Fixed Point Method

Description

We can use the fixed-point iteration to find the root of a function. Given a function $f(x)$ which we have set to zero to find the root ($f(x) = 0$), we rewrite the equation in terms of $x$ so that $f(x) = 0$ becomes $x = g(x)$ (note, there are often many $g(x)$ functions for each $f(x) = 0$ function). Next, we relabel each side of the equation as $x_{n + 1} = g(x_n)$ so that we can perform the iteration. Next, we pick a value for $x_1$ and perform the iteration until it converges towards a root of the function. If the iteration converges, it will converge to a root. The iteration will only converge if $|g'(root)| \lt 1$.

Algorithm

1. Select the initial value of $x$ that is in the interval where the function $g(x)$ was found to converge.

2. Evaluate $g(x)$ on the selected $x$.

3. Assign the result of $g(x)$ to $x$ for the next iteration.

4. Calculate the relative error, if it’s sufficiently small, the calculation stops, otherwise, continue:

   $$e_p = |\dfrac{x_{R (current)} - x_{R (previous)}}{x_{R (current)}}| \times 100$$