#forward substitution

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

#forward substitution

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$)$

use the forward substitution and backward substitution above to answer questions$1$ and $2$

Question $1$

$$ define a function

$$ name : cholesky factor

$$ inputs : A $(2$D numpy array$)$ that is positive definite

$$ output : R $(2$d numpy array$)$ the Cholesky factor of A

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

$$ note $2$ : use the bordered form of Cholesky decomposition

$$ note $3$ : call the forward substitution function to solve the triangular system in each step $($except possibly the first $2)$

Question $2$

$$ define a function

$$ name : solve$\_$system$\_$cholesky

$$ inputs : R $(2$D numpy array$)$ that is a Cholesky factor, 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 the Cholesky factor R of A is provided

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