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：我们提出一种两步牛顿法，用于改进经过一步消去过程（即单步消去奇异点）后奇异零点的近似解。针对方形解析系统的孤立奇异零点，该算法利用一个可逆线性算子，该算子通过结合雅可比矩阵及其核方向上的海森矩阵投影获得。我们证明了两步牛顿法应用于单步消去奇异点近似解时的二次收敛性。此外，该方法所需矩阵规模小于现有方法，因此效率更高。我们通过数值算例验证了该方法的有效性。