Newton Method
Newton's method finds roots via a first-order Taylor approximation: x_{k+1}=x_k - f(x_k)/f'(x_k); it converges quadratically but needs the derivative and a good initial guess.
Purpose
To find roots and extrema quickly, understanding its convergence conditions and limits.
Mechanism
At the current point, approximate f by its tangent and take the tangent's zero as the next iterate. Quadratic convergence: |e_{k+1}|≈C|e_k|^2. In optimization the update is x_{k+1}=x_k - H^(-1)∇f (the Newton step). It requires an invertible Hessian and an initial guess near the root.
Use cases
- Equation root finding
- Unconstrained optimization
- Numerical analysis