The Markowitz Efficient Frontier can be characterised by the portfolio with lowest variance of portfolio valuation, for a given return. That is you have the objective function:
$$ \min_w f(w) = w^T \Sigma w, \quad s.t. \quad \delta^T w = 1, \quad \mu^T w = r, \quad where \quad \Sigma = \sigma^T \rho \sigma,$$
(If you prohibit short selling there is a further constraint: $w \geq 0$)
Using Lagrange Multipliers the above formulation is analytically solvable for $r$ (with only equality constraints).
$$ L(w) = f(w) - \lambda_1 (\delta^T w -1) - \lambda_2 (\mu^T w - r)$$
$$ \nabla_w L = \nabla_w f -\lambda_1 \delta - \lambda_2 \mu = 2 \Sigma w -\lambda_1 \delta - \lambda_2 \mu$$
$$ \nabla_{\lambda}L = \begin{bmatrix} -\delta^T w+1 \\ - \mu^Tw+r \end{bmatrix} $$
Setting all derivatives to zero (Karush-Kuhn-Tucker conditions) gives the linear system;
$$ \begin{bmatrix} 2\Sigma & -\delta & -\mu \\ -\delta^T & 0 & 0 \\ \mu^T & 0 & 0 \end{bmatrix} \begin{bmatrix} w \\ \lambda_1 \\ \lambda_2 \end{bmatrix} = \begin{bmatrix} 0 \\ -1 \\ -r \end{bmatrix}$$
Therefore you can vary $r$, the targeted return and yield the weights for each efficient portfolio solving the above linear system.
If you prohibit short selling you need to use an optimisation solver due to the inequality constraint.