  The homotopy continuation method has been widely used in solving parametric
systems of nonlinear equations. But it can be very expensive and inefficient
due to singularities during the tracking even though both start and end points
are non-singular. The current tracking algorithms focus on the adaptivity of
the stepsize by estimating the distance to the singularities but cannot avoid
these singularities during the tracking. We present a stochastic homotopy
tracking algorithm that perturbs the original parametric system randomly each
step to avoid the singularities. We then prove that the stochastic solution
path introduced by this new method is still closed to the original solution
path theoretically. Moreover, several homotopy examples have been tested to
show the efficiency of the stochastic homotopy tracking method.
