One of the most famous methods for solving non-linear equations is the Newton-Raphson method. The Newton-Raphson method is a kind of open method which employs Taylor series for estimation the position of the root. For arbitrary function f(x), the Taylor series around a stsrting point can be written as follows:

 

where  is a point between x_i and x_i+1. If we neglect the second order term, the linear approximation of f(x) around x_i is as follows:

As it is shown in Fig. 1 the new guess of the root can be found from following equation:

or

 

 

Fig. 1.  The Newton-Raphson method

 

The convergence speed of the Newton-Raphson method is very high in the most of cases. Using Taylor series, it can be shown that the error is approximately proportional to the square of the previous error. In other words the Newton-Raphson method is qudraticly convergent.

The algorithm of the Newton-Raphson method is very simple as follows:

 

 

Step 1: Choose x₀ as a starting point.

Step 2: Let  

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

Step 4: End.

 

e: Acceptable approximated error.

o        Linear Interpolation Method                                                                          

o        Modified Methods                                                                                                

o        Newton-Raphson Method                                                                                

o        One point Interpolation Method