1. Solve a Large, Transient System Using LU Decomposition.

In the attached file, https://ucdenver.instructure.com/courses/399612/files/8474841/download?verifier=EjRK6kBSwOxdCaIrUhbcA2WmuvO70S3XktcOtzoQ&wrap=1 are an A matrix and b vector right-hand-side representing a model of a plate held initially at 100C, with edges at 0C [note that the model was developed using the finite difference method, but this is not important for your solution]. Solution of the matrix system is equivalent to taking a step in time; that is, the b vector may be seen as b(0), and the solution of A x = b is really solving A b(1) = b(0).

Use the MATLAB function lu() to perform LU decomposition of the A matrix, then perform 50 solves (i.e. 50 time steps), computing the b vector at each new time from the previous b vector. Use the MATLAB backslash function (\) to perform matrix solves with the L and U matrices provided from lu(). Overwrite the old b vector (don't save them).

At each timestep, visualize the current solution (b) using the following commands:

axis manual; axis([0 1 0 1 0 100]); drawnow; bb = reshape(b,N,N); surf(xx,yy,bb);

Note that N, xx, and yy are also provided in the .mat file above. Given that the u vector in the .mat file is u(0), print out the surface shape (into your submitted PDF) at the 10th, 20th, and 30th timestep.

2. Solve a Large System Using Iterative Methods.

For the following two sub-problems, use the built-in simple MATLAB matrix algebra rules, i.e. if you're multiplying a matrix L by a vector xi, it's just L*xi.

• Code up the Jacobi Method as presented in lecture and use it to solve the large matrix of problem 3 from HW4. Return the iteration count to solve this problem and a plot of the solution.
• Modify the Jacobi method to use the most current information (i.e. implement the Gauss-Seidel method) and solve problem 3 of HW4 once more. Again include the iteration count and solution plot.

Regarding convergence, I suggested the norm(x1-x0)/norm(x0) criterion today. Note further that you should use a lower tolerance with this, like 1e-4 (in class I said 1e-2 would be sufficient, but I mis-spoke).

EXTRA CREDIT

For 20 points extra credit, in problem 1, use tic() and toc() - RTM if you don't know these - to

• Time the LU Decomposition;
• Time the 50 solves using the L and U matrices by substitution;
• Time 50 solves using direct solution (i.e. Gauss elimination each time).
• Report the overall times, the per-solve time (i.e. the 50 solve times divided by 50), and comment.

Be sure to reset the initial vector (b) prior to the second solve, and, be ready for a long wait for the second set of solves (using Gauss elimination each time). This is why we prefer LU Decomposition.