deletions | additions       diff --git a/Numerical_Approach.md b/Numerical_Approach.md  index 35c0284..d60f727 100644  --- a/Numerical_Approach.md  +++ b/Numerical_Approach.md 
...
1. Sample an initial set of points in the internal degrees of freedom of each phase and compute the molar Gibbs energy.  2. We are converged if the largest change in chemical potentials from the previous iteration is less than some value. We cannot use the change in energy because of dilute cases.  3. Map the internal degrees of freedom to global composition space using the relation \(x_j = \sum_s b_s \frac{y_{s, j}}{1-y_{s, Va}}\), where \(y_{s, Va}\) is the site fraction of vacancies in sublattice \(s\).  4. Construct an initial \(J\)-simplex, where \(J\) is the number of components. Each vertex is located at a pure composition for each component, i.e., the first \(J\) simplex coordinates comprise a \(J \times J\) identity matrix and therefore bound composition space. Each vertex also has an energy coordinate. If any \(\mu_j\) are specified the corresponding energy coordinate for that vertex is fixed at that value. Otherwise, it can be set to any value, e.g., twice  the maximum energy sampled in the initial grid times a factor guaranteeing that (half if negative), as long as it causes  the simplex lies to lie  strictly above the energy surface. 5. Compute the driving force.  6. Identify a new candidate simplex and compute its chemical potentials.  7. Refine the internal degrees of freedom of the candidate simplex vertices subject to the new potentials by solving a Newton-Raphson sub-problem.