Recently my dad wrote a computer program that finds an approximate solution to the "crossed-ladders problem" http://en.wikipedia.org/wiki/Crossed_ladders_problem for inputed ladder lengths and height of ladder intersection. It uses what I just learned is called "fixed point iteration" to find the square of the width (approximately) and then after about ten cycles returns the square root of the result. We are both impressed with how efficiently it finds so precise an answer, but neither of us completely understands why it works, especially when if one takes a random polynomial function and attempts to find a solution to it by isolating the exponent=1 term, making an initial guess, plugging it into the other side and iterating, the result almost always diverges. So my question is: under what circumstances will fixed point iteration converge? I've played around with a few sample functions and right now I have the impression that it always works if the absolute value of the slope of the function (with the exponent=1 term removed and its coeffecient made into unity by diving both sides as necessary) never gets bigger than one. But I don't know if that's in fact true.