Hands-on- Solving Sets of Equations with MATLAB II

1. Factorization with Pivoting, $PA=LU$ ($A$ is a non-singular matrix, $P$ is permutation matrix).

   (a)

   Solve the following linear system by LU-decomposition of coefficient matrix;
   

   $\begin{array}{r} x_1+2x_2+4x_3+x_4=21\\ 2x_1+8x_2+6x_3+4x_4=52\\ 3x_1+10x_2+8x_3+8x_4=79\\ 4x_1+12x_2+10x_3+6x_4=82\\ \end{array}$

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

   
   

    $$

   LU-decomposition

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

   The obtained $X$ and $Y$ are the following $x$ and $y$.

    ${}$

   $Ax=b$

    ${}$

   $LUx=b$

    ${}$

   defining $y=Ux$ then solving two systems:

   1

   solve by hand $Ly=b$ for $y$ by using forward-substitution method

   2

   solve by hand $Ux=y$ for $x$ by using back-substitution method

    $$

   Check and compare your result with

   >> GEPivshow(A,B')
   

   (b)

   Download the files http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/luNopiv.m luNopiv.m, http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/luPiv.m luPiv.m.

   • Try the following commands to see $L$ and $U$ explicitly.

     >> [L,U] = luNopiv(A)
     >> [L,U,pv] = luPiv(A)
     

     are the results different? Why?

   (c)

   Now, assume that $B$ is changed as $BB=2*B$. We are given $L$ and $U$. Find new $X$ and $Y$ by hand. Check your results by

   >> BB=2*B
   >> [X,Y]=lufact(A,BB')
   >> GEPivshow(A,BB')
   

2. The http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter2/jacobi.m jacobi.m MATLAB code is given for Jacobi Iteration.
   • To solve the linear system $Ax=b$ by starting with an initial guess $x=P_0$ and generating a sequence ${P_k}$ that converges to the solution.
   • A sufficient condition for the method to be applicable is that $A$ is strictly diagonally dominant.
   • Analyze the given MATLAB code, then solve the following linear system by Jacobi iterations;
     
     $$\begin{array}{r} 4x-y+z=7\\ -2x+y+5z=15\\ 4x-8y+z=-21\\ \end{array}$$
     
     

   • Start by $P_0=(1,2,2)$; then answer: $x=2,y=4,z=3$ and number of iterations $k=19$.

     >> A=[? ? ?; ? ? ?; ? ? ?]
     >> B=[? ? ?]'
     >> P=[? ? ?]
     >> [k,X]=jacobi(A,B,P,10^-9,20)
     

   • Try some other starting sets such as $P_0=(0,0,0)$, $P_0=(2,2,2)$ and $P_0=(?,?,?)$, compare them. Which one hast the smallest value of iterations ($k$)?
3. Modify the code given in the previous item for Gauss-Seidel method. Solve the same linear system and compare your results. Is convergence accelerated?