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. 

　　Publication:     

　　Journal of Symbolic Computation, Volume 123, July–August 2024, 102278      

　　http://dx.doi.org/10.1016/j.jsc.2023.102278     

　　Author:     

　　Kisun Lee    

　　School of Mathematical and Statistical Science, Clemson University, 220 Parkway Drive, Clemson, SC 29634, USA    

　　Nan Li    

　　School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, Guangdong, China    

　　Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen 518060, Guangdong, China    

　　Lihong Zhi    

　　Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China    

　　University of Chinese Academy of Sciences, Beijing 100049, China    

　　Email: lzhi@mmrc.iss.ac.cn