Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions

Abstract

We consider the sensitivity of real zeros of polynomial systems with respect to perturbation of the coefficients, and extend our earlier probabilistic estimates for the condition number in two directions: (1) We give refined bounds for the condition number of random structured polynomial systems, depending on a variant of sparsity and an intrinsic geometric quantity called dispersion.(2) Given any structured polynomial system P , we prove the existence of a nearby well-conditioned structured polynomial system Q, with explicit quantitative estimates.Our underlying notion of structure is to consider a linear subspace E i of the space H di of homogeneous n-variate polynomials of degree d i , let our polynomial system P be an element of E := E 1 × · · · × E n−1 , and let dim(E) := dim(E 1 ) + · · · + dim(E n−1 ) be our measure of sparsity. The dispersion σ(E) provides a rough measure of how suitable the tuple E is for numerical solving.Part I of this series studied how to extend probabilistic estimates of a condition number defined by Cucker to a family of measures going beyond the weighted Gaussians often considered in the current literature. We continue at this level of generality, using tools from geometric functional analysis.

Computing the Homology of Semialgebraic Sets. II: General Formulas

“…But other choices are possible. In [7] a probabilistic analysis of κ(f ) in the case where f is square (n polynomials in n+1 homogeneous variables) is done which is valid for a broad class of probability distributions. A weak cost analysis, now valid for all distributions in this class, of the algorithm for counting zeros of square systems in [5] follows.…”

“…Can one develop a probabilistic analysis of κ aff (p) for more general distributions? For the class of distributions in [7], this reduces to developing a probabilistic analysis of κ(f ) in the underdetermined case.…”

Computing the Homology of Semialgebraic Sets. I: Lax Formulas

“…But other choices are possible. In [6] a probabilistic analysis of κ(f ) in the case where f is square (n polynomials in n + 1 homogeneous variables) is done which is valid for a broad class of probability distributions. A weak cost analysis, now valid for all distributions in this class, of the algorithm for counting zeros of square systems in [4] follows.…”

“…Can one develop a probabilistic analysis of κ aff (p) for more general distributions? For the class of distributions in [6], this reduces to developing a probabilistic analysis of κ(f ) in the underdetermined case.…”

The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

“…Smale started studying the probability that a polynomial is ill-conditioned [66]. This strategy was extended to linear algebra condition numbers [26,31,41], to systems of polynomial equations in diverse settings [42,62], to linear systems of inequalities [49], to linear and convex programming [2,68], eigenvalue and eigenvectors in the classic and other settings [7], to polynomial eigenvalue problems [6,8], and to other computational models [30], among others. As there is a substantive bibliography on this setting, we refer the reader to [28] for further details.…”