Schur-Complement Newton Correction for Nested Heidemann–Khalil Critical-Point Calculations of Multicomponent Mixtures

A Schur-complement Newton correction is proposed for the local calculation of mixture critical points with the nested Heidemann–Khalil formulation. The original method solves the determinant stability condition as an inner scalar problem in temperature and the cubic criticality condition as an outer scalar problem in molar volume. The proposed correction preserves this nested structure but treats the inner stability-limit equation as an implicit relation between temperature and molar volume. This gives a reduced Newton derivative obtained from the Schur complement of the local two-equation system. Three variants are evaluated: a full Schur derivative, a directional Schur derivative, and a secant Schur update. The Peng–Robinson equation of state is used for the thirty-two multicomponent benchmark mixtures originally reported by Peng and Robinson. Thirty repeated 32-mixture benchmark runs were performed. All local methods converged to the same critical coordinates. The full Schur variant was penalized by additional cubic-form derivative evaluations, whereas the directional and secant variants reduced the benchmark-level time index to It=0.773 and It=0.847, respectively. These values correspond to moderate but reproducible relative savings of 22.7% and 15.3%, respectively, with respect to the classical nested method. The method should therefore be interpreted as a local efficiency improvement for repeated vapor–liquid critical-point calculations, not as a global critical-point enumeration strategy.