Fixed point Method

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Fixed Point Method
A fixed point for a function is a number at which the value of the function does not change when the function is applied.
Definition 1: The number p is a fixed point for a given function g if g(p) = p.

Given a root-finding problem f ( p) = 0, we can define functions g with a fixed point at p in a number of ways, for example, as
g(x) = x ? f (x) or as g(x) = x + 3f (x).

Conversely, if the function g has a fixed point at p, then the function defined by
f (x) = x ? g(x)
has a zero at p.

Example 4: Determine any fixed points of the function g(x) = x2 ? 2.

Solution: A fixed point p for g has the property that
p = g(p) = p2 ? 2 which implies that 0 = p2 ? p ? 2 = (p + 1)(p ? 2).

A fixed point for g occurs precisely when the graph of y = g(x) intersects the graph of
y = x, so g has two fixed points, one at p = ?1 and the other at p = 2. These are shown in Figure 1.