Polylab: A MATLAB Toolbox for Multivariate Polynomial Modeling

TL;DR

Polylab is a MATLAB toolbox that enables multivariate polynomial modeling with symbolic-numeric support across CPU and GPU backends, including new features for variable handling and affine-normal computation.

Contribution

It introduces a unified MATLAB toolbox with GPU support, variable management, and advanced differentiation for polynomial modeling, enhancing prior capabilities.

Findings

MPOLY is optimal for lightweight workloads

MPOLY-HP excels in reduction-heavy tasks

Stochastic log-determinant is effective in large sparse regimes

Abstract

Polylab is a MATLAB toolbox for multivariate polynomial scalars and polynomial matrices with a unified symbolic-numeric interface across CPU and GPU-oriented backends. The software exposes three aligned classes: MPOLY for CPU execution, MPOLY_GPU as a legacy GPU baseline, and MPOLY_HP as an improved GPU-oriented implementation. Across these backends, Polylab supports polynomial construction, algebraic manipulation, simplification, matrix operations, differentiation, Jacobian and Hessian construction, LaTeX export, CPU-side LaTeX reconstruction, backend conversion, and interoperability with YALMIP and SOSTOOLS. Versions 3.0 and 3.1 add two practically important extensions: explicit variable identity and naming for safe mixed-variable expression handling, and affine-normal direction computation via automatic differentiation, MF-logDet-Exact, and MF-logDet-Stochastic. The toolbox has already been used successfully in prior research applications, and Polylab Version 3.1 adds a new geometry-oriented computational layer on top of a mature polynomial modeling core. This article documents the architecture and user-facing interface of the software, organizes its functionality by workflow, presents representative MATLAB sessions with actual outputs, and reports reproducible benchmarks. The results show that MPOLY is the right default for lightweight interactive workloads, whereas MPOLY-HP becomes advantageous for reduction-heavy simplification and medium-to-large affine-normal computation; the stochastic log-determinant variant becomes attractive in larger sparse regimes under approximation-oriented parameter choices.