Iteration for A Level Maths: staircase method & cobweb method

 E.g.

f(x) = x² - x - 4

When the above is rearranged in the form of x = g(x), it is said to be in iterative form. x² - x - 4 = 0 x = 1 + x/4 x_n+1 = 1 + x_n / 4 If x_n is known, x_n+1 can be calculated.

The initial value to trigger off the iterative process is found by looking for change in sign of the function by trial and improvement.

f(2) = - 2; < 0 f(3) = 2; > 0

That means there is a root between x = 2 and x = 3. Let's use x₀ as 2 to iterate the formula. As you can see, x approaches 2.56. The root is x ≈ 2.56(2dp)

Note: the simulation is made with Python:

The following two simulations show how iteration process works:

1. Cobweb method

As you can see, each iteration brings us closer to the root, weaving a pattern of a cobweb.

2. Staircase method - convergence to a root

Each iteration forms a part of a staircase and converges to a root.

2. Staircase method - divergence from a root

Although there is a root in the range, 2 < x < 3, if you start the iteration from x = 3, it is diverging from a root, instead of coming towards it.