Hands-on-Solving Sets of Equations with MATLAB I

1. Solve the following set of linear equations first by hand using Gaussian Elimination method and then by MATLAB as the following.

   
2. Upper Triangularization Followed by Back Substitution. To construct the solution to $Ax=b$, by first reducing the augmented matrix $[A\vert b]$ to upper-triangular form then performing back substitution.
   • reducing the augmented matrix $[A\vert b]$ to upper-triangular form; http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/uptrbk.m uptrbk.m.
   • back substitution; http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/backsub.m backsub.m.
   Analyze these MATLAB codes, then by using these codes solve the following linear system;
   $$\begin{array}{r} x_1+2x_2+x_3+4x_4=13\\ 2x_1+4x_3+3x_4=28... ... 4x_1+2x_2+2x_3+x_4=20\\ -3x_1+x_2+3x_3+2x_4=6\\ \end{array}$$
   Solution:
   save with the names http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/uptrbk.muptrbk.m and http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/backsub.mbacksub.m. Then;

   >> A=[? ? ? ?;? ? ? ?;? ? ? ?;? ? ? ?]
   >> B=[? ? ? ?]'
   >> uptrbk(A,B)
   

3. Factorization with Pivoting, $PA=LU$.
   • To construct the solution to $Ax=b$, where $A$ is a non-singular matrix.
   • Solve the following linear system by LU-decomposition of coefficient matrix;
     $$\begin{array}{r} x_1+2x_2+4x_3+x_4=21\\ 2x_1+8x_2+6x_3+4x... ...+10x_2+8x_3+8x_4=79\\ 4x_1+12x_2+10x_3+6x_4=82\\ \end{array}$$

   Solution:
   • Download the file http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/lufact.m lufact.m.

     >> A=[? ? ? ?;? ? ? ?;? ? ? ?;? ? ? ?]
     >> B=[? ? ? ?]
     

   • LU-decomposition

     >> [X,Y]=lufact(A,B')