NUMERICAL METHODS

When attempting to solve the equation f(x) = 0, it would be wonderful if we could rewrite the equation in a form which gives explicitly the solution, in a manner similar to the familiar solution method for a quadratic equation. While this does not occur for the vast majority of equations we must solve, we can always find a way to re-arrange the equation f(x) = 0 in the form:
X= G(X)

Finding a value of x for which x = g(x) is thus equivalent to finding a solution of the equation f(x) = 0.
The function g(x) can be said to define a map on the real line over which x varies, such that for each value of x, the function g(x) maps that point to a new point, x on the real line. Usually this map results in the points x and X being some distance apart. If there is no motion under the map for some x = xp, we call xp a fixed point of the function g(x). Thus we have xp = g(xp), and it becomes clear that the fixed point of g(x) is also a zero of the corresponding equation f(x) = 0.

Suppose we are able to choose a point x0 which lies near a fixed point, xp, of g(x), where of course, we do not know the value of xp (after all, that is our quest here). We might speculate that under appropriate circumstances, we could use the iterative scheme:
xn+1 = g(xn)
where n=0,1,2,3,... , and we continue the iteration until the difference between successive xn is as small as we require for the the precision desired. To that level precision, the final value of xn approximates a fixed point of g(x), and hence approximates a zero of f(x).