+function x_star = conjugate_gradient_method(A, b, x0, tolerance)
+ ##
+ ## Solve,
+ ##
+ ## Ax = b
+ ##
+ ## or equivalently,
+ ##
+ ## min [phi(x) = (1/2)*<Ax,x> + <b,x>]
+ ##
+ ## using Algorithm 5.2 in Nocedal and Wright.
+ ##
+ ## INPUT:
+ ##
+ ## - ``A`` -- The coefficient matrix of the system to solve. Must
+ ## be positive definite.
+ ##
+ ## - ``b`` -- The right-hand-side of the system to solve.
+ ##
+ ## - ``x0`` -- The starting point for the search.
+ ##
+ ## - ``tolerance`` -- How close ``Ax`` has to be to ``b`` (in
+ ## magnitude) before we stop.
+ ##
+ ## OUTPUT:
+ ##
+ ## - ``x_star`` - The solution to Ax=b.
+ ##
+ ## NOTES:
+ ##
+ ## All vectors are assumed to be *column* vectors.
+ ##
+ zero_vector = zeros(length(x0), 1);
+
+ k = 0;
+ xk = x0;
+ rk = A*xk - b; # The first residual must be computed the hard way.
+ pk = -rk;
+
+ while (norm(rk) > tolerance)
+ alpha_k = step_length_cgm(rk, A, pk);
+ x_next = xk + alpha_k*pk;
+ r_next = rk + alpha_k*A*pk;
+ beta_next = (r_next' * r_next)/(rk' * rk);
+ p_next = -r_next + beta_next*pk;
+
+ k = k + 1;
+ xk = x_next;
+ rk = r_next;
+ pk = p_next;
+ end
+
+ x_star = xk;
+end