Staircase and cobweb diagrams

In a nutshell

An iterative formula can take the form of a staircase or a cobweb diagram when sketched on a graph. These are visual indications of convergence to a root, or divergence away from it. 

Staircase diagrams

​$$f(x)=x^2-2x-1$$ can produce the iterative formula $$x=\sqrt{2x+1}$$. To begin, plot two lines on a graph corresponding to each side of this equation: $$y=\sqrt{2x+1}$$​  and $$y=x$$.

Start with $$x_0$$ as the origin. To create a staircase diagram, draw up from $$x_0$$ until meeting the curve, then across from the curve to the line, then up from the line until meeting the curve again and so on. Every point the staircase diagram meets the line $$y=x$$ is a new $$x_n$$ number: $$x_0, x_1,x_2,...$$ A diagram is shown below:

As you can see, the 'stairs' get closer and closer to the point of intersection between $$y=\sqrt{2x+1}$$ and $$y=x$$. That intersecting point lies at $$x_n$$, which is a root of the original equation $$f(x)=x^2-2x-1$$. The staircase diagram converges onto the root.

Cobweb diagrams

Cobweb diagrams have a the same principles as a staircase diagram. Consider the function $$f(x)=2x^3-4x^2-1$$, with an iterative formula $$g(x)=2+\dfrac{1}{2x^2}$$. Plot $$g(x)$$ and the line $$y=x$$ onto a graph.

Making $$x_0=0.5$$ and following the same rules as the staircase diagram, this diagram emerges:

The diagram circles around the root, forming what's known as a cobweb diagram. The result is the same: through iteration you land on the intersection of $$y=g(x)$$ and $$y=x$$, which is the point whose $$x$$-coordinate is the root of $$f(x)=2x^3-4x^2-1$$.​