In a recent joint work with Anton Leykin and Ailing Zhao, we propose a deflation operator to restore the quadratic convergence of Newton's method when approaching an isolated singular solution of a polynomial system. The number of deflations needed to obtain a regular Jacobian matrix is bounded by the multiplicity of the singular solution. Practical experiments on a large class of examples showed that our deflation algorithm can be seen as a reconditioning method as it allows the accurate computation of isolated singularities with standard floating point arithmetic.
Our deflation algorithm is inspired by earlier work of Ojika, Watanabe, and Mitsui, from the mid eighties. More recently, Lecerf proposed a symbolic deflation method which restores the quadratic convergence of Newton's method. At ISSAC'05, Dayton and Zeng presented a duality analysis of our deflation algorithm, giving tighter bounds on the number of deflations and special case algorithms (for corank equal to one), using our deflation as a preprocessing step to compute the generators for the dual space which reveal the multiplicity structure of the singularity.
Workshop on Geometry and Symmetry in Numerical Computation and Numerical Analysis in honor of Gene Allgower's 70th Birthday, August 8-10, 2005. Colorado State University, Fort Collins, Colorado.