Lemma 7.10.
We bound
‖N(f, X) − Z‖ = ∥ ∥X − Z − Df(X) † f(X) ∥ ∥
≤ ‖Df(X) † ‖‖f(X) − Df(X)(X − Z)‖.
The Taylor expansions of f and Df around Z are respectively:
f(X) = Df(Z)(X − Z) + ∑ k≥2
1
k! Dk f(Z)(X − Z) k
(9.2)