Numerical methods to find the root of a continous function from $R$ to $R^2$ I have a continuous function from $R$ to $R^2$, which I want to find a root of. I tried finding a root of the sum of squares, but since that function is always positive I can't use bisection, and any other method I tried didn't converge.
What root finding methods could work on a function from $R$ to $R^2$?
Alternatively, are there methods which always converge but don't require $f(a)f(b) \le 0$?

A function $t \mapsto C(t) = (x(t), y(t))$ mapping $\mathbb R$ to $\mathbb R^2$ is a parametric curve in the $xy$ plane. If you are looking for a “root”, you are seeking a value of $t$ at which this curve passes through the origin. In floating-point arithmetic, it’s highly unlikely that a general curve will pass through the origin, so direct root-finding is not a good approach.
At the point you are seeking, the distance from the curve to the origin is a local minimum. So, you can find the point using a standard minimization algorithm.
If you really want to solve the problem by root-finding, instead, note that the curve tangent at the desired point will be perpendicular to the vector pointing to the origin. In other words $C(t)\cdot C’(t) = 0$. You can solve this equation by root-finding. Of course, not all solutions will correspond to local minima, so you’ll need some checking.
Alternatively, you can use your favorite algorithm to find the roots of $t \mapsto x(t)$ and $t \mapsto y(t)$, and then check to see if there are any roots in common.
