Nonlinear Equations - UFRJ

90 [CH. 7: NEWTON ITERATION
t 1
t 2
t 3
t 0
Figure 7.2: y = h γ (t)
are well-defined for all i, lim i→∞ t i = 0, and
|t i |
|t 0 | = u ( ) 2 i −1
i u0
≤
.
u 0 ψ(u 0 )
Moreover,
|t i |
|t 0 | ≤ 2−2i +1
for all i if and only if u 0 ≤ 3−√ 7
2
.
Proof. We just compute
h ′ γ(t) =
ψ(γt)
(1 − γt) 2
th ′ γ(t) − h γ (t) = − γt2
(1 − γt) 2
N(h γ , t) = − γt2
ψ(γt) .