Solving Equations By Iteration
Sometimes we cannot solve an equation exactly. If we cannot solve an equation exactly we may be able to solve it approximately using iteration.
Example: Solve the equation
We can rearrange this equation.
From this we can get the iteration rule
If we take
We could go on. The iterates seem to be getting closer together. In fact because the original equation is a quadratic we can solve it exactly.
The iteration is converging to the first of this. An iteration can only converge to one solution at a time, and iteration is not guaranteed. The solution that the the iterations converge to may be different if the original equation is rearranged to give a different iteration formula.