# The Fixed Point Method for Approximating Roots

We saw on the Fixed Points page that $\alpha$ is a fixed point of $g$ if $\alpha = g(\alpha)$. We will now discuss why fixed points are important in finding roots. Suppose that we want to solve $f(\alpha) = 0$. Then we can define a function $g$ with a fixed point at $\alpha$ in many different ways, and as a result, if $\alpha$ is a fixed point of $g$, then $f$ will have a zero at $x = \alpha$. For example, consider the function $f(x) = x^2 - 3x + 1$. Suppose that we want to find a root of $f$. Of course, the easiest method would be applying the quadratic formula, however, let's put that aside for the time. We want to find $\alpha$ such that $f(\alpha) = 0$, that is:

(1)Now suppose that we let $g(\alpha) = -\frac{1}{\alpha - 3}$ for $\alpha \neq 3$ (clearly $3$ is not a root of the original function since f(3) \neq 0 $]]). Then we have that:

(2)So $\alpha$ is a fixed point of $g$, and furthermore, $f(x) = g(x) - x = 0$, so $f(\alpha) = g(\alpha) - \alpha = 0$ and $\alpha$ is a root of $f$. So if we can approximate a fixed point of $g$, then we consequentially approximate a root of $f$.

Let $f$ be a continuous function and suppose that there exists a root $\alpha$ of $f$ on the interval $[a, b]$, that is, find $\alpha \in [a, b]$ such that $f(\alpha) = 0$. If we can rewrite $f$ as $x = g(x)$ for some function $g$ (note that the form $x = g(x)$ may not be unique), then it is possible that the recursive sequence $x_{n+1} = g(x_n)$ will converge, and if this sequence does converge, say to some $\alpha$, then:

(3)We are now ready to look at the Fixed Point Method for finding roots of a function.

Theorem 1 (The Fixed Point Method): Suppose that $f$ is a continuous function on $[a, b]$ and that we want to solve $f(x) = 0$ in the form $x = g(x)$ where $g$ is continuous on $[a, b]$. A fixed point $\alpha$ of $g$ is therefore a root of $f$.Step 1: Let $x_0$ be an initial approximation.Step 2: Generate a sequence $\{ x_n \}$ where $x_{n+1} = g(p_n)$ for $n ≥ 0$.If the sequence $\{ x_n \}$ converges, then $\lim_{n \to \infty} x_n = \alpha$. |

In Theorem 1 above, the Fixed Point Method may not converge. The following graph represents a scenario for which the sequence $\{ x_n \}$ converges to the fixed point of interest.

Meanwhile, the next graph represents a scenario for which the sequence $\{ x_n \}$ diverges.