Iterative Methods

Forming Iteration Formulas by Rearranging Equations

Iteration formulas are formed by rearranging equations and isolating a single variable.

There are many ways to rearrange equations, and not all will give you an iteration formula that converges to give a root. So an exam question will most likely ask you to show that an equation can be rearranged into a specific form.

Example: 

Show that $2x^3-3x-5=0$ can be rearranged to: $x=sqrt[3]{dfrac{3x+5}{2}}$

First add $3x+5$ to both sides to give:

$2x^3=3x+5$

Next divide both sides by $2$:

$x^3=dfrac{3x+5}{2}$

Finally take the cube root of both sides:

$x=sqrt[3]{dfrac{3x+5}{2}}$

Therefore, the iteration formula is: $x_{n+1}=sqrt[3]{dfrac{3x_{n}+5}{2}}$ to find approximate roots.

Using Iterations to Draw Diagrams

After you have used an iteration method, you can form a sequence of iterations using $x_{n+1}=f(x_n)$ and then plot these points on a diagram to show whether the sequence converges or diverges.

Forming iteration diagrams

1) Sketch the graphs of $y=x$ and $y=f(x)$, where $f(x)$ is the iterative formula. The root of the original equation is the point of intersection of the two graphs.

2) From your starting point $x_0$ draw a vertical line until it meets $y=x$

3) Next draw a horizontal line from this point to the line $y=f(x)$. This point is the first iteration, $x_1$.

4) After this, draw a vertical line from this point to the line $y=x$ and then a horizontal line to $y=f(x)$. Repeat this step for the remaining iterations.

5) If after each step points are getting closer to the roots, the sequence is converging. If after each step the points are getting further away from the root, the sequence is diverging.

There are two types of diagrams – staircase diagrams and cobweb diagrams.

In convergent staircase diagrams, the iterations increasingly get closer to the root.

In convergent cobweb diagrams, the iterations alternate between going above and below the root, progressively getting closer.