Linear algebraic problems

A fundamental problem in numerical analysis is the solution of a system of linear equations  

$${\bf A} \cdot {\bf x} = {\bf b},$$ (241)

where ${\bf A}$ is a square matrix and ${\bf x}$, ${\bf b}$ are vectors of $n\times n$ and n components, respectively. The formal solution  

$${\bf x} = {\bf A}^{-1} \cdot {\bf b}$$ (242)

is rarely used numerically, as it requires the determination of the inverse matrix ${\bf A}^{-1}$which is computationally very expensive if it is to be calculated through Kramer's rule (and also numerically unstable). Instead, we calculate directly the vector(s) ${\bf x}$ for given right-hand-side vector(s) ${\bf b}$ using either direct or iterative approaches.