Iterative method to solving the Colebrook-White equation

[theBEAST]

Homework Statement

In our fluid mechanics class we were taught that we could use the following equation to solve for the Darcy friction factor f:

To do this by hand:
1. Guess a value for 1/sqrt(F), guess 3
2. Get the right hand side result of the equation using 3
3. Use that result for the next value of 1/sqrt(F)
4. Continue using the result for the next value.
5. To find F, just divide one by that value squared.

This iterative approach works but I am not too sure why. Can anyone explain why it works? I'm guessing it requires some knowledge of mathematical proofs?

 

[Chestermiller]

You are trying to solve an equation of the form x = f(x) using successive substitution. The successive substitution scheme is $$x^{n+1}=f(x^n)$$ where n signifies the n'th iteration. If we also consider the previous iteration, we have $$x^n=f(x^{n-1})$$. If we subtract the two equations, we have:
$$x^{n+1}-x^n=f(x^n)-f(x^{n-1})$$
If we expand the rhs in a taylor series about xⁿ, we obtain:
$$x^{n+1}-x^n=f'(x^n)(x^n-x^{n-1})$$
In order for the scheme to converge, the magnitude of the changes in x from one iteration to the next must be getting smaller. If x is in the close vicinity of the solution, this means the, in order for the scheme to converge, $$|f'(x)|<1$$
That is, the absolute value of the derivative of the function f must be less than 1 for the scheme to converge.