Nonlinear Equations - UFRJ

98 [CH. 7: NEWTON ITERATION
The constant α 0 is the largest possible with those properties.
This theorem appeared in [77]. The value for α 0 was found by
Wang Xinghua [84]. Numerically,
α 0 = 0.157, 670, 780, 786, 754, 587, 633, 942, 608, 019 · · ·
Other useful numerical bounds, under the hypotheses of the theorem,
are:
r 0 ≤ 1.390, 388, 203 · · · and r 1 ≤ 0.390, 388, 203 · · · .
The proof of Theorem 7.15 follows from the same method as the
one for Theorem 7.5. We first define the ‘worst’ real function with
respect to Newton iteration. Let us fix β, γ > 0. Define
h βγ (t) = β − t +
γt2
1 − γt = β − t + γt2 + γ 2 t 3 + · · · .
We assume for the time being that α = βγ < 3−2 √ 2 = 0.1715 · · · .
This guarantees that h βγ has two distinct zeros ζ 1 = 1+α−√ ∆
4γ
and
ζ 2 = 1+α+√ ∆
4γ
with of course ∆ = (1 +α) 2 −8α. An useful expression
is the product formula
h βγ (x) = 2 (x − ζ 1)(x − ζ 2 )
γ −1 . (7.9)
− x
From (7.9), h βγ has also a pole at γ −1 . We have always 0 < ζ 1 <
ζ 2 < γ −1 .
The function h βγ is, among the functions with h ′ (0) = −1 and
β(h, 0) ≤ β and γ(h, 0) ≤ γ, the one that has the first zero ζ 1 furthest
away from the origin.
Proposition 7.16. Let β, γ > 0, with α = βγ ≤ 3 − 2 √ 2. let h βγ be
as above. Define recursively t 0 = 0 and t i+1 = N(h βγ , t i ). then
with
t i = ζ 1
1 − q 2i −1
1 − ηq 2i −1 , (7.10)
η = ζ 1
ζ 2
= 1 + α − √ ∆
1 + α + √ ∆ and q = ζ 1 − γζ 1 ζ 2
ζ 2 − γζ 1 ζ 2
= 1 − α − √ ∆
1 − α + √ ∆ .