Nonlinear Equations - UFRJ

[SEC. 9.4: THE ALPHA THEOREM 131
Those will be compared to
ˆβ i = β(h βγ , t i )) and ˆγ i = γ(h βγ , t i )).
Induction hypothesis: β i ≤ ˆβ i and for all l ≥ 2,
‖Df(X i ) † D l f(X i )‖ ≤ − h(l) βγ (t i)
h ′ βγ (t i) .
The initial case when i = 0 holds by construction.
assume that the hypothesis holds for i. We will estimate
So let us
β i+1 ≤ ‖Df(X i+1 ) † Df(X i )‖‖Df(X i ) † f(X i+1 )‖ (9.4)
and
γ i+1 ≤ ‖Df(X i+1 ) † Df(X i )‖ ‖Df(X i) † D k f(X i+1 )‖
. (9.5)
k!
By construction, f(X i ) + Df(X i )(X i+1 − X i ) = 0. The Taylor
expansion of f at X i is therefore
Df(X i ) † f(X i+1 ) = ∑ k≥2
Passing to norms,
while we know from (7.14) that
ˆβ i+1 = − h βγ(t i+1 )
h ′ βγ (t i)
From Lemma 9.4,
Df(X i ) † D k f(X i )(X i+1 − X i ) k
k!
‖Df(X i ) † f(X i+1 )‖ ≤ β2 i γ i
1 − γ i
= β(h βγ, t i ) 2 γ(h βγ , t i )
1 − γ(h βγ , t i )
‖Df(X i+1 ) † Df(X i )‖ ≤ (1 − β iγ i ) 2
.
ψ(β i γ i )
= ˆβ 2 i ˆγ i
1 − ˆγ i