2023/05/18 by Lee, Kisun, Li, Nan, Zhi, Lihong
#14Q99 #65D18 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA) #Symbolic Computation (cs.SC)
paper · doi:10.48550/arxiv.2305.10803
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.