Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams

Pearson Edexcel A-Level Mathematics: Simple Iterative Methods, Cobweb and Staircase Diagrams

Overview

Iteration is the numerical escape route when an equation will not give up a tidy algebraic answer. In Pearson Edexcel A-Level Mathematics 9MA0, section 9.2, candidates must turn an equation into the form (x=g(x)), repeatedly calculate (x_{n+1}=g(x_n)), and explain the resulting behaviour using a cobweb or staircase diagram. The calculation, graph and written conclusion all assess the same idea: the current approximation becomes the next input.

This is an excellent marks-per-minute topic when the method is organised. A question may ask you to calculate a few terms, choose or use an initial value, draw steps between (y=g(x)) and (y=x), or decide whether the iteration converges. It connects directly to roots, graphs, recurrence relations and the numerical methods that follow. The hard part is not pressing a calculator button. It is showing the examiner that you understand what the numbers and the graph mean. Vertical to the curve, horizontal to the diagonal is the construction to remember.

Specification anchor: Pearson Edexcel A-Level Mathematics (9MA0), topic 9.2, requires simple iterative methods of the form (x_{n+1}=f(x_n)), roots of (x=f(x)), and cobweb and staircase diagrams to show convergence geometrically.

Key Concepts

1. From a root equation to a fixed-point equation

A root problem normally begins as (F(x)=0). If exact algebra is awkward or impossible, rearrange it into (x=g(x)). A value (\alpha) satisfying (\alpha=g(\alpha$)) is a fixed point. Provided the rearrangement is valid for the root being sought, that fixed point gives a root of the original equation.$

For example, (x^2-x-1=0) can be written as (x^2=x+1). For the positive root, (x=\sqrt{x+1}), giving (x_{n+1}=\sqrt{x_n+1}). Notice the restriction: the positive square root targets the positive root. A rearrangement is not a decorative algebra step. It chooses the function that will generate the approximations.

Why it works: at a fixed point, putting (\alpha) into (g) returns (\alpha). The process has stopped changing. Iteration attempts to make the changing values (x_0,x_1,x_2,\ldots) settle at that value.

2. The iteration routine

Use the starting value stated in the question, or a sensible value suggested by a graph or bracket. Then calculate in order:

[
x_1=g(x_0),\quad x_2=g(x_1),\quad x_3=g(x_2),\ldots
]

Keep full calculator accuracy in the stored answer. Write only the requested accuracy in your table or final answer. A candidate who rounds every intermediate term to three decimal places can shift later terms enough to lose the accuracy mark.

For (x_{n+1}=\sqrt{x_n+1}) with (x_0=1):

[
x_1=\sqrt{2}=1.414213\ldots,\qquad
x_2=\sqrt{2.414213\ldots}=1.553774\ldots
]

The values rise towards (1.618\ldots). This looks promising, but do not write only “the answer is 1.618”. State that the iterates appear to converge to the fixed point. If the question asks you tojustify a rounded root, use the original function and an appropriate sign-change interval from topic 9.1 as well.

3. The indispensable line (y=x)

A graph of (y=g(x)) alone is not an iteration diagram. You also need the diagonal line (y=x). It converts an output height into the next input position.

Start at (x_0) on the x-axis. Move vertically to the curve (y=g(x)). The height reached is (x_1=g(x_0)). Move horizontally to (y=x). At that point, the horizontal coordinate is also (x_1), ready to be used as the next input. Repeat vertical, then horizontal.

Memory hook: VCHD = Vertical to the Curve, Horizontal to the Diagonal.

This is why beginning horizontally is an error. You cannot transfer an output until you have first generated one.

4. Staircase behaviour

$A staircase diagram normally occurs when successive values approach the fixed point from the same side. Using (x_{n+1}=\sqrt{x_n+1}) with (x_0=0.5), the approximations lie below the positive fixed point and rise towards it. The repeated vertical and horizontal movements make narrowing steps.$

A full-mark interpretation is specific: “The successive iterates move closer to the intersection of (y=\sqrt{x+1}) and (y=x), so the iteration converges to the fixed point (\alpha\approx1.618).” Credit is given for identifying the intersection and describing the shrinking steps, not for a vague statement that the graph “goes up”.

