First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 87
Application to Newton’s Method
Recall Newton’s iteration
p n = p n−1 − f(p n−1)
f ′ (p n−1 ) .
Put g(x) =x − f(x) . Then the fixed-point iteration p f ′ (x) n = g(p n−1 ) is Newton’s method. We
have
g ′ (x) =1− [f ′ (x)] 2 − f(x)f ′′ (x)
= f(x)f ′′ (x)
[f ′ (x)] 2 [f ′ (x)] 2
and thus
Similarly,
g ′ (p) = f(p)f ′′ (p)
[f ′ (p)] 2 =0.
g ′′ (x) = (f ′ (x)f ′′ (x)+f(x)f ′′′ (x)) (f ′ (x)) 2 − f(x)f ′′ (x)2f ′ (x)f ′′ (x)
[f ′ (x)] 4
which implies
g ′′ (p) = (f ′ (p)f ′′ (p)) (f ′ (p)) 2
= f ′′ (p)
[f ′ (p)] 4 f ′ (p) .
If f ′′ (p) ≠0, then Theorem 46 implies Newton’s method has quadratic convergence with
lim
n→∞
which was proved earlier in Theorem 32.
p n+1 − p
(p n − p) = f ′′ (p)
2 2f ′ (p)
Exercise 2.7-1: Use Theorem 38 (and Remark 39) toshowthatg(x) =3 −x has a
unique fixed-point on [1/4, 1]. Use Corollary 42, part (4), to find the number of iterations
necessary to achieve 10 −5 accuracy. Then use the Julia code to obtain an approximation,
and compare the error with the theoretical estimate obtained from Corollary 42.
Exercise 2.7-2: Let g(x) =2x − cx 2 where c is a positive constant. Prove that if the
fixed-point iteration p n = g(p n−1 ) converges to a non-zero limit, then the limit is 1/c.