Pure maths · Solid · Dissertation

When does iteration converge?: a maths EPQ idea

Why it works as an EPQ

It starts from the A Level numerical methods topic and asks a precise question you can answer with a theorem, graphs and computed evidence.

Scope and difficulty

Solid. Solid. One clear theorem with proof, then experiments; Newton–Raphson makes a strong final comparison.

The maths

Builds on these A Level topics: Numerical methods · Differentiation · Sequences and series.

You would learn:

• The mean value theorem
• Order of convergence
• Newton–Raphson as a fixed-point method

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Show with cobweb and staircase diagrams how |g′(x)| near the root decides what happens.
2. Prove convergence when |g′| < 1 near the root.
3. Measure errors from a spreadsheet and estimate the order of convergence.
4. Compare rearrangements of the same equation and Newton–Raphson.

Pitfalls

• Only showing examples, with no proof.
• Stopping iterations by eye instead of using a clear error tolerance.

Where to start reading

• Search for: fixed point iteration convergence mean value theorem
• Your A Level numerical methods notes, then a first-year numerical analysis text

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

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Your EPQ must be your own work. These pages coach: ideas, structure, checklists and planning. Submitting text, proofs, code or analysis written by someone else or by an AI tool as your own is malpractice. How to use help and AI honestly.