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.
An $n \times n$ matrix is diagonally dominant if and only if;

$\displaystyle \vert a_{ii}\vert > \sum_{j=1,j\neq i}^n \vert a_{ij}\vert,~~~~ i = 1 , 2,\ldots, n$
Example. Given matrix & After reordering;

$\displaystyle 6x_1 - 2x_2 + x_3$ $\displaystyle =$ $\displaystyle 11$

$\displaystyle x_1 + 2x_2 - 5x_3$ $\displaystyle =$ $\displaystyle -1$

$\displaystyle -2x_1 + 7x_2 + 2x_3$ $\displaystyle =$ $\displaystyle 5$

$\&$

$\displaystyle 6x_1 - 2x_2 + x_3$ $\displaystyle =$ $\displaystyle 11$

$\displaystyle -2x_1 + 7x_2 + 2x_3$ $\displaystyle =$ $\displaystyle 5$

$\displaystyle x_1 + 2x_2 - 5x_3$ $\displaystyle =$ $\displaystyle -1$
The solution is (for both cases?).
Before we begin our iterative scheme we must first reorder the equations so that the coefficient matrix is diagonally dominant.