GaussElimination

%% Q1 
%% Coded By Dr. Raghavan Ranganathan, Ranga Teja 
clc
clear
close

a = [4 1 -1 0 0 0 0 0; 1 6 -2 1 -1 0 0 0; 0 1 5 0 -1 1 0 0; 0 2 0 5 -1 0 -1 -1; 0 0 -1 -1 6 -1 0 -1; 0 0 -1 0 -1 5 0 0; 0 0 0 -1 0 0 4 -1; 0 0 0 -1 -1 0 -1 5];
b = [3 -6 -5 0 12 -12 -2 2]'; % column vector

%% Matlab way of solving the linear system
x_direct = a \ b;
n = size(c,1);
m = zeros(n,n); % Matrix containing multipliers

if det(a) == 0
    print('Can''t evaluate the linear system')
else
    for k = 1 : n-1 % Need to iterate over n-1 steps
        
        for i = k+1 : n
            m(i,k) = a(i,k)/a(k,k); % Define the multipliers
        end
        
        for i = k+1 : n
            for j = k : n
                a(i,j) = a(i,j) - m(i,k)*a(k,j);
            end
            b(i) = b(i) - m(i,k)*b(k);
        end
        
    end
end

%% Backward substitution:

x(n) = b(n)/a(n,n);
    for i = n-1 : -1 : 1
        rsum = 0;
        for j = i+1 : n
            rsum = rsum + a(i,j)*x(j);
        end
        x(i) = (b(i)-rsum)/a(i,i);
    end
    
 %% compare the solutions:
 
 x_direct - x