Newton's method is an efficient way to calculate roots of a function.
Definition 2.8.1 (Newton's method). Let $\vec f:U \to \R^n$ be a differentiable map. Newton's method consists of starting with some guess $a_0$ for a solution of $\vec f(x) = \vec 0$l Then linearlize the equation at $a_0$: replace the increment to the function, $\vec f(x) - \vec f(a_0)$, by a linear function of the increment, $D\vec f(a_0)$. Now solve the corresponding linear equation:
$$ \vec f(a_0) + D\vec f(a_0)=\vec 0. $$
This is a system of $n$ linear equations in $n$ unknowns. We can write it as
$$ D\vec f(a_0)=-\vec f(a_0). $$
If $[D\vec f(a_0)]$ is invertible, which will usually be the case, then
$$ x = a_n - [D\vec f(a_n)]^{-1}f(a_n). $$
Newton's method doesn't work for every function. In the single variable case, it is clear when it does not work.
To find at which points Newton's method converges, we create an explicit bound on how good an approximation $[D\vec f(x_0)]\vec h$ is to $\vec f(x_0 + \vec h) - \vec f(x_0)$.
Definition 2.8.4 (Lipschitz condition for a derivative). Let $U \subset\R^n$ be open and let $f: U \to \R^n$ be a differentiable mapping. The derivative $[Df(x)]$ satisfies a Lipschitz condition on a subset $V \subset U$ with Lipschitz ratio $M$ if for all $x,y\in V$,
$$ \underbrace{\Big|[Df(x)] - [Df(y)]\Big|}{\text{distance between derivatives}} \leq M \underbrace{|x-y|}{\text{distance between points}}. $$
<aside> 💡 A function whose derivative satisfies a Lipschitz condition is certainly continuously differentiable.
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Definition 2.8.7 (Second partial derivative). Let $U \subset \R^n$ be open, and let $f: U \to \R$ be a differentiable function. If the function $D_if$ is itself differentiable, then its partial derivative with respect to the $j$th variable, $D_j(D_if)$ is called the second partial derivative of $f$.
Proposition 2.8.9 (Derivative of a $C^2$ mapping is Lipschitz). Let $U \subset \R^n$ be an open ball, and $f: U \to \R^n$ a $C^2$ mapping. If