se Jupyter notebook to write a Python function

myGE$61($A$,$b$,$TOL,info$)$

to implement the algorithm $6\mathrm{.}1$ of Gaussian Elimination with Backward Substitution on

page $368.$ Your function should return $[$x$,$info$],$ where vector x is the solution of Ax $=$ b$.$

TOL is a tiny real number $($say$,10^14)$ for verifying whether a number is zero in coding:

in the outline, use $|$a$\_$pi$|>$TOL to replace a$\_$pi $=0$ and $|$a$\_$Gaussian Elimination with Backward Substitution

To solve the $nn$ linear system

$E_1$:$,a_{11}{x}_1+a_{12}{x}_2+$cdots$+a_{1n}{x}_n=a_{1,n+1}$

$E_2$:$,a_{21}{x}_1+a_{22}{x}_2+$cdots$+a_{2n}{x}_n=a_{2,n+1}$

vdots,vdots,vdots,vdots

$E_n$:$a_{n1}{x}_1+a_{n2}{x}_2+$cdots$+a_{}{x}_n=a_{n,n+1}$

INPUT number of unknowns and equations $n$; augmented matrix $A=[a_{ij}],$ where $1$

$in$ and $1jn+1.$

OUTPUT solution $x_1,x_2,$dots,$x_n$ or message that the linear system has no unique solu$-$

tion.

Step $1$ For $i=1,$dots,$n-1$ do Steps $2-4.($Elimination process.$)$

Step $2$ Let $p$ be the smallest integer with $ipn$ and $a_{}0.$

If no integer $p$ can be found

then OUTPUT $(\text{'}$no unique solution exists'$)$;

STOP.

Step $3$ If $p$i then perform $(E_p)harr(E_i).$

Step $4$ For $j=i+1,$dots,$n$ do Steps $5$ and $6.$

Step $5$ Set $m_{ji}=\frac{a_{ji}}{a_{ii}}.$

Step $6$ Perform $(E_j-m_{ji}{E}_i)(E_j)$;

Step $7$ If $a_{}=0$ then OUTPUT $(\text{'}$no unique solution exists'$)$;

STOP.

Step $8$ Set $x_n=\frac{a_{n,n+1}}{a_{}}.($Start backward substitution.$)$

Step $9$ For $i=n-1,$dots,$1$ set $x_i=\frac{a_{i,n+1}-{\displaystyle \underset{j=i+1}{\overset{n}{}}}{a}_{ij}{x}_j}{a_{ii}}.$

Step $10$ OUTPUT $(x_1,$dots,$x_n)$; $($Procedure completed successfully.$)$

STOP.nn$|$