deletions | additions       diff --git a/Numerical_Approach.md b/Numerical_Approach.md  index a3b36ca..53bb0a6 100644  --- a/Numerical_Approach.md  +++ b/Numerical_Approach.md 
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\[ F(T, P, f^{i}, y^i_{s, j}) = \sum_i f^{i}G^{i}_{m}(T, P, y^i_{s, j})\]  \[ L(T, P, f^{i}, y^i_{s, j}, \lambda_k) = F(T, P, f^{i}, y^i_{s, j}) - \sum_k \lambda_k c_k(T, P, f^{i}, y^i_{s, j})\]  \[ \left[ \begin{array}{ccc}  W_k & -A^T_k \\\ \\  A_k & 0 \end{array} \right]\left[\begin{array}{ccc}  p_k \\\ \\  p_\lambda \end{array} \right] =   \left[\begin{array}{ccc}  -\nabla F_k + A^T_k \lambda_k \\\ \\  -c_k \end{array} \right]\]  \(p\) is the Newton step direction for the next iteration. \(W_k\) is the Hessian matrix of \(L\) with respect to all degrees of freedom except \(\lambda_k\). \(A_k\) is the Jacobian matrix of the constraints, \([\nabla c_1, \nabla c_2, ..., \nabla c_m]^T\).  The key advantage of this approach is you can optimize the degrees of freedom of all phases, including phase fractions and chemical potentials, simultaneously.