A Subdivision Solver for Systems of Large Dense Polynomials

TL;DR

This paper introduces a SageMath package that efficiently solves large, dense polynomial systems over real numbers using interval analysis, adaptive precision, and certified existence and non-existence tests.

Contribution

The paper presents a novel subdivision solver for large dense polynomial systems that combines interval analysis, adaptive precision, and certification techniques within SageMath.

Findings

Robust solver effectively handles high-degree, large coefficient polynomials.

Certifies the existence or non-existence of solutions within a domain.

Adaptive precision heuristic improves computational efficiency.

Abstract

We describe here the package {\tt subdivision\\_solver} for the mathematical software {\tt SageMath}. It provides a solver on real numbers for square systems of large dense polynomials. By large polynomials we mean multivariate polynomials with large degrees, which coefficients have large bit-size. While staying robust, symbolic approaches to solve systems of polynomials see their performances dramatically affected by high degree and bit-size of input polynomials.Available numeric approaches suffer from the cost of the evaluation of large polynomials and their derivatives.Our solver is based on interval analysis and bisections of an initial compact domain of $\R^n$ where solutions are sought. Evaluations on intervals with Horner scheme is performed by the package {\tt fast\\_polynomial} for {\tt SageMath}.The non-existence of a solution within a box is certified by an evaluation scheme that uses a Taylor expansion at order 2, and existence and uniqueness of a solution within a box is certified with krawczyk operator.The precision of the working arithmetic is adapted on the fly during the subdivision process and we present a new heuristic criterion to decide if the arithmetic precision has to be increased.