Iterative Methods

Solving an equation by an iterative method is carrying out a process to get closer to a value. We are given a rough approximated initial value, and then become more and more accurate. This gets us closer to the real value and is necessary when we do not have a straightforward way of solving for the roots of the equation.

Using the iterative method

An iterative method can be used to find a value of $x$ when $f\left(x\right)=0$. To perform this iteration we first need to rearrange the function.

The basis of this is we need to rearrange $f\left(x\right)=0$ to $x=g\left(x\right)$. Therefore we need to make $x$ the subject of $f\left(x\right)=0$. However, it does not matter if we have $x$ terms remaining on the other side.

This is because we can perform $x_{n+1}=g\left(x_n\right)$, meaning we take our value of $x$ and iterate it.

Let's look at two key worked examples below.

Worked examples of iteration calculation

Let's look at a more difficult example.

How do we see iterative methods on a graph?

Iterative methods are all about getting closer and closer to a root of an equation. We use them when we cannot directly solve Equations with any other methods.

The higher the value of $n$ in $x_n$, the closer we are to the root of this equation as we are performing this process more and more times.

We can see this on a graph in two ways: a staircase diagram or a cobweb diagram.

Staircase diagrams

A staircase diagram works for a function that directly converges to a root meaning that from $x_0$we are either directly increasing or decreasing towards our root with each iterative value.

Let's look at our initial example $f\left(x\right)=x^2-3x-5$.

We will solely be focusing on the positive root (the one on the right).

A graph of $f\left(x\right)=x^2-3x-5$

We know from rearranging that $x=\sqrt{3x+5}$when$f\left(x\right)=0$ . Therefore if we sketch the lines $y=x$ and $y=\sqrt{3x+5}$, the intersection is the root of this equation.

An intersection between the lines $y=x$ and$y=\sqrt{3x+5}$

Let's now plot our points for $x_0,x_1,x_2$ and $x_3$.

Values $x_0,x_1,x_2,x_3$ marked.

Alt-text: Values marked on graph.

This is a more zoomed-in version and we can see that these points are slowly converging to the intersection. If we add a few lines, we can see our staircase.

A staircase diagram attaching all of the aforementioned points.

If we continue and use more values we will get closer and closer to that intersection. We will never actually reach it but we can get it in view of high accuracy.

Cobweb Diagrams

A cobweb diagram is when we are converging on a root in more than one direction, meaning our values become both too high and too low around the root.

Our second example $f\left(x\right)=x{e}^{-x}-x+3$ demonstrates this. We will be focusing on the negative root, the one to the left.

Image Caption

A sketch of the graph $f\left(x\right)=x{e}^{-x}-x+3$

Once again, from rearranging we know that $x=-\ln (1-\frac{3}{x})$when $f\left(x\right)=0$. Therefore we can sketch the lines $y=x$ and $y=-\ln (1-\frac{3}{x})$, and their intersection is the solution to the equation $f\left(x\right)=0$.

Intersection between the lines $y=x$ and $y=-\ln (1-\frac{3}{x})$,

Now let's plot our points for $x_0,x_1,x_2$ and $x_3$.

Points $x_0,x_1,x_2,x_3$ marked.

We can clearly see these points are not in order. Therefore we do not have a staircase, we have a cobweb and are converging on the value from different directions. This is what it looks like.

A cobweb diagram attaching all marked points.

We can see this is like a cobweb, and is converging towards a value from values greater than and less than the intersection.