Linear Algebra

This section will cover some of the main methods that can be used to solve sets of linear equations of the form \[A \boldsymbol{x} = \boldsymbol{b} \quad \text{where} \quad A \in \mathbb{R}^{N \times N}, \quad \boldsymbol{x} \in \mathbb{R}^N, \quad \boldsymbol{b} \in \mathbb{R}^N.\]

This can be done by using a Direct Method if the solution of the system can be obtained in a finite number of steps or an Iterative Method if the solution, in principle, requires an infinite number of steps.

The choice between direct and iterative methods may depend on several factors, primarily the predicted theoretical efficiency of the scheme, but also the particular type of matrix (such as systems that are sparse, diagonally dominant, tridiagonal and so forth) and the memory storage requirements.