deletions | additions       diff --git a/Numerical_Approach.md b/Numerical_Approach.md  index 36c6e1d..94a9a54 100644  --- a/Numerical_Approach.md  +++ b/Numerical_Approach.md 
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-\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 phases, including phase fractions and chemical potentials,  simultaneously. 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 system of equations must be rebuilt every time the stable set of phases changes. However, this approach can be useful if the stable set of phases is already known.