A symbolic-numeric method to solve a system of polynomial equations constructs in a first stage a homotopy connecting the system that must be solved to a system with similar structure that is easier to solve. In the second stage, numerical algorithms are applied to track the solution paths defined by the homotopy. At a singular solution, the matrix of all partial derivatives is rank deficient.
In theory, using random complex numbers, singularities do not occur, except at the end of the path, when the system that must be solved has singular solutions. In practice, an apriori step size control mechanism takes into account the curvature of the path and the distance to the nearest singularity, applying the ratio theorem of Fabry. Extrapolation methods are effective in locating and approximating the singularities at the end of a solution path.
This talk is based based on joint work with Nathan Bliss (Linear Algebra and Applications 2018), with Simon Telen and Marc Van Barel (SISC 2020), and on ongoing work with Kylash Visvanathan (CASC 2022).
Symbolic-Numeric Computing Seminar, 6 November 2023