Question $4(20$pts$)$: Solve a Large, Transient System Using LU Decomposition $(20$ pts$)-$ Partial credit possible $$

In the attached file TransientHeatFlux.mat darr are an A matrix and $b$ vector right$-$hand$-$side representing a model of a plate held initially at $250$ C$,$ with front edges at $OC$ and back edges insulated $[$note that the model was developed using the finite difference method, but this is not important for your solution$].$ The solution $"x"$ to this system is the temperature $T$ of various positions in the plate at a particular time $t_i.$ Solution of the matrix system is equivalent to taking a step in time; that is$,$ solving the system $Ax=b$ is really solving $AT(t_{i+1})=T(t_i).$ Taking this to the next step, the current solution vector $x=T(t_{i+1})$ becomes the right$-$hand side vector $"$b$"$ to obtain the solution at the next time $x=T(t_{i+2}),$ and so on$.$

Use the MATLAB function lu$()$ or equivalent python function scipy.linalg.lu$\_$factor$()$ to perform LU decomposition of the A matrix, then perform $75$ solves $($i$.$e$.75$ time steps$),$ computing the b vector at each new time from the previous b vector. Use the MATLAB backslash function $()$ or equivalent python function scipy.linalg.lu$\_$solve$()$ to perform matrix solves with the $L$ and $U$ matrices obtained. Note you may also need to track a pivot$/$permutation matrix if the decomposition routine performs pivoting $($row swapping$).$ If needed, be sure to use the solving sequence discussed in lecture $(1)Lz=b,$ then $(2)Ux=z.$ Overwrite the old $b$ vector $($you don't need to save them, but you will plot them at certain times $-$ see below$).$

NOTE: $A$ is a $6,4006,400$ element matrix $(40,960,000$ entries!$).$ It may take a while to LU decompose and even more to solve via na$$ve Gauss Elimination. Don't "look" at this A matrix, it will scroll in the command window until the end of time...

At each time step, visualize the current solution $($b$)$ using the following commands:

$($MATLAB$)($python$)$Note that N$,$xx$,$ and yy are also provided in the $.$mat file above, although depending on how the $.$mat file is loaded your use of these variables in python may be slightly different. Given that the b vector in the $.$mat file is b$(0),$ print