Methods for solving linear algebraic equations can be categorized as direct and iterative schemes. There are several methods which directly solve equation (5). Prominent among these are such as Cramer's rule, Gaussian elimination and QR factorization. As indicated later, the Cramer's rule is unsuitable for computer implementation and is not discussed here. Among the direct methods, we only present the Gaussian elimination here in detail.

3.1 Gaussian Elimination and LU Decomposition

The Gaussian elimination is arguably the most used method for solving a set of linear algebraic equations. It makes use of the fact that a solution of a special system of linear equations, namely the systems involving triangular matrices, can be constructed very easily. For example, consider a system of linear equation given by the following matrix equation                 --------(16)Here, is a upper triangular matrix such that all elements below the main diagonal are zero and all the diagonal elements are non-zero, i.e. for all i. To solve the system , one can start from the last equation                                                     --------(17)and then proceed as follows