User:Math3243/sandbox

Linear Systems

=Introduction=

We propose to make a ``Wikipedia'' article on the topic of \textit{systems of linear equations}. A \textit{systems of linear equations} shown below, or \textit{linear system} for short, is one of the most widely used objects in mathematics. Methods for solving linear systems of high dimension are an active area of research.

$$       a_{11}x_1  + a_{12}x_2 + \cdots +  a_{1n}x_n = b_1 a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m

$$

In particular, the efficiency of methods, and number of iterations needed for solving such a linear system is of high interest. This is because many advanced computer programs (often related to computer graphics) require solving linear equations of very high dimension.\\

We will look at the history of linear systems, and their connection to applied mathematics. We will look at direct methods for solving linear systems and their matrix representation. In particular we will look at the number of iterations needed to solve a linear system using different methods. Finally, we will look at current research on this topic, the most recent advances, and the fastest known methods for solving linear systems.

=History=

=Matrix Representation=

=Special Types of Matrices=

Diagonally Dominant Matrices
=Special method for solving linear system=