Question

Problem 2 [25 points] (Coding, pen and paper) Write the code to perform Jacobi and Gauss- Seidel methods for solving the linear systern in Problem i. Using rk 2 < 10-6 as the stopping criteria for both methods. Print out the residual norm rkl2 for all iterations and the final solution.

Answer 1

%%%%%%%%%% Note: Problem 1 you have not given so I have taken one example problem

function xrt=gaus_jocob_Sidel()
clc;
clear all;

% 10x1 + x2 - x3 = 18
%
% x + 15x2 + 2x3 = -12
%
% -x1 + x2 + 20x3 = 17

A = [6 -1 2 1 ; 1 6 1 -1;0 1 3 1;1 -2 1 5];
b = [3; 2; -6;1];
n=length(b);
% error tolerance

disp('Exact solution by Matirx inverse')
y=inv(A)*b %%%Exact

%initial guess:
x0 = ones(n,1);
tol=1e-6; % Stoping (tolerence)
iter=1;

disp(' Solution, number of iteration and error by Gauss Jacobi method')
[x_jacobian, iter, err1]=gausjocobi(A,b,x0,tol)
disp(' Solution, number of iteration and error by Gauss Seidel method')
[x_gauss, iter, err1]=gausssidel(A,b,x0,tol)

% [x_Sor, iter]=Sor(A,b,x0,tol)
function [x_jacobian, iter, err1]=gausjocobi(A,b,x0,tol)

% Jacobi method
%---------------

%A is matrx
%b is right side matrix
%x0 is inital vector
%tol is tolerence

xnew=x0;
iter=1;
error=1;
while (error>tol )
xold=xnew;

for i=1:length(xnew)
off_diag = [1:i-1 i+1:length(xnew)];
xnew(i) = 1/A(i,i)*( b(i)-sum(A(i,off_diag)*xold(off_diag)) );
end

error=norm(xnew-xold);
err1(iter)=error;
iter = iter+1;
end
x_jacobian=xnew;

end

function [x_gauss, iter, err1]=gausssidel(A,b,x0,tol)
xnew=x0;
iter=1;
err=1;
while (err>tol)
lambda=1;
xold=xnew;
for i=1:n
I = [1:i-1 i+1:n];
xnew(i) = (1-lambda)*xnew(i)+lambda/A(i,i)*( b(i)-A(i,I)*xnew(I) );
end
err = norm(xnew-xold);
err1(iter)=err;
iter = iter+1;
  
end
x_gauss=xnew;
end

end

%%%%%%%% Solution

Exact solution by Matirx inverse

y =

1.305429864253394
0.635746606334842
-2.438914027149322
0.680995475113122

Solution, number of iteration and error by Gauss Jacobi method

x_jacobian =

1.305429958624094
0.635746583563940
-2.438914093841731
0.680995535312764

iter =

16

err1 =

Columns 1 through 3

3.933615809065920 1.574193895221642 0.486147264969486

Columns 4 through 6

0.175559380981398 0.056338328280127 0.019173026432052

Columns 7 through 9

0.006157919146378 0.002042336090436 0.000629963356748

Columns 10 through 12

0.000208036056003 0.000059033489543 0.000019697795727

Columns 13 through 15

0.000004870769552 0.000001681352088 0.000000334066886

Solution, number of iteration and error by Gauss Seidel method

x_gauss =

1.305429870055986
0.635746615687181
-2.438914040669446
0.680995480397565

iter =

12

err1 =

Columns 1 through 2

3.609349522843711 1.127507478402957

Columns 3 through 4

0.094708804346742 0.016763100058599

Columns 5 through 6

0.005555852012273 0.001033868860446

Columns 7 through 8

0.000112855725824 0.000015284172615

Columns 9 through 10

0.000005320753452 0.000001066916329

Column 11

0.000000125915926

>>