Gaussian Elimination

It is the most commonly used algorithm for solving linear systems, based on decomposition of a matrix and solving the triangular system. The most general form of Gaussian elimination is:

1. Factorize into , where
   = permutation matrix (an identity matrix with some permutation applied to the rows)
   = unit lower triangular (all the diagonal entries are equal to one)
   = non-singular upper trianglar (assuming is non-singular)
   
   This corresponds to solving .
2. (permuted entries of ).
3. (forward substitution)
4. (back substitution)