JuliaCon 2026

Homotopy path tracking is widely used to solve systems of nonlinear equations in scientific computing and applied mathematics. The basic idea is to start from a system whose solutions are already known and continuously deform it into the system of interest, while tracking each solution along the deformation. This process defines smooth solution paths that connect the known solutions to the target solutions.

In practice, solution paths are tracked using floating-point approximations step by step from the initial system to the target system. This approach may cause a jump to another solution path, resulting in an incorrect solution. Because detecting path jumping is generally hard, certifying only the final approximation in this case is insufficient. Certified homotopy tracking not only tracks the path but also proves that the entire path is uniquely contained within a rigorously validated compact region. It guarantees that the computed endpoint corresponds to the intended solution path.

CertifiedHomotopyTracking.jl implements certified homotopy tracking using interval arithmetic to construct a rigorous compact region that uniquely contains a solution path. Systems of equations are defined using Symbolics.jl and compiled via straight-line programs, and interval arithmetic is handled through Nemo.jl.

As a key feature, the package supports the certified computation of monodromy groups for parametrized systems. Such systems frequently arise in applied mathematics, particularly in data-driven problems. When one tracks a loop in parameter space, the system's solutions are permuted, thereby inducing correspondences among them. To prove these correspondences, certified tracking along the entire loop is essential. The package employs a homotopy graph approach to compute the resulting monodromy group, and automates group-theoretic analysis through GAP.jl.

The talk consists of a brief review of backgrounds, an overview of the certified tracking algorithms, and a software demonstration. We emphasize how the package interoperates with other nonlinear algebra packages in Julia, such as HomotopyContinuation.jl and Pandora.jl, enabling users to invoke certified computation within existing workflows whenever rigorous guarantees are required.

The value of the package lies in its integration and application of multiple Julia packages for nonlinear algebra. We expect the package to provide a unified framework that broadens the practical use of certified numerical methods.