deletions | additions       diff --git a/Numerical_Approach.md b/Numerical_Approach.md  index 9e8631f..890d33e 100644  --- a/Numerical_Approach.md  +++ b/Numerical_Approach.md 
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The disadvantages are most apparent in large multi-component systems and systems with miscibility gaps;  in principle \(c+2\) copies of *each phase* in the system need to be entered or else some procedure for adding and removing composition sets is required, in which case the Hessian must be rebuilt every time the stable set of phases changes.  \[ L(T, P, f^{i}, y^i_{s, j}, \lambda_k) = \sum_i f^{i}G^{i}_{m}(T, P, y^i_{s, j}) - \sum_k \lambda_k c_k(T, P, f^{i}, y^i_{s, j})\]  \[ \left[ \begin{array}{ccc}  W_k & -A^T_k \\\  A_k & 0 \end{array} \right]\left[\begin{array}{ccc}  p_k \\\  p_\lambda \end{array} \right]\left[\begin{array}{ccc}  -\nabla f_k + A^T_k \lambda_k \\\  -c_k \end{array} \right]\]  However this approach can be useful if one is already close to the solution.  \[ \chi(\lambda) = \left| \begin{array}{ccc}