JuliaCon 2022 (Times are UTC)

PHCpack.jl: Solving polynomial systems via homotopy continuation
07-27, 14:50–15:00 (UTC), Red

PHCpack is a software package for solving polynomial systems via homotopy continuation methods. Our interface exports the functionality of PHCpack either via the executable or the shared object file, via its C interface. The software is free and open source and we have a cloud server that hosts the application at phcpack.org.
Our talk will also explore a specific application area in the design of mechanisms.

Systems of many polynomial equations in several variables occur in various areas of science and engineering, such as mechanism design, Nash equilibria, computer vision, etc. Use cases of PHCpack can be found in more than one hundred scientific papers. In addition to the need in applications, theorems from algebraic geometry have led to efficient algorithms to compute all isolated solutions and to compute the degrees and dimensions of all positive dimensional solution sets. PHCpack contains many of the first implementations of algorithms in numerical algebraic geometry.

PHCpack allows users to provide polynomial systems in a variety of formats to the solver, including symbolically. The Julia Interface to PHCpack obtains the symbolic input from the user and using native Julia Dataframes and Data Structures, processes and returns the results numerically via PHCpack.

As one approach, using only the phc executable file, one can call the relevant features of PHCpack from a Julia program. Alternatively, we have compiled a C interface into a shared object, which can be imported into a Julia session.

As a use case, we consider the design of a 4-bar mechanism. The 4-bar mechanism traces a curve and given sufficiently many points on the curve that one wants the mechanism to trace, one can compute all necessary parameters of the mechanism. This computation requires the solution of a system of many equations and variables.

All the code is available in public github repositories.

PhD Candidate UIC (Department of Mathematics, Statistics and Computer Science)

Jan Verschelde obtained his PhD in Computer Science in 1996 at the Katholieke Universiteit Leuven. After postdoctoral studies at Michigan State University and the Mathematical Sciences Research Institute in Berkeley, he joined the Department of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago, where he teaches courses in symbolic computing, numerical analysis, computational geometry, industrial math & computation, scientific software, and supercomputing. He is the main developer of PHCpack, a software package to solve polynomial systems by homotopy continuation.