Iterative Methods

• Gaussian elimination and its variants are called direct methods. An entirely different way to solve many systems is through iteration. In this, we start with an initial estimate of the solution vector and proceed to refine this estimate.
• The two methods for solving $Ax = b$ that we shall discuss in this section are the Jacobi Method and the Gauss-Seidel Method.
• An $n \times n$ matrix $A$ is diagonally dominant if and only if;
  
  $$\vert a_{ii}\vert > \sum_{j=1,j\neq i}^n \vert a_{ij}\vert, i = l , 2,\ldots, n$$
  
  

• Although this may seem like a very restrictive condition, it turns out that there are very many applied problems that have this property.
• i.e.,
  
  $$\begin{array}{l} 6x_1 - 2x_2 + x_3 = 11 \\ x_l + 2x_2 - 5x_3 = -1\\ -2x_1 + 7x_2 + 2x_3 = 5\\ \end{array}$$
  
  

• The solution is $x_1= 2, x_2 = 1, x_3 = 1$. However, before we begin our iterative scheme we must first reorder the equations so that the coefficient matrix is diagonally dominant
  
  $$\begin{array}{l} 6x_1 - 2x_2 + x_3 = 11 \\ -2x_1 + 7x_2 + 2x_3 = 5\\ x_l + 2x_2 - 5x_3 = -1\\ \end{array}$$