Solving non-linear equations and systems

Let f be a continuous application of $ℝ$ in itself. The problem we'll be looking at in this paragraph is that of finding x such that f (x ) = 0. We then say that x is a root of f. Another, completely equivalent way of posing the same problem is to find x such that x = F (x ). We then say that x is a fixed point of F. In the following, when we use the letter f (in a theorem or algorithm), this implicitly means that the problem to be solved is put in the form f (x ) = 0. When we use the letter F, it means that our problem is written in the form x = F (x ). These two formulations are equivalent because, if it's in the form f (x ) = 0, we also have x = x + af (x ) = F (x ) with any a ¹ 0. Conversely, if x = F (x...