The Newton-Raphson method which is employed for solving a single non-linear equation can be extended to solve a system of non-linear equations. Using multi-dimensional Taylor series, a system of non-linear equations can be written near an arbitrary starting point Xi = [ x1 , x2 ,… , xn ] as follows:



To solve the system of non-linear equations the function F(X) should equal zero at the position Xi+1. It means:

Solving the above system of linear equations, we have:

It means:

The stability of the Newton-Raphson method is very sensitive to the starting point. A good knowledge about the behavior of every function of the system of non-linear equations is very important for choosing a suitable starting point as near as possible to the accurate position of the root.

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


Step 1: Choose X0 = [ x1 , x2 ,… , xn ]  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.

System of Nonlinear Equations                                 
           (Source Code in C++)                                               

o        Newton-Raphson Method                                                                        

                  o        One point Iterpolation Method