Root finding

The roots (aka zeros) of a function are values of function arguments forwhich the function is zero:

Find $x$ such that:

It can become complicated when we consider vector $\mathbf{x}$ and even $\mathbf{f}$, but consider a special case of finding the roots of $\mathbf{f}(\mathbf{x})$ is our familiar linear system, $\mathbf{A} \mathbf{x} -\mathbf{b} = \mathbf{0}$. Solving linear systems is a special case of root finding, which is generalized to nonlinear functions.

The roots of general nonlinear functions are more complicated than linear functions. We can seldom solve for the roots directly (notable exception being polynomial functions), and instead will need to search for them iteratively. Iterative searches are subdivided into open and closed methods depending if we can bound the problem beforehand.