2.1 General remarks

Given $f(x)$ , introduce finite differences
$\Longrightarrow$Vector $\mbox{$\bf f$} \,\equiv\,(f_{k}\,;\;k=1,\dots,M)$


Similarly, given $f(x,y)$ or $f(x,t)$
$\Longrightarrow$Matrix $\mbox{${\bf F}$}\equiv[f_{i,j}]\equiv[f(x_{i},y_{j})\,;\;i=1,\dots M;\,j=1,\dots N]$


Approximate the various differentials by differences:
$\Longrightarrow$ Convert Partial Differential Equations (PDEs) into Systems of Linear Equations $\mbox{${\bf A}$} \cdot \mbox{$\bf x$}=\mbox{$\bf b$}$.


As a rule $\mbox{${\bf A}$}$ has a simple structure: sparse, diagonally dominated, positive definite, etc.

Fundamental manipulations:

• Invert a matrix:
  

  $$\mbox{${\bf A}$} \Longleftrightarrow \mbox{${\bf A}$}^{-1}$$

  
  

• Find the solution to the system of equations:
  

  $$\mbox{${\bf A}$} \cdot \mbox{$\bf x$} \, = \, \mbox{$\bf b$}$$

  
  

• Find the eigenvalues $\lambda_{i}$ and the eigenvectors $\mbox{$\bf a$}_{i}$ of a quadratic matrix:
  

  $$\left. \begin{array}{r@{\quad=\quad}l} \left\vert \mbox{${\bf A}$... ...lambda_{i}\,\mbox{${\bf I}$})\cdot\mbox{$\bf a$}_{i}\,&\,0 \end{array} \right\} \;\;\; i=1,\dots N$$