A study of convergence on the Newton-homotopy continuation method

The traditional Newton method is a well-known and popular method for solving non-linear equation. It has high efficient in the convergence speed. However, we always need to guess the initial value in the iteration process. Good initial guess value can solve the equation quickly. Bad initial guess value usually will yield divergence. Homotopy continuation method is a kind of perturbation method. It can guarantee the answer by a certain path if we choose the auxiliary homotopy function, or call start system, well. This paper presents some useful rules for the choice of the auxiliary homotopy function to avoid the problem of divergence of traditional Newton method.