IA idea · Numerical methods & error analysis

Press cos again and again: when does fixed-point iteration converge?

AA SLAA HL Solid Also in: Pure maths, Fractals & chaos

Research question

Why does repeatedly pressing cos on a calculator settle on 0.739…, which equations of the form x = g(x) can be solved this way, and how does the speed of convergence depend on g′ at the solution?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

It starts with a calculator curiosity and leads to a clean criterion (|g′| < 1) that you can explain with cobweb diagrams and then test on equations you choose.

The mathematics you'll need

Iteration and recurrence
Cobweb diagrams
The condition |g′(x*)| < 1 for convergence (justified with the mean value idea or a linear approximation)
Rate of convergence and the ratio of successive errors
Rearranging an equation into different x = g(x) forms

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explore the cos iteration and draw cobwebs.
Try several rearrangements of an equation and record which converge.
Justify the |g′| < 1 criterion.
Measure error ratios and compare with g′.
Reflect on choosing a good rearrangement.

Pitfalls that cost marks

Calculator in degrees instead of radians.
No explanation of why the criterion works.
Testing only functions that behave well.

Showing personal engagement

Find the strangest equation you can solve this way.
Predict which rearrangement converges fastest.
Explain the calculator trick to a younger student.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Iteration and recurrence; Cobweb diagrams
AA HL	Good fit	Iteration and recurrence; Cobweb diagrams
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Choose the function or equation yourself and predict how each method will behave before you run it. Hunting for the cases that break a method is engagement an examiner can see.

Reflection (D)

Reflect on error: how it changes with step size or iterations, why a method converges slowly or not at all, and how you know your 'exact' comparison value is correct. For this idea, start with: calculator in degrees instead of radians — say how it affects your answer.

Use of mathematics (E)

SL: The trapezoidal rule or a simple iteration applied correctly, errors tabulated against step size and explained, with any method outside the syllabus explained step by step.

HL: Convergence analysed rather than observed: an error bound derived with calculus or a series, an order of convergence measured and justified, or Euler's method studied against an exact solution.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Investigate what happens when |g′| = 1, or accelerate convergence with Aitken's method.

Extending it for HL

Derive an error bound with a Maclaurin series or calculus, then show your numerical results follow it.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete numerical methods exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).