First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 82
hence g(x) ∈ I. Now use Theorem 40, setting [a, b] to [p−ɛ, p+ɛ], to conclude the fixed-point
iteration converges.
To prove convergence is linear, we note
|p n+1 − p| = |g(p n ) − g(p)| ≤|g ′ (ξ n )||p n − p| ≤k|p n − p|
which is the definition of linear convergence (with k being a positive constant less than 1).
We can actually prove something more:
|p n+1 − p|
lim
n→∞ |p n − p|
= lim
n→∞
|g(p n ) − g(p)|
|p n − p|
= lim
n→∞
|g ′ (ξ n )||p n − p|
|p n − p|
= lim
n→∞
|g ′ (ξ n )| = |g ′ (p)|.
The last equality follows since g ′ is continuous, and ξ n → p, which is a consequence of ξ n
being between p and p n ,andp n → p, asn →∞.
Example 45. Let g(x) =x + c(x 2 − 2), which has the fixed-point p = √ 2 ≈ 1.4142. Pick
a value for c to ensure the convergence of fixed-point iteration. For the picked value c,
determine the interval of convergence I =[a, b], that is, the interval for which any p 0 from
the interval gives rise to a converging fixed-point iteration. Then write a Julia code to test
the results.
Solution. Theorem 44 requires |g ′ (p)| < 1. We have g ′ (x) =1+2xc, and thus g ′ ( √ 2) =
1+2 √ 2c. Therefore
|g ′ ( √ 2)| < 1 ⇒−1 < 1+2 √ 2c