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 ,… x_n ) = 0

f₂ ( x_{1 ,} x₂ ,… x_n ) = 0

.

.

.

f_n ( x₁ , x₂ ,… x_n ) = 0.

 

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

 

x₁ = g₁ ( x₁ , x₂ ,… x_n )

x₂ = g ₂ ( x_{1 ,} x₂ ,… x_n )

.

.

.

x_n = g _n ( x₁ , x₂ ,… x_n ) .

 

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,…,x_n] as a starting point.

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

Step 3: If ||x-x₀||<e  then let root = X, else X₀ = X ; go to step 2.

Step 4: End.