One Point Iteration

The simple one point iteration method which is introduced for solving non-linear equations now can be extended for solution of the system of non-linear equations. Suppose an arbitrary system of non-linear equations which can be define as follows:

 

f1 ( x1 , x2 ,… xn ) = 0

f2 ( x1 , x2 ,… xn ) = 0

.

.

.

fn ( x1 , x2 ,… xn ) = 0.

 

This system of non-linear equation can easily be rewritten as follows:

 

x1 = g1 ( x1 , x2 ,… xn )

x2 = g 2 ( x1 , x2 ,… xn )

.

.

.

xn = g n ( x1 , x2 ,… xn ) .

 

Choosing a starting point, the new guess of roots can be found from above equations. Unfortunately, the convergence condition of one point iteration method is very restrictive as follows:

 

  The algorithm of one point iteration method is for solving system of non-linear equations similar to the simple one point iteration method algorithm. This algorithm can be written as follows:

 

Step 1: Choose X0 = [x1,x2,…,xn] as a starting point.

Step 2: Let  X = G(X0) where G=[g1,g2,…,gn].

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                                                
                                                                                                         
                         
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