IA idea · Numerical methods & error analysis

When does Newton–Raphson fail? Cycles, divergence and basins of attraction

AA HLAA SL Ambitious Also in: Fractals & chaos, Pure maths

Research question

For a cubic with three real roots, which starting values lead to which root, where does the method cycle or diverge, and why do the boundaries between these regions look so complicated?

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

Most students only see the method work. Mapping exactly where it fails turns a routine technique into a real exploration that connects with fractals.

The mathematics you'll need

Newton–Raphson iteration
Tangents with zero gradient and divergence
A 2-cycle found by solving g(g(x)) = x
Basins of attraction mapped numerically
HL: complex starting values and fractal basins

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; map basins in a spreadsheet or a short program.

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

Choose a cubic and find its roots exactly.
Map which root each starting value reaches.
Find and explain a cycle and a divergent start.
Zoom in on a boundary and describe what you see.
Reflect on what this means for using the method in practice.

Pitfalls that cost marks

Pretty pictures without explanation.
Not verifying a cycle algebraically.
Starting values chosen without a plan.

Showing personal engagement

Design a function where the method cycles.
Predict the basins before mapping them.
Compare with bisection on the same function.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Newton–Raphson iteration; Tangents with zero gradient and divergence
AA HL	Good fit	Newton–Raphson iteration; Tangents with zero gradient and divergence
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: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: pretty pictures without explanation — 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

Use complex starting values and map the basins in the Argand plane.

Extending it for HL

This idea already has HL mathematics in it: complex starting values and fractal basins. 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).