Search Algorithm, Numerical Methods

Hello Everyone!

I have the following problem. Suppose that a value x0 generates some function f(x). However, f(x) must satisfy some initial condition:

f(x0)=0.

The problem is that x0 is not known a priori (I'm talking about a free boundary problem). So, we choose an initial value x0,1 and next find f1.

To test if x0 is the "right value" we evaluate: f1(x0,1), and see if this function is equal or different from zero. Of course, if f1(x0,1) is equal to zero then x0,1=x0 and the problem is done. If f1(x0,1) is different from zero we need to choose some value x0,2 and start the process again.

My question is: how to choose the x0,i's? Of course, we need this to converge to the right solution.

Any suggestion?

Thanks in advance.