Question

Using MatLab！！！！
1.b

1. Program the Gauss-Seidel method and test it on these examples: ( 3x + y +z = 5 a. { x + 3y - 2 = 3 ( 3x + ” – 5z = -1 | 3x + y +z = 5 b. {3x + y — 5z = -1 1 x + 3y – 2 = 3 Analyze what happens when these systems are solved by simple Gaussian elimination without pivoting.

Answer 1

% Solving Ax=b using Gauss Siedel Method
A=[3 1 1; 3 1 -5;1 3 -1];
b=[5;-1;3];
Ab=[A b];
% Rearranging to get diagonaly dominant matrix
As=[Ab(1,:);Ab(3,:);Ab(2,:)];
% Initializing
n=3;
x=zeros(n,1);
error=zeros(n,1);
% G-S Iterations
for iter =1:10
for k=1:n
xold=x(k);
num=As(k,end)-As(k,1:k-1)*x(1:k-1)-As(k,k+1:n)*x(k+1:n);
x(k)=num/As(k,k);
error(k)=abs(x(k)-xold);
end
disp(['Iter' ,num2str(iter),' Error= ',num2str(max(error))]);
disp(x)
end

Before running the below code please define Matrix A and vector b in command window. As shown below

>>A=[3,1,1;3,1,-5;1,3,-1];
>> b=[5;-1;3];

% Gauss elimination method with out pivoting;
function x = Gauss(A, b)

[n, n] = size(A);   
[n, k] = size(b);
x = zeros(n,k);   
for i = 1:n-1
m = -A(i+1:n,i)/A(i,i); % multipliers
A(i+1:n,:) = A(i+1:n,:) + m*A(i,:);
b(i+1:n,:) = b(i+1:n,:) + m*b(i,:);
end;

% Use back substitution to find unknowns
x(n,:) = b(n,:)/A(n,n);
for i = n-1:-1:1
x(i,:) = (b(i,:) - A(i,i+1:n)*x(i+1:n,:))/A(i,i);
end

As we can see after running these code we are not getting the solution without pivoting. So pivoting is necessary to get the solution.