I2PureStretch

Iteration

Q: What is a fixed point?

A value of x where g(x)=x exactly — geometrically, where the curve y=g(x) crosses the line y=x. It corresponds to a root of the original equation f(x)=0.

Q: Why do some rearrangements converge and others do not?

Convergence depends on the gradient g'(x) near the fixed point. If |g'(x)| 1, it moves further away.

Rearrange an equation $f (x) = 0$ into the form $x = g (x)$ , feed a starting guess through $g$ repeatedly, and — remarkably — the results often converge onto the exact root, getting closer with every repetition. This is fixed-point iteration.

32 min Video by TLMaths Numerical Methods

The big picture

The change-of-sign method only traps a root inside an interval; $iteration$ can pin it down to as many decimal places as you like. Rearranging $f (x) = 0$ into $x = g (x)$ turns root-finding into a self-feeding process: substitute a guess into $g$ , get a better guess, and repeat. Whether this converges (and how fast) depends entirely on the gradient of $g$ near the root — which is exactly what cobweb and staircase diagrams visualise.

A-Level Maths I2-01: Locating Roots: Introducing the x=g(x) MethodWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Rearrange $f (x) = 0$ into an iterative form $x = g (x)$
Generate a sequence of iterates using $x_{n + 1} = g (x_{n})$
Sketch and interpret staircase and cobweb diagrams
Recognise conditions for convergence and divergence

Rearranging into x = g(x)

An equation $f (x) = 0$ can usually be rearranged algebraically (isolating one $x$ ) into the form $x = g (x)$ . A root of $f$ then corresponds to a $fixed point$ of $g$ : a value where $g (x) = x$ exactly.

1Isolate

x

x^{3} = x + 1

, so

x = 3 x + 1

Answer

x = g (x) = 3 x + 1

Tip — Many rearrangements are possible for the same equation — some converge quickly, some diverge entirely. Choosing a good one is part of the skill (covered further below).

Generating the iteration

Starting from an initial guess $x_{0}$ , repeatedly apply $g$ : $x_{1} = g (x_{0})$ , $x_{2} = g (x_{1})$ , and so on. If the sequence converges, it converges to a root of the original equation.

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

The iterative formula.

x_{1} = 3 1 + 1 = 32 \approx 1.2599

x_{2} = 3 1.2599 + 1 = 3 2.2599 \approx 1.3103

x_{3} = 3 1.3103 + 1 = 3 2.3103 \approx 1.3212

Answer

x_{1} \approx 1.2599

x_{2} \approx 1.3103

x_{3} \approx 1.3212

(converging toward the true root

\approx 1.3247

)

Cobweb and staircase diagrams

Plotting $y = g (x)$ alongside $y = x$ visualises the iteration process: draw a vertical line from $x_{0}$ to the curve $y = g (x)$ (giving $x_{1}$ ), then a horizontal line across to $y = x$ , then vertical again to the curve, repeating. A $staircase$ pattern (moving steadily in one direction) shows convergence when $0 < g^{'} (x) < 1$ near the root; a $cobweb$ pattern (spiralling around the fixed point) shows convergence when $- 1 < g^{'} (x) < 0$ .

Why this works

If $∣ g^{'} (x) ∣ > 1$ near the root, the iteration DIVERGES — the staircase or cobweb spirals outward rather than in, moving further from the root with each step.

Tip — To check whether an iteration will converge without drawing a full diagram, estimate $∣ g^{'} (x) ∣$ near the suspected root: if it is less than 1, expect convergence; if greater than 1, expect divergence.

Think like an examiner

When asked to comment on convergence, refer to the gradient of $g (x)$ near the root — $∣ g^{'} (x) ∣ < 1$ for convergence, $∣ g^{'} (x) ∣ > 1$ for divergence.
Always work to at least one more decimal place than the final accuracy required, to avoid rounding errors compounding across iterations.
A staircase diagram converges monotonically from one side; a cobweb diagram spirals in from both sides alternately — describe which you see accurately.

Common misconceptions

Assuming every rearrangement

x = g (x)

of an equation will converge.

Convergence depends on the gradient of

g

near the root — some valid rearrangements diverge and are useless for iteration, even though they are algebraically correct.

Rounding intermediate iterates too aggressively.

Carry extra decimal places through each iteration step to avoid compounding rounding errors — only round the FINAL answer to the required accuracy.

Confusing a cobweb diagram with a staircase diagram.

A staircase moves in steadily on one side (positive gradient near the root); a cobweb spirals in, alternating sides (negative gradient near the root).

Fixed-point iteration

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

The iterative formula.

∣ g^{'} (x) ∣ < 1 near the root \Rightarrow converges

Condition for convergence.

Stretch yourself

The equation $x^{2} - 5 x + 2 = 0$ is rearranged as $x = \frac{x ^{2} + 2}{5}$ . Using $x_{0} = 0.5$ , find $x_{1}$ , $x_{2}$ , $x_{3}$ to 4 d.p., and comment on convergence.

Hint — Substitute repeatedly into $g (x) = \frac{x ^{2} + 2}{5}$ , then estimate $g^{'} (x)$ near the resulting value to comment on convergence.

Questions students ask

What is a fixed point?

A value of

x

where

g (x) = x

exactly — geometrically, where the curve

y = g (x)

crosses the line

y = x

. It corresponds to a root of the original equation

f (x) = 0

Why do some rearrangements converge and others do not?

Convergence depends on the gradient

g^{'} (x)

near the fixed point. If

∣ g^{'} (x) ∣ < 1

there, the iteration draws closer with each step; if

∣ g^{'} (x) ∣ > 1

, it moves further away.

What is the difference between a staircase and a cobweb diagram?

Both visualise the same iteration process. A staircase pattern (positive gradient near the root, between 0 and 1) approaches from one side only; a cobweb pattern (negative gradient, between −1 and 0) spirals in, alternating sides.

Key takeaways

Rearrange $f (x) = 0$ into $x = g (x)$ ; a root becomes a fixed point of $g$ .
Generate iterates using $x_{n + 1} = g (x_{n})$ from a starting value.
Staircase diagrams show convergence for $0 < g^{'} (x) < 1$ ; cobweb diagrams for $- 1 < g^{'} (x) < 0$ .
If $∣ g^{'} (x) ∣ > 1$ near the root, the iteration diverges.

How this fits the course

Build on

Locating Roots

The Newton-Raphson Method

Leads to

The Newton-Raphson Method

Test yourself

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