Gaussian Elimination

Abstract

In this chapter we will deal with the problem of solving systems of linear equations. These take the form

$$\displaystyle{ \begin{array}{ccccccccc} \alpha _{11}\xi _{1} & +& \alpha _{12}\xi _{2} & +&\ldots &+& \alpha _{1n}\xi _{n} & =& \beta _{1}\\ \vdots & & & & & & \vdots & & \vdots \\ \alpha _{m1}\xi _{1} & +&\alpha _{m2}\xi _{2} & +&\ldots &+&\alpha _{mn}\xi _{n}& =&\beta _{m} \end{array} }$$

(or more briefly Ax = b), where $A = (\alpha _{ij})_{1\leq i\leq m,\,1\leq j\leq n} \in \mathbb{R}^{m\times n}$ and $b = (\beta _{1},\ldots,\beta _{m})^{\top } \in \mathbb{R}^{m}$ are given and one wishes to determine $x = (\xi _{1},\ldots,\xi _{n})^{\top } \in \mathbb{R}^{n}$. In other words, one wishes to solve the following numerical computational problem: