Multiphase flash calculations and phase stability analysis are central in compositional reservoir and chemical process simulators. For instance, in some simulations, a huge amount of phase equilibrium calculations is required (the most important part of the computational effort). Moreover, a single failure may cause significant error propagations leading to false solutions. Thus, it is imperative that calculation algorithms are efficient and highly robust. The most difficult regions in mixture phase envelopes are in the vicinity of singularities: critical points for flash calculations and the stability test limit locus for stability analysis. For these conditions, all algorithms have difficulties to converge. Traditionally, a number of successive substitution iterations (SSI) are performed before switching to the second-order Newton method (many SSI iteration may be required before switch very close to singularities). The Trust-Region method has the advantage of performing a Newton step whenever the Hessian is definite positive; otherwise, the Trust-Region corrects the Hessian matrix by adding a diagonal element to make it positive definite, thus a descent direction is guaranteed. The Trust-Region limits the solution within a trust-radius, which is updated automatically at each iteration level, depending on the quality of the quadratic approximation. If the function is convex, the trust-radius enables larger changes in iteration variables, otherwise restricted steps are used to ensure a progress towards the solution. The proposed Trust-Region algorithm, as well as a hybrid methodology that combines SSI, Newton and Trust-Region steps, are tested for multiphase flash calculations and stability analysis on a variety of mixtures involving hydrocarbon components, carbon dioxide and hydrogen sulfide, exhibiting complicated phase envelopes. The proposed method compares favorably to the widely used SSI-Newton methods with various independent variables. The more difficult a test point is, the more spectacular the algorithm acts from both efficiency and reliability perspectives. (C) 2013 Elsevier B.V. All rights reserved.