Improved two-step Newton's method for computing simple multiple zeros of polynomial systems

Nan Li, Lihong Zhi

Abstract: Given a polynomial system f that is associated with an isolated singular zero ξ whose Jacobian matrix is of corank one, and an approximate zero x that is close to ξ, we propose an improved two-step Newton’s method for refining x to converge to ξ with quadratic convergence. Our new approach is based on a closed-form basis of the local dual space and a recursive reduction of the simple multiple zero. By avoiding solving several least-squares problems which appeared in the previous methods, an overall 2 ×-5 × acceleration is achieved. The proof of the quadratic convergence of proposed iterations is also simplified significantly. Numerical experiments demonstrate up to 100 × speed-up when we replace the least-squares-solving calculations with closed-form solutions for refining approximate singular solutions of large-size problems (1000 equations and 1000 variables).