Iterative Methods

• Gaussian elimination and its variants are called direct methods.
• An entirely different way to solve many systems is through iteration.
• In this way, we start with an initial estimate of the solution vector and proceed to refine this estimate.
• The two methods for solving $Ax=b$ are
  1. the Jacobi Method ,
  2. the Gauss-Seidel Method.

• An $n \times n$ matrix $A$ is diagonally dominant if and only if;
  $$\vert a_{ii}\vert > \sum_{j=1,j\neq i}^n \vert a_{ij}\vert, i = 1 , 2,\ldots, n$$

• 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.,
  $$\begin{array}{l} 6x_1 - 2x_2 + x_3 = 11\\ x_1 + 2x_2 - 5x_3 = -1\\ -2x_1 + 7x_2 + 2x_3 = 5\\ \end{array}$$

• 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.