I am curious on how the Newton algorithm would work to solve an equation of the type: $f(x_1,\dots,x_n)=0$.
As far as I understand, in dimension $1$, one solves $f(x)=0$ by starting with some $x_0$, and building the sequence $(x_n)$ defined by $x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}$.
My concern is that I don't see how to generalize this in multiple dimension. Indeed, if I start the algorithm with $x^0=(x_1^0,\dots,x_n^0)$, then I will approximate my equation $f(x_1,\dots,x_n)=0$ by $f(x_1^0,\dots,x_n^0)+\displaystyle\sum_{i=0}^n \frac{\partial f(x^0)}{\partial x_i}(x^1_i-x^0_i)=0$, and then I don't see how I can find $x^1$ as this equation will have an infinite number of solutions...
Please let me know if I am missing something obvious, or if there is a smart trick to choose $x^1$.
Thanks!