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The advantage of this approach is you can solve for the chemical potentials and site fractions of all phases simultaneously.  The disadvantages of this are most apparent in large multi-component systems and systems with miscibility gaps;  the Hessian matrix will be sparse and will have to be rebuilt every time the stable set of phases changes.  Nevertheless this method is worth mentioning because it can be used to compute derivatives with respect to user-specified conditions at equilibrium, when the equilibrium configuration is already known {TODO: ref}. This is useful, for example, when coupling thermodynamic codes to \[ \chi(\lambda) = \left| \begin{array}  \lambda -  a diffusion simulation. & -b & -c \\  -d & \lambda - e & -f \\  -g & -h & \lambda - i \end{array} \right|.\]