Strong molecular basis imbedded into equations of state such as statistical associating fluid theory (SAFT) enhanced their popularity for phase behavior prediction of complex fluids and their mixtures despite added computational challenges. The compressibility expressions for the SAFT equation of state are seventh-order and ninth-order polynomials for pure components and mixtures, respectively. Reliable computation of all the possible solutions of the SAFT compressibility equation and identification of the physically meaningful solutions to use is a challenging but necessary task. A global fixed-point homotopy continuation based method which can reliably compute all the possible solutions of the SAFT equation of state is described. The global fixed-point homotopy continuation guarantees that all the solutions of a nonlinear equation can be located on a single homotopy path when it is forced to start from a single starting point. This method is very desirable for the SAFT equation of state since the number and nature of density roots for a given temperature and pressure is not known a priori. The method shows that it is possible to locate all the solutions of the SAFT compressibility equation on a single path if a starting point is selected from a criterion which minimizes the number of real roots of the global fixed-point homotopy function. Examples covering both pure components and mixtures are presented to illustrate the methodology.