Nonlinear Equations - UFRJ

[SEC. 7.3: ESTIMATES FROM DATA AT A POINT 97
Definition 7.13 (Approximate zero of the second kind). Let f : D ⊆
E → F be as above. An approximate zero of the first kind associated
to ζ ∈ D, f(ζ) = 0, is a point x 0 ∈ D, such that
1. The sequence (x) i defined inductively by x i+1 = N(f, x i ) is
well-defined (each x i belongs to the domain of f and Df(x i ) is
invertible and bounded).
2.
‖x i+1 − x i ‖ ≤ 2 −2i +1 ‖x 1 − x 0 ‖.
3. lim i→∞ x i = ζ.
For detecting approximate zeros of the second kind, we need:
Definition 7.14 (Smale’s β and α invariants).
β(f, x) = ‖Df(x) −1 f(x)‖ and α(f, x) = β(f, x)γ(f, x).
The β invariant can be interpreted as the size of the Newton step
N(f, x) − x.
Theorem 7.15 (Smale). Let f : D ⊆ E → F be an analytic map
between Banach spaces. Let
Define
α ≤ α 0 = 13 − 3√ 17
.
4
r 0 = 1 + α − √ 1 − 6α + α 2
4α
and r 1 = 1 − 3α − √ 1 − 6α + α 2
.
4α
Let x 0 ∈ D be such that α(f, x 0 ) ≤ α and assume furthermore that
B(x 0 , r 0 β(f, x 0 )) ⊆ D. Then,
1. x 0 is an approximate zero of the second kind, associated to some
zero ζ ∈ D of f.
2. Moreover, ‖x 0 − ζ‖ ≤ r 0 β(f, x 0 ).
3. Let x 1 = N(f, x 0 ). Then ‖x 1 − ζ‖ ≤ r 1 β(f, x 0 ).