Solving Non-Linear Algebraic Equations by a Scalar Newton-homotopy Continuation Method

In this paper, a scalar Newton-homotopy continuation method with the incorporation of a Manifold-Based Exponentially Convergent Algorithm (MBECA) for solving non-linear algebraic equations is proposed. To conduct a scalar-based homotopy continuation method, we first convert the vector function to a scalar function by taking the square norm of the vector function and then, by introducing a time variable τ , a scalar Newton-homotopy function can be constructed. To improve the convergence and the accuracy of the scalar Newton-homotopy method, we use the scalar Newton-homotopy method to compute a rough solution and then use it as the initial guess for the MBECA. Taking the advantages of the global convergence from the scalar Newton-homotopy method and the characteristics of fast convergence from the MBECA, we expand the ability of the Newton-homotopy method to solve a large class of problems effectively and accurately. In addition, the proposed scalar Newton-homotopy method does not need to calculate the inverse of the Jacobian matrix and thus has great numerical stability. Results obtained show that the proposed method is highly efficient to find the true roots and it can also significantly improve the accuracy as well as the convergence.