The convergence locus (CL) and stability test limit locus (STLL) are important underlying properties of multicomponent system phase diagrams. In the pressure-temperature plane, the CL separates the region where the negative flash has non-trivial solutions. The mathematical domain of flash calculations is significantly wider than the physical domain; if a negative flash is performed, the equilibrium constants are continuously derivable when a phase boundary is crossed. Criticality criteria are met for the phase compositions resulting from the negative flash and the minimum eigenvalue of a quadratic form evaluated with these compositions (which are intrinsically stable) is used to locate the CL. An efficient negative flash procedure is also proposed. The STLL is important because in its vicinity the number of iterations for phase stability testing increases dramatically and divergence may occur. Outside the STLL the tangent plane distance function has only a trivial solution; between STLL and the phase boundary, there is a non-trivial positive solution. The spinodal criterion is met at the STLL for trial phase compositions. The CL and STLL are located at given pressure or temperature based on rigorous thermodynamic criteria in only few Newton iterations. The proposed method avoids repeated expensive negative flash calculations in the vicinity of the CL and phase stability calculations in the vicinity of the STLL. Results are presented for representative hydrocarbon mixtures with different amounts of classical contaminants and different shapes of phase envelopes. (c) 2007 Elsevier B.V. All rights reserved.