  Many engineering problems are described by systems of nonlinear equations,
which may exhibit multiple solutions, in a challenging situation for
root-finding algorithms. The existence of several solutions may give rise to
complex basins of attraction for the solutions in the algorithms, with severe
influence in their convergence behavior. In this work, we explore the
relationship of the basins of attractions with the critical curves (the locus
of the singular points of the Jacobian of the system of equations) in a phase
equilibrium problem in the plane with two solutions, namely the calculation of
a double azeotrope in a binary mixture. The results indicate that the conjoint
use of the basins of attraction and critical curves can be a useful tool to
select the most suitable algorithm for a specific problem.
