Solving Nonlinear Equations

Intro

We mainly focus on single variable with a single equation. We are interested in $f(x^{*}) = 0$ or equivalently $g(x) = x$ because $f(x) = g(x) - x = 0$.

What if there are multiple variables? Is there a reduction algorithm? Or just linear algebra? How to do an efficient brute force to find all roots?

Backstory

Intermediate Value Theorem

Taylor Theorem

Newton's Method

Secant Method

Iterative Method

Bisection Method