Flash calculations for use in compositional simulation are more difficult and time-consuming as the number of equilibrium phases increase beyond two. Because of its complexity many simulators do not even attempt to incorporate three or more hydrocarbon phases even though such cases are important in many low-temperature gas floods, or for high temperatures where hydrocarbons can partition into water. Multiphase flash algorithms typically use successive substitution (SS) followed by Newton's method. For Np-phase flash calculations, (Np-1) Rachford-Rice (RR) equations are solved in every iteration step in SS, and depending on the choice of independent variables, in Newton's method. Solution of RR equations determines both compositions and amounts of phases for a fixed overall composition and set of K-values. A robust algorithm for RR is critical to obtain convergence in multiphase compositional simulation, and has not been satisfactorily developed unlike the traditional two-phase flash. In this paper, we develop a robust and efficient algorithm for RR equations for multiphase compositional simulation. We focus primarily on systems with three equilibrium hydrocarbon phases, although other applications are possible. We derive a function, whose gradient vector consists of RR equations. This correct solution to the RR equations is formulated as a minimization of the non-monotonic convex function using the independent variables of (Np-1) phase mole fractions. The key to obtaining a robust algorithm is that we specify non-negative constraints for the resulting equilibrium phase compositions, which are described by a very small region with no poles. The minimization uses Newton's direction with a line search technique to exhibit super-linear convergence. We show a case in which a previously developed method cannot converge while our algorithm rapidly converges in a few iterations. We implement the algorithm both in a stand-alone flash code and in UTCOMP, a multiphase compositional simulator, and show that the algorithm is guaranteed to converge when a multiphase region exists as indicated by stability analysis.