In this assignment you will implement partial row pivoting on a coefficient matrix during LU factorization. In

notebook #$21,$ we show an algorithm to transform a square coefficient matrix into upper triangular form $U.$ Then

we show a modified algorithm that transforms a square coefficient matrix into $L+U+$I form, where $L$ is a canonical

lower triangular matrix, $U$ is the upper triangular form from the first algorithm, and I is the identity. Consider the

coefficient matrix

$A=\left[\begin{array}{cccc}1& 0& 3& 0\\ 0& 1& 3& -1\\ 3& -3& 0& 6\\ 0& 2& 4& -6\end{array}\right]$

$(20$ points$)$ Apply the initial upper triangular algorithm to this system. What issue do you observe?

Demonstrate the problem in your notebook and explain the problem in a Markdown cell.

$(20$ points$)$ Implement partial row pivoting in the upper triangular algorithm. The idea is to search all rows below

the current row $i$ in the row iteration to find the row $j$ such that if rows i and $j$ were swapped $($exchanged$),$ the

swapped in row would have the largest diagonal element in absolute value.

$(20$ points$)$ Implement partial row pivoting in the $L+U+$I form algorithm and save the indices of the rows that are

pivoted in a data structure.

$(20$ points$)$ Show $A=LU.$ Note you need to first apply the same pivots to $A.$

$(20$ points$)$ Use factors $L$ and $U$ to solve $Ax=b$ for $x$ where

$b=\left[\begin{array}{c}6\\ 0\\ 4\\ 3\end{array}\right]$

using the forward and backward substitution methods shown in notebook #$21.$ code with python