What properties does a numerical approximation method have when it converges regardless of initial value? For example, sometimes the function does not allow Newton's method to work if we choose an initial value too far from value we want to go to. But sometimes, regardless of the initial value, it will always converge.
Is there anything special about this? Anything special having to do with the method or function?
Numerical methods that converge regardless of initial value are usually called globally convergent.
It is not common that a numerical method is globally convergent, but here is a theorem that guarantees globally convergence for Newton's method when the function is convex:
If $f: \mathbb{R} \to \mathbb{R}$ is $C^2$, increasing, convex, and has a zero, then Newton's method finds this zero regardless of the starting point chosen.
A nontrivial study of how to choose a finite set of initial points that guarantee the convergence for Newton's method for complex polynomials is given in the paper
How to find all roots of complex polynomials by Newton's method by Hubbard et al.
Invent. Math. 146 (2001), no. 1, 1–33. pdf
