10.2PureStretch

Iteration

Rearranging f(x) = 0 into the form x = g(x) lets you generate a sequence xₙ₊₁ = g(xₙ) that may converge to a root. Cobweb and staircase diagrams show how the iteration behaves.

26 min Video by Zeeshan Zamurred Numerical Methods

Edexcel A level Maths: 10.2 Iteration (Cobweb and Staircase Diagrams)Watch the full walkthrough before the notes below.

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What you'll be able to do

Rearrange f(x) = 0 to x = g(x)
Run an iterative formula
Interpret cobweb and staircase diagrams
Recognise convergence and divergence

The iterative formula

Rearrange the equation into $x = g (x)$ , then iterate $x_{n + 1} = g (x_{n})$ from a starting value $x_{0}$ . If the sequence settles down, it converges to a root.

x_{n + 1} = g (x_{n})

Iterative formula.

x_{1} = 2 \approx 1.414

x_{2} \approx 2.414 \approx 1.554

x_{3} \approx 1.598

AnswerConverging towards ≈ 1.618.

Tip — Keep extra decimal places between steps to avoid rounding error.

Cobweb and staircase

On a graph of $y = x$ and $y = g (x)$ , the iteration is traced by horizontal and vertical steps. A $staircase$ pattern (steady approach) or $cobweb$ (oscillating) shows convergence; moving away shows divergence.

Formula recap

x_{n + 1} = g (x_{n})

Iteration.

x = g (x) \Leftrightarrow f (x) = 0

Rearranged equation.

Common mistakes to avoid

Rounding too early between iterations.

Carry full precision; round only the final answer.

Assuming every rearrangement converges.

Some forms of g(x) diverge — a different rearrangement may be needed.

Key takeaways

Rearrange to x = g(x), iterate xₙ₊₁ = g(xₙ).
Cobweb/staircase diagrams show convergence or divergence.
Keep full precision between steps.

Test yourself

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