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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 Numerical Methods
Edexcel A level Maths: 10.2 Iteration (Cobweb and Staircase Diagrams)Curated by Kepler, not produced by us. Watch it 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
1

The iterative formula

Rearrange the equation into x=g(x)x = g(x), then iterate xn+1=g(xn)x_{n+1} = g(x_n) from a starting value x0x_0. If the sequence settles down, it converges to a root.

xn+1=g(xn)x_{n+1} = g(x_n)
Iterative formula.
Example 1Use xn+1=xn+1x_{n+1} = \sqrt{x_n + 1} with x0=1x_0 = 1 (3 steps).
1x1=2≈1.414x_1 = \sqrt{2} \approx 1.414.
2x2≈2.414≈1.554x_2 \approx \sqrt{2.414} \approx 1.554.
3x3≈1.598x_3 \approx 1.598.
Answer
Converging towards ≈ 1.618.

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

2

Cobweb and staircase

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

Formula recap

xn+1=g(xn)x_{n+1} = g(x_n)
Iteration.
x=g(x)⇔f(x)=0x = 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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. An iterative formula has the form:

    • xn+1=g(xn)x_{n+1} = g(x_n)
    • xn+1=f(xn)=0x_{n+1} = f(x_n) = 0
    • xn=g(xn+1)x_n = g(x_{n+1})
    • x=0x = 0
    Show answer and explanation

    Answer: xn+1=g(xn)x_{n+1} = g(x_n)

    Next from current.

  2. To set up iteration you rearrange f(x)=0 into:

    • x=g(x)x = g(x)
    • y=g(x)y = g(x)
    • f′(x)=0f'(x) = 0
    • x2=g(x)x^2 = g(x)
    Show answer and explanation

    Answer: x=g(x)x = g(x)

    Fixed-point form.

  3. A staircase or cobweb diagram shows:

    • convergence/divergence
    • the derivative
    • the area
    • asymptotes
    Show answer and explanation

    Answer: convergence/divergence

    Behaviour of the iteration.

  4. You should round:

    • only the final answer
    • every step
    • never
    • x₀ only
    Show answer and explanation

    Answer: only the final answer

    Avoid accumulated error.

  5. If the iterates move away from the root, the sequence:

    • diverges
    • converges
    • is periodic
    • is constant
    Show answer and explanation

    Answer: diverges

    Try another g(x).

Next step

Keep practising

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