Generate the appropriate matrix$-$vector system to solve

on the interval

using

discretization points. This requires you to define

and load the matrix accordingly.

Test the method for solving this problem using the true solution

$.$ This means

$.$

Include plots of the approximate versus the true solution in your submission. $($Save as $.$png or $.$jpg files.$)$ Provide a picture to the discussion forum.

Solve the system using

$,$

$,$

$,$

$,$

$,$

$,$ and

discretization points. Calculate the error and note the convergence rate. To compute the error, use

errorVals$($i$,1)=$ norm$($sin$($pi$*$xvals$)-$ uapprox, inf$)$

where uapprox is the solution obtained after solving the matrix$-$vector system. Several codes loaded to Canvas $->$ MATLAB Codes $->$ Module $4$ will be useful here: uDoublePrime.m Download uDoublePrime.m$,$ uDoublePrimeSP.m Download uDoublePrimeSP.m$,$ and boundaryProbConv.m Download boundaryProbConv.m$.$

Use both uDoublePrimeSP.m and uDoublePrime.m to generate the matrices for this problem. Note any differences between the two methods used to generate the matrices, especially with regards to solution times and ability to store the matrices.

Summarize your work in a document and upload to Canvas. Do not forget to include supporting code.