We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.

翻译：我们提出了一种两步牛顿法，用于精化退化过程一步终止（即单步退化奇点）的奇异零点近似。给定方形解析系统的孤立奇异零点，该算法利用了通过结合雅可比矩阵及沿雅可比核方向的 Hessian 投影所获得的可逆线性算子。我们证明了两步牛顿法在应用于单步退化奇异零点近似时具有二次收敛性。此外，该算法所需的矩阵规模小于现有方法，从而具有更高的效率。我们通过示例和实验展示了该方法的有效性。