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:
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) \begin{align} \alpha = g(\alpha) \end{align} 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) \begin{align} \quad \lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty} g(x_n) \\ \quad \alpha = g(\alpha) \end{align} We are now ready to look at the Fixed Point Method for finding roots of a function.  Theorem 1 (The Fixed Point Method): Suppose thatf$is a continuous function on$[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. 