Please don't use any AI

$($MATLAB programming$)$ Consider the system of $2n+1$ equations:

$($a$)(6$ points$)$ Write two function M$-$files to solve this system using the GENP and GEPP approaches, respectively. Your programs

have to make use of the structure of this system so that they do not do unnecessary computations and do not waste storage.

How many flops are involved in your programs?

How many memory locations are needed?

$($b$)(3$ points$)$ Write a script M$-$file to test your two programs on the same data: random elements $a_i,d_i$ produced by MATLAB

built$-$in function randn $($note that $a_{n+1}$ should be equal to $d_{n+1}),$ and $b_i$ defined by $b_i=d_i+a_{2n+2-i}$ for $in+1,$ and

$b_{n+1}=d_{n+1}.$ Notice that the exact solution $x=[1,1,$dots,$1{]}^T.$ Let $x_{np}$ and $x_{pp}$ be the computed solutions by your two programs,

respectively. Compute $||x-x_{np}|\frac{{|}_2}{|}|x|{|}_2,||x-x_{pp}|\frac{{|}_2}{|}|x|{|}_2$ and $lon||A|{|}_2||A^{-1}|{|}_2.$ Here for the test purpose in MATLAB, $A$ is used

to store the coefficient matrix. In your test, you may take $n=9.$ Comment on your results.

Note: Check MTALAB built$-$in functions or constants norm, cond, ones and eps, which can be used in your programs.

$($c$)(2$ points$)$ Now you change $d_1$ in your previous test example to ${10}^{-15},$ and correspondingly you change $b_1$ so that $b_1$ still

satisfies $b_1=d_1+a_{2n+1}.$ Do the same thing as in $3($b$).$

$($d$)(2$ points$)$ Now you make other changes as well in your test example: change $a_1$ to ${10}^{-8},$ and correspondingly change $b_{2n+1}$

so that

$b_{2n+1}=a_1+d_{2n+1}$ still holds. Again do the same thing as in $3($b$).$

Print out the data and computed results.