Daniel J. Bates

Adaptive Multiprecision Path Tracking

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

The solution set

Software for Numerical Algebraic Geometry: A Paradigm and Progress Towards its Implementation

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Efficient path tracking methods

of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point

Toward a Salmon Conjecture

\mathbf{p}

Khovanskii–Rolle Continuation for Real Solutions

on the set

Numerical Algebraic Geometry for Optimal Control Applications

V

Numerical Decomposition of the Rank-Deficiency Set of a Matrix of Multivariate Polynomials

, this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of

Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

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Bertini_real: Software for One- and Two-Dimensional Real Algebraic Sets

which pass through