Newton-Raphson

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 xi and xi+1. If we neglect the second order term, the linear approximation of f(x) around xi 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.

Roots of Equations                                        
   (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                                                                                                       
                                                                                 

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