Hands-on- Solving Sets of Equations with MATLAB II

1. The following 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.

   function [k,X]=jacobi(A,B,P,delta, max1)
   % Input  - A is an N x N nonsingular matrix
   %        - B is an N x 1 matrix
   %        - P is an N x 1 matrix; the initial guess
   %	 - delta is the tolerance for P
   %	 - max1 is the maximum number of iterations
   % Output - X is an N x 1 matrix: the jacobi approximation to
   %	        the solution of AX = B
   
   N = length(B);
   for k=1:max1
      for j=1:N
         X(j)=(B(j)-A(j,[1:j-1,j+1:N])*P([1:j-1,j+1:N]))/A(j,j);
      end
      err=abs(norm(X'-P));
      relerr=err/(norm(X)+eps);
      P=X';
         if (err<delta)|(relerr<delta)
        break
      end
   end
   X=X';
   

   • Analyze the MATLAB code above, then by using solve the following linear system;
     
     $$\begin{array}{r} 4x-y+z=7\\ 4x-8y+z=-21\\ -2x+y+5z=15\\ \end{array}$$
     
     
     • Start by $P_0=(1,2,2)$; then answer: $x=2,y=4,z=3$ and number of iterations $k=19$.
     • Try some other starting sets and compare them.
   Solution:
   save with the name jacobi.m. Then;

   >> A=[4 -1 1; 4 -8 1;-2 1 5]
   >> B=[7 -21 15]'
   >> P=[1,2,2]
   >> [k,X]=jacobi(A,B,P,10^-9,20)
   

2. Modify the code given in the previous item for Gauss-Seidel method. Solve the same linear system and compare your results. Is convergence accelerated?