Nonlinear Equations - UFRJ

94 [CH. 7: NEWTON ITERATION
2 63
2 31
i = 3
i = 4
2 7
2 3
2
2 15 3− √ 7
i = 2
i = 1
2
5− √ 17
4
Figure 7.3: Values of log 2 (u i /u 0 ) in function of u 0 for i = 1, . . . , 4.
Therefore, we need to consider inexact Newton iteration. An obvious
modification of the proof of Theorem 7.5 gives us the following
statement:
Theorem 7.12. Let f : D ⊆ E → F be an analytic map between
Banach spaces. Let ζ be a non-degenerate zero of f. Let
√
14
0 ≤ 2δ ≤ u 0 ≤ 2 − ≃ 0.129 · · ·
2
Assume that
1.
( )
u 0
B = B ζ, ⊆ D.
γ(f, ζ)
2. x 0 ∈ B, and the sequence x i satisfies
‖x i+1 − N(f, x i )‖γ(f, ζ) ≤ δ
3. The sequence u i is defined inductively by
u i+1 =
u2 i
ψ(u i ) + δ.