function x_tilde = conjugate_gradient(A,b,x0)
	n = size(A)(2);
	if(nargin == 2)
		x_tilde = zeros(n,1);
	else
		x_tilde = x0;
	endif
	residue_old = b - A*x_tilde;
	conjugate_direction = residue_old;
	for i=1:n
		A_conjugate_direction = A*conjugate_direction;
		size_residue_old = residue_old'*residue_old;
		alpha = (size_residue_old)/(conjugate_direction'*A_conjugate_direction);
		x_tilde = x_tilde + alpha*conjugate_direction;
		residue_new = residue_old - alpha*A_conjugate_direction;
		if(residue_new'*residue_new <10e-6)
			break
		endif
		beta = (residue_new'*residue_new)/(size_residue_old);
		conjugate_direction = residue_new + beta*conjugate_direction;
		residue_old = residue_new;
	endfor
endfunction