import numpy as np

#from copy import copy

$1)$ Use the following forward substitution and back substitution functions

def forward$\_$subs$($G$,$b$)$:

rows,cols $=$ G$.$shape

x $=$ np$.$zeros$(($rows$,),$dtype $=$ float$)$

for i in range$($rows$)$:

Gx $=0\mathrm{.}0$

for j in range$($i$)$:

Gx $+=$ G$[$i$,$ j$]*$ x$[$j$]$

x$[$i$]=($b$[$i$]-$ Gx$)/$ G$[$i$,$ i$]$

return x

def back$\_$subs$($G$,$b$)$:

rows,cols $=$ G$.$shape

x $=$ np$.$zeros$(($rows$,),$dtype $=$ float$)$

for i in range$($rows $-1,-1,-1)$:

Gx $=0\mathrm{.}0$

for j in range$($i $+1,$ cols$)$:

Gx $+=$ G$[$i$,$ j$]*$ x$[$j$]$

x$[$i$]=($b$[$i$]-$ Gx$)/$ G$[$i$,$ i$]$

return$($x$)$

$2)$ define a function

$$ name : PLU$\_$decomp

$$ inputs : A $(2$D numpy array$)$

$$ output : P $(2$d numpy array$),$ L $(2$d numpy array$),$ U $(2$d numpy array$)$

$$ note $1$ : the function should work for square matrices of any size

$$ note $2$ : use LU decomposition with pivoting

$$ note $3$ : PA $=$ LU

def PLU$\_$decomp$($A$)$:

# add code here

return$($P$,$L$,$U$)$

$3)$ define a function

$$ name : solve$\_$system$\_$PLU

$$ inputs : P $(2$d numpy array$),$ L $(2$D numpy array$),$ U $(2$D numpy array$),$ b $(1$d numpy array$)$

$$ output : x $(1$d numpy array$)$

$$ note $1$ : the function should work for square matrices of any size

$$ note $2$ : the function should solve the equation Ax$=$b where

$$ PA$=$LU is the LU decomposition $($with pivoting$)$ of A

$$ note $3$ : call the forward subs and back subs functions to solve

$4)$ def solve$\_$system$\_$PLU$($P$,$L$,$U$,$b$)$:

x $=$ np$.$zeros$($b$.$shape$)$

# add code here

return$($x$)$