5. Cobweb behaviour

A cobweb diagram uses the identical construction but usually shows values alternating either side of the fixed point. The same original equation (x^2-x-1=0) can be rearranged as (x=1+\frac1x), giving (x_{n+1}=1+\frac1{x_n}). Starting at (x_0=2.7) gives (1.370\ldots, 1.730\ldots, 1.578\ldots, 1.634\ldots). The approximations jump across (1.618\ldots) while closing in.

Do not say that cobwebs and staircases are two different numerical methods. Both are graphical records of fixed-point iteration. The shape tells you how the sequence approaches the intersection: same side for a staircase; alternating sides for a cobweb.

6. Convergence, divergence and choosing a rearrangement

Different valid rearrangements of the same equation can behave very differently. A useful iteration has steps that close in towards the intersection. A divergent iteration has steps that move away or fail to settle. It is therefore possible for the algebra to be correct but the iteration to be unsuitable.

For this specification point, the diagram and the numerical values are the key evidence. You may know the useful diagnostic (|g'(\alpha)|<1) from wider work with functions, but it is not a replacement for drawing or interpreting the required geometry. In an Edexcel answer, follow the command word. If asked to draw, show the route. If asked to explain, refer to the iterates and the intersection.

Mathematical Relationships

There is no dedicated iterative-method formula to copy from the Edexcel formula booklet. Candidates are expected to understand the notation and use the rearrangement supplied or derived in the question.

Graph and Data Skills

• Plot (y=x) accurately with a ruler. It must be a straight diagonal through the origin with gradient 1.
• Plot or use the given curve (y=g(x)) on the same scale and label both graphs clearly.
• Begin at the stated (x_0) on the x-axis. First move vertical to (y=g(x)), then horizontal to (y=x).
• Use arrows or clear step directions. A chain of unlabelled right angles may not communicate the iteration process.
• Describe convergence by saying the steps get closer to the intersection. Describe divergence by saying the steps move away or do not settle.

Practical Applications

Iteration is how computers turn equations with no convenient exact solution into usable approximations. Engineers can use repeated calculations inside a model, programmers use recurrence relations in numerical algorithms, and economists may repeatedly update a predicted equilibrium. In the exam, however, the application is mathematical: the answer is an approximation and its reliability depends on the behaviour of the chosen iteration.

Examiner's Marking Lens

Treat a fixed-point question as three linked pieces of evidence. First comes the algebra: the stated or derived (g(x)) must genuinely represent the equation being solved. Second comes the numerical evidence: each line must feed the current approximation into the formula, with sensible accuracy. Third comes the geometrical evidence: the labelled path between the curve and the line must support the written conclusion. Strong candidates keep these pieces joined together rather than treating the graph as an unrelated sketch.

For a short calculation question, the first substitution is often the most valuable line to show. It proves that you understand both the index notation and the rule. On a diagram question, labels are marks-protectors: write (y=x), write (y=g(x)), and identify (x_0). Use a ruler for the diagonal and for the vertical-horizontal movements. The exact artistic shape of a hand-drawn cobweb is less important than an unambiguous sequence of steps which begins in the correct place and heads towards or away from the intersection.

When you interpret a graph, use a complete mathematical sentence. For example: “The iterates alternate about the fixed point but the distance to the intersection decreases, so the sequence converges.” This earns more credit than “it gets closer”. It identifies the behaviour, gives evidence and reaches a conclusion. Conversely, if the steps become larger, say that the iteration diverges and is unsuitable from that starting value or with that rearrangement. There is no required practical or unit conversion for this pure-mathematics topic; accuracy, notation and graphical communication are the assessed skills.

Podcast Revision Companion

Use the ten-minute audio companion after reading this guide. Pause during the recall quiz and answer before the tutor reveals the response. Then draw one staircase and one cobweb diagram from memory.