Norms, Condition Numbers

Abstract

In the following chapter we are going to discuss the solution of a system of linear equations

$$\displaystyle{ Ax = b }$$

(2.1)

where A is an n × n real matrix: $A \in \mathbb{R}^{n\times n},\,b \in \mathbb{R}^{n}$ is a given vector and $x \in \mathbb{R}^{n}$ is the unknown vector. In practice both vector b and matrix A are often given with uncertainties: they are perturbed by errors, e.g. by rounding errors. Hence, in this chapter we study the magnitude of the error in the vector x caused by errors in A and b. As a result of this examination we will understand the following: on what does it depend that on a given computer a linear system with a given error can be solved with an acceptable error, or not?