Paramotopy: Parameter homotopies in parallel

TL;DR

Paramotopy is a parallel, optimized software implementation that enhances parameter homotopy methods in numerical algebraic geometry, enabling efficient solutions of polynomial systems across parameter spaces with adaptive singularity handling.

Contribution

It introduces a novel, parallel implementation of parameter homotopies with adaptive techniques and flexible problem specification, improving efficiency and usability.

Findings

Allows simultaneous solutions on parameter space meshes

Supports arbitrary polynomial systems in parameterized families

Automates handling of singular points in parameter space

Abstract

Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently used for solving a parameterized family of polynomial systems at multiple parameter values. Parameter homotopies have recently been useful in several areas of application and have been implemented in at least two software packages. This article describes Paramotopy, a new, parallel, optimized implementation of this technique, making use of the Bertini software package. The novel features of this implementation, not available elsewhere, include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on an automatically generated (or manually provided) mesh in the parameter space of coefficients, front ends and back ends that are easily specialized to particular classes of problems, and adaptive techniques for solving polynomial systems near singular points in the parameter space. This last feature automates and simplifies a task that is important but often misunderstood by non-experts.