Discussion Overview
The discussion revolves around proving that a sequence converges quadratically to a root of multiplicity greater than one for a given function. The focus is on the mathematical formulation of the iteration process and the application of Taylor's series to derive the convergence properties.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant defines a root of multiplicity $m>1$ and presents the iteration formula $x_{k+1}=x_k-mf(x_k)/f'(x_k)$, seeking to prove quadratic convergence under the condition that $f^{(m)}(x_*)≠0$.
- Another participant requests the original poster to share their working or thoughts to better assist them, indicating a collaborative approach to problem-solving.
- A third participant suggests that the problem resembles a previous discussion and encourages the original poster to adapt that solution, implying that there may be established methods to tackle similar problems.
- A later reply expresses disappointment in the original poster's progress, suggesting that the problem may be more challenging than initially thought.
Areas of Agreement / Disagreement
There is no consensus on the proof or the approach to take, as participants express varying levels of confidence and understanding of the problem. Multiple viewpoints and suggestions are presented without resolution.
Contextual Notes
Participants have not provided complete working solutions or assumptions, and the discussion lacks specific mathematical steps that might clarify the convergence proof.