function [x,A] = lusolve_nopivoting(A,b)
% x = lusolve_nopivoting(A,b) solve the linear system Ax=b using Gaussian elimination
% without pivoting.
%
% [x,LU] = lusolve_nopivoting(A,b) in addition returns a matrix LU which contains 
% the LU-decomposition of A.
%
% INPUT: A (Coefficient matrix, assumed invertible, and solvable without row
%           interchanges.)
%        b (Right hand side.)
%
% OUTPUT: x (Solution vector.)
%         A (The resulting matrix from the forward elimination procedure.)

%% initialization and preparation
% check if the dimensions of the problem somehow match and return an error
% message if not
n = length(b);
dimA = size(A);
if(length(dimA) > 2 || dimA(1)~= n || dimA(2) ~= n)
    error('The dimensions of A and b do not match.')
end

% initialize the solution x as zero vector of the correct size
x = zeros(size(b));

%% forward elimination
for k=1:n-1
    % compute the multipliers and store them in the matrix A
    A(k+1:end,k) = A(k+1:end,k)/A(k,k);
    % subtract multiples of line k from the next lines
    A(k+1:end,k+1:end) = A(k+1:end,k+1:end)-A(k+1:end,k)*A(k,k+1:end);
end

%% back substitution
% process the right hand side in the same way as the matrix A
for k=1:n-1
    b(k+1:end) = b(k+1:end)-A(k+1:end,k)*b(k);
end

% finally compute the solution of the equation
for i=n:-1:1
    x(i) = (b(i)-A(i,i+1:end)*x(i+1:end))/A(i,i);
end