Authors: Alexander Demin, Christina Katsamaki, Fabrice Rouillier

Title: Certified Symbolic-Numeric Algorithms for Critical Engineering Applications

Abstract:

Many critical engineering applications, particularly in robotics or robust control theory, rely heavily on solving complex systems of non-linear and polynomial equations. Driven by the strict certification and safety requirements of these domains, our work focuses on developing reliable mathematical algorithms to tackle these demanding industrial challenges.

In such highly critical fields, purely numerical root-finding approaches, while computationally fast, often lack the rigorous error bounds necessary to certify a system's behavior. Conversely, purely symbolic methods provide exactness but frequently suffer from prohibitive computational complexity when scaled to multi-variable real-world problems. Furthermore, real-world engineering systems inherently involve physical uncertainties, such as manufacturing tolerances or measurement errors. Incorporating these bounded uncertainties is notoriously difficult when relying solely on classical exact algebraic computation tools.

To bridge this gap, we present a hybrid symbolic-numeric approach that leverages the strengths of both paradigms. Our algorithmic pipeline relies first on exact algebraic elimination techniques—such as Resultants, Gröbner bases, Rational Univariate Representations (RUR), and Discriminant varieties—to reduce the dimensionality and complexity of the initial systems. We then couple these symbolic representations with certified numerical evaluation strategies. By utilizing arbitrary-precision interval arithmetic, we implement validated numerical methods—including interval Newton algorithms and the Kantorovich theorem—to efficiently isolate real roots and track solution paths via continuation methods, rigorously validating the existence and uniqueness of solutions within strictly bounded regions.

To achieve both optimal performance and broad usability, the computational core of our framework is made of implementations in several different languages, predominantly C and Julia. A key feature of our approach is strong interoperability: we provide seamless, high-level interfaces connecting these cores with versatile environments like Julia and computer algebra systems such as Maple, with a MATLAB extension currently under development. Ultimately, our primary goal is to deliver simple, "black-box" functions that can be readily used by non-specialists in engineering. Achieving this level of accessibility requires extensive underlying work to automatically select the most appropriate mathematical models, navigate the multitude of available algorithmic strategies, and fine-tune the numerous parameters imposed by the blending of symbolic and numerical techniques.

In this talk, we will demonstrate the efficiency and ease-of-use of our symbolic-numeric pipeline through concrete industrial applications in robotic kinematics and robust control analysis. Some of our tools and examples are available at https://pace.gitlabpages.inria.fr/pace.jl/.