The combination of successive substitution and the Newton method provides a robust and efficient algorithm to solve the nonlinear isofugacity and mass balance equations for two-phase split computations. The two-phase Rachford-Rice equation may sometimes introduce complexity, hut the Newton and bisection methods provide a robust solution algorithm. For three-phase split calculations, the literature shows that the computed three-phase region is smaller than measured data indicate. We suggest that an improved solution algorithm for the three-phase Rachford-Rice equations can address the problem. Our proposal is to use a two-dimensional bisection method to provide good initial guesses for the Newton algorithm used to solve the three-phase Rachford-Rice equations. In this work, we present examples of various degree of complexity to demonstrate powerful features of the combined bisection-Newton method in three-phase split calculations. To the best of our knowledge, the use of the bisection method in two variables has not been used to solve the three-phase Rachford-Rice equations in the past. (C) 2010 American Institute of Chemical Engineers AIChE J, 57: 2555-2565, 2011