Nonlinear Equations - UFRJ

102 [CH. 7: NEWTON ITERATION
Passing to norms,
‖Df(x i ) −1 f(x i+1 )‖ ≤ β2 i γ i
1 − γ i
The same argument shows that
From Lemma 7.9,
− h βγ(t i+1 )
h ′ βγ (t i)
= β(h βγ, t i ) 2 γ(h βγ , t i )
1 − γ(h βγ , t i )
‖Df(x i+1 ) −1 Df(x i )‖ ≤ (1 − β iγ i ) 2
.
ψ(β i γ i )
Also, computing directly,
We established that
β i+1 ≤ β2 i γ i(1 − β i γ i )
ψ(β i γ i )
h ′ βγ (t 2
i+1) (1 − ˆβˆγ)
h ′ βγ (t =
i) ψ( ˆβˆγ)
. (7.14)
≤ ˆβ 2 i ˆγ i(1 − ˆβ iˆγ i )
ψ( ˆβ iˆγ i )
Now the second part of the induction hypothesis:
= ˆβ i+1 .
Df(x i ) −1 D l f(x i+1 ) = ∑ k≥0
1 Df(x i ) −1 D k+l f(x i )(x i+1 − x i ) k
k!
k + l
Passing to norms and invoking the induction hypothesis,
‖Df(x i ) −1 D l f(x i+1 )‖ ≤ ∑ k≥0
and then using Lemma 7.9 and (7.14),
− h(k+l) βγ
(t i ) ˆβ i
k
k!h ′ βγ (t i)
‖Df(x i+1 ) −1 D l f(x i+1 )‖ ≤ (1 − ˆβ iˆγ i ) 2 ∑
ψ( ˆβ
− h(k+l) βγ
(t i ) ˆβ i
k
iˆγ i ) k!h ′ βγ (t i) .
k≥0