deletions | additions       diff --git a/Numerical_Approach.md b/Numerical_Approach.md  index daff149..463ad08 100644  --- a/Numerical_Approach.md  +++ b/Numerical_Approach.md 
...
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 of each point 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 (half if negative), as long as it causes the simplex to lie strictly above the energy surface.  5. Compute the driving force, \(\Delta G = G_z - \sum_j \mu_j x_j\) by computing the distance of every point in the system to the current simplex. Find the point \(z\) for which \(\Delta G\) is maximized. The \(\mu_j\) here are the hyperplane coefficients of the candidate simplex.  6. Replace each vertex of the candidate simplex by the point \(z\), one at a time, until a new simplex satisfying the mass balance constraints is found. This is the new candidate simplex.  7. Recompute the chemical potentials of the candidate simplex.  8. Refine the internal degrees of freedom of the candidate simplex vertices subject to the new potentials by solving a Newton-Raphson sub-problem.