Newton-Raphson Method and Application in Fractals
Project Overview
This project investigated the Newton-Raphson method, a fundamental numerical technique for finding roots of functions. We explored its theoretical foundations, practical applications, and failure cases, documenting our findings in a detailed LaTeX report. A key focus was the method’s sensitivity to initial conditions and its behaviour near critical points, where divergence and oscillation can occur. Additionally, we applied the method to complex functions, generating fractal visualisations of basins of attraction using Python. Our work was presented at Exeter University, sparking discussions on the method’s strengths, limitations, and mathematical beauty.
Challenge
Understanding the Newton-Raphson method’s convergence properties and failure cases, and effectively visualising its fractal behaviour in the complex plane.
Solution
We combined numerical analysis, theoretical research, and Python-based simulations to model and illustrate the method’s behaviour, highlighting its applications in fractals and chaotic systems.
Results & Impact
Our project provided insights into the Newton-Raphson method's practical applications and limitations, producing striking visualisations of fractals. Presenting our findings at Exeter University led to engaging discussions with mathematicians on iterative methods and chaos theory.
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