Iterative Methods

This lesson covers iterative methods for solving equations — rearranging f(x) = 0 into the form x = g(x) and using iteration — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to understand convergence and divergence, and interpret staircase and cobweb diagrams.

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Why Iterative Methods?

Many equations cannot be solved exactly using algebraic methods. For example:

• x³ + 3x - 7 = 0
• eˣ = 3x
• x = cos x

In these cases, we use numerical methods to find approximate solutions. Iterative methods generate a sequence of values x₁, x₂, x₃, ... that (hopefully) converge to the root.

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The Fixed-Point Iteration Method

The Idea

1. Start with the equation f(x) = 0.
2. Rearrange it into the form x = g(x).
3. Choose a starting value x₀.
4. Apply the iteration formula: x_{n+1} = g(x_n) repeatedly.
5. If the sequence converges, the limit is a root of the original equation.

Why Does This Work?

If the sequence converges to a limit L, then as n → ∞: x_{n+1} → L and x_n → L

So the iteration formula x_{n+1} = g(x_n) becomes L = g(L).