Analysis and calculus · Maths EE idea · Solid
Where Newton's method sends you
A research question to start from
For a cubic with three real roots, how does the starting value determine which root Newton's method converges to, and why are the boundaries so complicated?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A simple iteration with surprisingly rich behaviour; you can prove local convergence and explore the rest.
Mathematics you would need
- Newton–Raphson iteration
- Quadratic convergence
- Fixed points
- Graphical analysis of iterations
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Prove quadratic convergence near a simple root.
- Map starting values to roots for a chosen cubic.
- Explain points where the method fails or cycles.
- Discuss the structure of the boundaries.
Scope and difficulty
Solid. Solid; extending to the complex plane is optional.
Pitfalls
- Pretty fractal pictures with no analysis.
- Claims about chaos without definition.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Newton's method basins of attraction cubic; quadratic convergence proof. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).