Step $1$: LU decomposition of A into A$=$ LU $($This step is done using Gauss Elimination$)$

Step $2$: Ax $=$ b is now LUx $=$ b or Ly $=$ b where y $=$ Ux

Step $3$: Solve for y in Ly $=$ b$,$ using forward substitution.

Step $4$: Solve for x in Ux $=$ y using back$-$substitution

Note: This is a very efficient implementation of Gauss Elimination. In the above algorithm, the

subdiagonal entries of L are given by lik$=$mik. The diagonal and superdiagonal entries of U

replace those of A$.$