My title - Departamento de Matemática da Universidade do Minho

5 ÓRBITAS REGULARES E PERTURBAÇÕES 38
Proof. We may assume, without loss of generality, that the root we are looking for is the origin,
so that p(0) = 0. Now, suppose we are at x n after n iterations. Taylor’s formula with Lagrange
estimate of the error around x n says that
p(x) = p(x n ) + p ′ (x n ) · (x − x n ) + 1 2 p′′ (y) · (x − x n ) 2
for some y between x and x n . Taking x = 0 (the root!) and dividing by p ′ (x n ) we get
0 = p(0) = p(x n ) − p ′ (x n ) − x n + 1 2 p′′ (y) · x 2 n i.e. x n − p(x n)
p ′ (x n ) = 1 p ′′ (y)
2 p ′ (x n ) · x2 n
But the l.h.s. is x n+1 . There follows that the distance ε n = |x n − 0| between the n-th iterate and
the root satisfies the iterative bound
|x n+1 | ≤ 1 p ′′ (y)
2 ∣p ′ (x n ) ∣ · |x n| 2
Since p ′ (0) ≠ 0 (and polynomials have continuous derivatives), there is an interval I =] − ε, ε[
around the root 0 where M = sup x∈I |p ′′ (x)| < ∞ and δ = inf x∈I |p ′ (x)| > 0. Let K = M/2δ. If
the initial guess is sufficiently near the root 0 (so near that |x 0 | < 1 and also K|x 0 | < 1), then the
x n ’s converge to 0 and the convergence is quadratic, i.e.
|x n+1 | ≤ K · |x n | 2
Exercícios.
• Verifique que o método de Newton aplicado ao polinómio quadrático z 2 − a, com a > 0,
corresponde ao algoritmo de Heron!
• Estime √ 17 ...
• Escreva a receita do método de Newton para resolver z n − a = 0, com a > 0 e n ≥ 2.
• Utilize o método de Newton para estimar raizes de
Iteração de funções racionais.
z 2 + 1 + z z 3 − z − 1 z 5 + z + 1 z 3 − 2z − 5