Assignment 2 - Solving Sets of Equations

1. Solve the following linear system by using Gauss-Jordan Method;
   
   $$\begin{array}{r} x_1+2x_2+x_3+4x_4=13\\ 2x_1+4x_3+3x_4=28... ... 4x_1+2x_2+2x_3+x_4=20\\ -3x_1+x_2+3x_3+2x_4=6\\ \end{array}$$
   
   
   • What is the solution vector?
   • Which method is better? Why?
   Hint: Modify the MATLAB code for Upper Triangularization Followed by Back Substitution (uptrbk.m).
2. Modify the MATLAB code for $PA=LU$:Factorization with Pivoting (lufact.m) so that $L,U$ and $P$ are output, then by using solve the following linear system;
   
   $$\begin{array}{r} x_1+2x_2+4x_3+x_4=21\\ 42x_1+8x_2+6x_3+4... ...+10x_2+8x_3+8x_4=79\\ 4x_1+12x_2+10x_3+6x_4=82\\ \end{array}$$
   
   
   • What is the solution vector?
   • Output $L,U$ and $P$ matrices.
   Hint: You can check your results by using MATLAB as;

   >>[L,U,P]=lu(A,b)
   >>inv(P)*L*U
   

3. Solve the following linear system by using Gauss-Seidel Iteration;
   
   $$\begin{array}{r} 4x-y+z=7\\ 4x-8y+z=-21\\ -2x+y+5z=15\\ \end{array}$$
   
   
   • Start by $P_0=(1,2,2)$.
   • Tabulate the iterations. Compare with the Jacobi Iteration.
   Hint: Modify the MATLAB code for Jacobi Iteration (jacobi.m).