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.

同伦连续法已广泛用于求解非线性方程的参数系统。但是，即使起点和终点都不是奇异的，但由于跟踪过程中的奇异性，它可能非常昂贵且效率低下。当前的跟踪算法通过估计到奇异点的距离来关注步长的适应性，但是在跟踪过程中无法避免这些奇异点。我们提出了一种随机同态跟踪算法，该算法每步随机扰动原始参数系统，以避免奇异性。然后，我们证明了这种新方法引入的随机解路径在理论上仍接近于原始解路径。此外，已经测试了几个同伦实例，以显示随机同伦跟踪方法的效率。