Real-world problems very rarely translate into equations whose solutions have a simple algebraic form that can be computed directly. Approximation methods are therefore usually essential. Curiously enough, Rolle's theorem, or its main corollary, the mean value theorem, makes it possible to devise successive-approximation methods that converge to the solution of certain real-world problems while also providing a good estimate of the error in the approximation.
Using recurrence relations -------------------------------------
Suppose we need to solve numerically an equation of the form f(x) = 0, and that it can be transformed into an equivalent equation of the form x=ϕ(x)x =\phi(x). Starting from an arbitrary initial value x0, we can construct a sequence of values *(xk) using the recurrence relation xk+1=ϕ(xk)x_{k+1} = \phi(x_k). If the resulting sequence converges, its limit is necessarily a solution of the original equation f(x) = 0. Rolle's theorem gives a sufficient condition for convergence. The sequence converges if the successive differences |xk – xk*–1| tend to 0 as k tends to infinity. The mean value theorem then shows that there is a point x~\tilde x between xk–1 and xk such that
ϕ′(x~)=ϕ(xk)−ϕ(xk−1)xk−xk−1\phi'(\tilde x)=\frac{\phi(x_k)-\phi(x_{k-1})}{x_k-x_{k-1}}
From this we obtain: