Solving Multivariate Polynomial Systems and Rectangular Multiparameter Eigenvalue Problems with MacaulayLab

TL;DR

MacaulayLab is a MATLAB toolbox that offers robust algorithms for solving complex multivariate polynomial systems and eigenvalue problems, demonstrating versatility and improved performance over existing software.

Contribution

The paper introduces MacaulayLab, a new MATLAB toolbox that unifies solutions for polynomial systems and eigenvalue problems, independent of polynomial basis and capable of handling positive-dimensional solution sets.

Findings

MacaulayLab effectively solves multivariate polynomial systems and eigenvalue problems.

It outperforms other software like PNLA, PHCpack, and MultiParEig in various tests.

The toolbox is freely available online with extensive test problems.

Abstract

We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result of several years of research and algorithmic development. We demonstrate how the software works and compare its performance with other software packages, such as PNLA, PHCpack, and MultiParEig. Some core features of MacaulayLab are the fact that it solves two key problems via one common approach, works independently of the chosen polynomial basis and monomial order, and is capable of dealing with positive-dimensional solution sets at infinity. The toolbox (including its future updates) and a large collection of test problems are freely available online.