For n $=2,3,....,$ generate a Hilbert matrix H and a n$-$vector b$=$Hx using x$,$ a n$-$vector with all

entries equal to $1.$ You can generate these via a function generatorHb. Using the

GaussElimination, forwardSubstitution, and backwardSubstitution functions developed in

Assignment $2,$ solve the system the system $2=$ to obtain the approximate solution $2.$ Find

the $\backslash$infty $$norm of the residual $=2$ and of the error $=2,$ where x is the true

solution, i$.$e$.,$ the n$-$vector with all entries equal to $1.$ How large can you take n before the error is

$100\%?$ Also monitor the cond$($H$)$ and see how this varies as a function of n$.$

Note:

$1.$ Your PDf submission should include the complete algorithm package including the

following functions: GaussElimination, forwardSubstitution, backwardSubstitution,

generatorHb that work for a general $\backslash$times $$ matrix and n$-$vector.

$2.$ You cannot use a built$-$in Gauss elimination or LU decomposition functions of the

program.

$3.$ For each n$,$ your program should print the following vectors: $2,,$ r

$4.$ As part of your solution, you should present the following table in your PDF file

n $$& Cond$($H$)$

$2$

$3$

:

$5.$ Finally, you should also identify the n at which the error goes to $100\%$

To pass this challenge project, you must:

$1.$ The data in point $3$ and the table in point $4$ in the Note above should be included in your

before your code in the pdf file.

$2.$ Include the complete code for all the functions. Do not print images of the code or paste

code with a black background.

$3.$ Obtain the correct answers for $2,,$ r for the different values of n$.$