function x = gauss_no_pivot(A, b)
% gauss_no_pivot computes the solution to AX=b
%  Assumes that gaussian elimination can be done without pivots
%  and that a solution exits (matrix is not singular) 
AA = [A b]; %% create augmented matrix 
[n,m] = size(A); %% n = #rows, m = # columns
x = zeros(n,1);  %% initialize a vector for solution x
if n ~= m; error('A is not a square matrix');
elseif n ~= size(b); error('b must be the same size as A');
else %% Carry out row reduction to upper triangular form
    for k = 1:n-1  %% for each row (except last)
        if AA(k,k) == 0, error('Null diagonal element'); 
        end
        for i = k+1:n  %% for row i below row k
            m = AA(i,k)/AA(k,k);  %% m = scalar for row i
            for j = k:n+1   %% subtract m*row k from row i -> row i
                AA(i,j) = AA(i,j) - m*AA(k,j);
            end
        end
    end  %% At this point, AA should be in upper triangular form
    x(n) = AA(n,n+1)/AA(n,n); %% Start of back-substitution
    for i = n-1:-1:1  %% for each row going from row n-1 to row 1
        s=0;  %% Variable to hold a partial sum
        for j = i+1:n
            s = s + AA(i,j)*x(j);  %% add up all entries except (i,i)
        end
        x(i) = (AA(i,n+1)-s)/AA(i,i);  %% solve for x(i)
    end
end