Quadratic Newton Iteration for Systems with Multiplicity

Abstract

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m-adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.

Numerically intersecting algebraic varieties via witness sets

“…Dayton and Zeng [5] used a construction called a Macaulay matrix to help understand the number of iterations required to reach nonsingularity, and this lead to a higherorder deflation method in [13] and the closedness-subspace method [33]. (On another track, a symbolic deflation method was presented in [12].) It was observed in [31] that the LVZ deflation method, when applied to a witness point isolated by slicing an irreducible component with a generic linear space, yields a way of treating nonreduced solution components.…”

“…Starting with the points in N 1 , the next step is to track 40 paths using the homotopy H 1,0 defined by (12). This yields the set S 0 consisting of seven points, three of which are the endpoint of a unique path and the other four are the endpoint of seven paths.…”

“…Using the homotopy H 1,0 defined by (12) and tracking the 114 paths emanating from the points in N 1 , one obtains S 0 containing six points and N 0 containing 66 points. The remaining 42 paths diverged to infinity.…”

A method for tracking singular paths with application to the numerical irreducible decomposition

“…where H is as defined in (6). For nonsingular components, this can be tracked by conventional methods to give a new sample on the component.…”

“…For polynomial systems, related recent work on dealing with components of solutions of multiplicity at least two is described in [3] and in [5,6]. A semi-numerical approach to restore the quadratic convergence of Newton's method by deflation can be found in [13] and [14].…”