Approximate Radical for Clusters: A Global Approach Using Gaussian Elimination or SVD

Abstract

We introduce a matrix of traces, attached to a zero dimensional ideal $$\tilde{I}$$ . We show that the matrix of traces can be a useful tool in handling systems of polynomial equations with clustered roots. We present a method based on Dickson’s lemma to compute the “approximate radical” of $$\tilde{I}$$ in $${\mathbb{C}}[x_1,\ldots, x_m]$$ which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is “global” in the sense that it works simultaneously for all clusters: the problem is reduced to the computation of the numerical nullspace of the matrix of traces, a matrix efficiently computable from the generating polynomials of $$\tilde{I}$$ . To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if $$\tilde{I}$$ has k distinct zero clusters each of radius at most ɛ in the ∞-norm, then k steps of Gaussian elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ɛ2. We also show that the (k + 1)-th singular value of the matrix of traces is proportional to ɛ2. The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ɛ2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.