Nonlinear Equations - UFRJ

96 [CH. 7: NEWTON ITERATION
Exercise 7.3. Consider the following series, defined in C 2 :
g(x) =
∞∑
x i 1x i 2.
i=0
Compute its radius of convergence. What is its domain of absolute
convergence ?
Exercise 7.4. The objective of this exercise is to produce a nonoptimal
algorithm to approximate √ y. In order to do that, consider
the mapping f(x) = x 2 − y.
1. Compute γ(f, x).
2. Show that for 1 ≤ y ≤ 4, x 0 = 1/2 + y/2 is an approximate
zero of the first kind for x, associated to y.
3. Write down an algorithm to approximate √ y up to relative
accuracy 2 −63 .
Exercise 7.5. Let f be an analytic map between Banach spaces, and
assume that ζ is a non-degenerate zero of f.
1. Write down the Taylor series of Df(ζ) −1 (f(x) − f(ζ)).
2. Show that if f(x) = 0, then
γ(f, ζ)‖x − ζ‖ ≥ 1/2.
This shows that two non-degenerate zeros cannot be at a distance
less than 1/2γ(f, ζ). (Results of this type appeared in [28], but some
of them were known before [55, Th.16]).
7.3 Estimates from data at a point
Theorem 7.5 guarantees quadratic convergence in a neighborhood of
a known zero ζ. In practical situations, ζ is not known. A major
result in alpha-theory is the criterion to detect an approximate zero
with just local information. We need to slightly modify the definition.