The fixed-point iteration method

The fixed-point iteration method rewrites the equation $f(x)=0$ in the form $g(x)=x$. For any particular function $f(x)$, There are many ways to do this, but one procedure that will always give the right form is to add $x$ to both sides of the original equation, thus giving $f(x)+x=x$ and then identifying $g(x)$ as $f(x)+x$. Thus in our example above we have

$$g(x)=f(x)+x=\cos x = x$$ (3)

$\phantom{0}$There are other ways in which our equation could be written in the form $g(x)=x$. Find one of these.

Fixed-point iteration then works by choosing a first guess for the root, say $x_0$, which can be obtained from a graph. Once such a first guess is obtained, (hopefully) improved estimates are found using the iteration

$$x_{n+1} = g(x_n), \quad n=0,1,2,\dots$$ (4)

where $g(x)=\cos x$ for our case.

First we will use MATLAB as a simple calculator to see how this procedure works. In the MATLAB window enter the initial $x=0.75$.

Then use the following command to calculate the first improved iteration:

x = cos(x)

The new value obtained is $x_1$ (it should be 0.7317). Repeat the command three times in order to obtain $x_4$.

$\phantom{0}$ Record the value of $x_4$ to four significant figures.