Nonlinear Equations - UFRJ

92 [CH. 7: NEWTON ITERATION
The Taylor expansions of f and Df around 0 are respectively:
f(x) = x + ∑ k≥2
1
k! Dk f(0)x k
and
Df(x) = I + ∑ k≥2
1
k − 1! Dk f(0)x k−1 . (7.7)
Combining the two equations, above, we obtain:
f(x) − Df(x)x = ∑ k≥2
k − 1
D k f(0)x k .
k!
Using Lemma 7.6 with d = 2, the rightmost term in (7.6) is
bounded above by
‖f(x) − Df(x)x‖ ≤ ∑ k≥2(k − 1)γ k−1 ‖x‖ k =
γ‖x‖ 2
(1 − γ‖x‖) 2 . (7.8)
Combining Lemma 7.9 and (7.8) in (7.6), we deduce that
‖N(f, x)‖ ≤
γ‖x‖2
ψ(γ‖x‖) .
By induction, u i ≤ γ‖x i ‖. When u 0 ≤ (3 − √ 7)/2, we obtain as
in Lemma 7.10 that
‖x i ‖
‖x 0 ‖ ≤ u i
u 0
≤ 2 −2i +1 .
We have seen in Lemma 7.10 that the bound above fails for i = 1
when u 0 > (3 − √ 7)/2.
Notice that in the proof above,
lim
i→∞
u 0
ψ(u i ) = u 0.
Therefore, convergence is actually faster than predicted by the
definition of approximate zero. We proved actually a sharper result: