Jonathan D. Hauenstein

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.

Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems

Smale’s α-theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α-theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

Regenerative cascade homotopies for solving polynomial systems

Abstract A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.

Witness sets of projections

Abstract Elimination is a basic algebraic operation which geometrically corresponds to projections. This article describes using the numerical algebraic geometric concept of witness sets to compute the projection of an algebraic set. The ideas described in this article apply to computing the image of an algebraic set under any linear map.

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

The solution set

Isosingular Sets and Deflation

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Cell decomposition of almost smooth real algebraic surfaces

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

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

\mathbf{p}

Energy-landscape analysis of the two-dimensional nearest-neighbor φ4 model

on the set

Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis.

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