====== Math 753/853 LU decomposition with pivoting ======

The LU decomposition with pivoting is the factorization of a matrix $A$ into an lower-triangular matrix $L$ and an upper-triangular matrix $U$ with a permutation matrix $P$ applied to $A$

\begin{equation*}
PA = LU
\end{equation*}

The permutation matrix $P$ is crucial to suppressing accumulation of rounding errors in the computation. It's so important that **nobody ever uses LU decomp without pivoting (A=LU) in practice.**

Here a links to a few good online resources for the LU decomposition
  * [[https://en.wikipedia.org/wiki/LU_decomposition|Wikipedia]]. Decent general description, an example, and applications to $Ax=b$ and computing $A^{-1}$.
  * [[http://mathworld.wolfram.com/LUDecomposition.html|Wolfram Alpha]]. Pretty terse description of algorithm and application to $Ax=b$. 
  * [[http://nbviewer.jupyter.org/url/www.maths.usyd.edu.au/u/olver/teaching/MATH3976/notes/12.ipynb|Sheehan Olver's MATH3976 Julia notebook]]. Substantially more detailed example than either of the above.