First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 2. SOLUTIONS OF EQUATIONS: ROOT-FINDING 77
2.6 Fixed-point iteration
Many root-finding methods are based on the so-called fixed-point iteration; a method we
discuss in this section.
Definition 37. Anumberp is a fixed-point for a function g(x) if g(p) =p.
We have two problems that are related to each other:
• Fixed-point problem: Find p such that g(p) =p.
• Root-finding problem: Find p such that f(p) =0.
We can formulate a root-finding problem as a fixed-point problem, and vice versa. For
example, assume we want to solve the root finding problem, f(p) =0. Define g(x) =x−f(x),
and observe that if p is a fixed-point of g(x), that is, g(p) =p − f(p) =p, then p is a root
of f(x). Here the function g is not unique: there are many ways one can represent the
root-finding problem f(p) =0as a fixed-point problem, and as we will learn later, not all
will be useful to us in developing fixed-point iteration algorithms.
The next theorem answers the following questions: When does a function g have a fixedpoint?
If it has a fixed-point, is it unique?
Theorem 38. 1. If g is a continuous function on [a, b] and g(x) ∈ [a, b] for all x ∈ [a, b],
then g has at least one fixed-point in [a, b].
2. If, in addition, |g(x) − g(y)| ≤λ|x − y| for all x, y ∈ [a, b] where 0