One-Point Iteration

The simple one point iteration method is a kind of open methods. In this method the root of equation isn’t searched within an interval, but it is searched by using a single start point in an open area. Different to bracketing methods which are always convergent, the open methods can be convergent or divergent, but when they are convergent, their convergent speed is usually better than bracketing methods.

Assume f(x) is an arbitrary function of x. It is clear that the roots of function f(x) can be found by solving the equation f(x)=0. Using some mathematical manipulation, this equation can be rewritten in the form of x=g(x). Choosing a start point, simple one point iteration method employs this equation for finding a new guess of the root as it is illustrated in Fig. 1. That means:

xi+1 = g(xi)

Fig. 1.  The simple point iteration method


It can be shown that if   in the area of search, this method is convergent. The algorithm of simple one point iteration method is very easy as follows:



Step 1: Choose x0 as a starting point.

Step 2: Let  x = g(x0).

Step 3: if |x-x0|<e  then let root = x, else x0 = x ; go to step 2.

Step 4: End.


e: Acceptable approximated error.
Roots of Equation                                       
   (Source Code in C++)                                                      

o        Bisection Method                                                         

o        Linear Interpolation Method                                           

o        Modified Methods                                                        

o        Newton-Raphson Method                                               

o        One point Interpolation Method                                       

o        Secant Method