Solving linear systems using LU

• Consider a general linear system $Ax=b$.
• Assume we have $A=LU$ where $L$ is lower triangular and $U$ is upper triangular.

We can solve $Ax=b$ in two steps:

1. $Lz=b$
2. $Ux=z$

Both steps involve triangular systems.

The computational cost of solving $Ax=b,$ assuming we have $A$ in the form $A=LU,$ is $O(n^2)$.

How can we get $L$ and $U$ starting from $A$?