1. The Problem 🌱
Slides 1-3Finding where a function crosses zero (Root) or hits bottom (Minimum) is fundamental.
- Root Finding: Solve $f(x)=0$.
- Minimization: Solve $f'(x)=0$ (where slope is zero).
- Methods are surprisingly similar!
"It's just walking downhill."
2. Root Finding Methods 🕵️
Slides 4-15Method 1: Bisection 🍰
Robust but slow. Divide the interval in half.
Method 2: Brent's Method 🚀
Combines Bisection with Parabolic Interpolation. Much faster.
Method 3: Newton-Raphson 🍎
Uses derivatives. $x_{new} = x - f(x)/f'(x)$. Quadratic convergence.
Method 4: SciPy `opt.newton` 🛠️
The professional way.
3. Minimization 📉
Slides 16-25Golden Section Search 🏆
Like Bisection, but for minimums. Uses the Golden Ratio.
Brent's Method (Min) 🐆
Parabolic interpolation for the minimum.
Multivariate Minimization (SciPy) 🌐
Finding minimum in 3D space ($x,y,z$).
4. Curve Fitting (The Higgs) ⚛️
Slides 26-35Fitting theoretical models to data is just Minimizing the Chi-Square error!
5. Genetic Algorithms 🧬
Slides 40+What if the function is ugly, discontinuous, or has many traps? Gradient methods fail. We use Evolution: Selection, Crossover, Mutation.