Abstract:

A polynomial homotopy is a family of polynomial systems, typically depending on one parameter. The homotopy defines solution paths, which are tracked by numerical continuation methods. Robust path tracking methods [1] apply the ratio theorem of Fabry and Pade approximants, respectively to detect nearby singular solutions and to approximate the next point on the path.

As in [2], we consider the problem of tracking paths moving to a singularity, which may be isolated or not, which may be at infinity or not. For slowly converging power series expansions of the solution paths, the number of terms required to accurately approximates the coordinates of the singular solutions may become the computational bottleneck.

Extrapolation methods can remove this computational bottleneck, allowing to reach the same accuracy with far fewer terms in the series.

References

1. S. Telen, M. Van Barel, and J. Verschelde. A robust numerical path tracking algorithm for polynomial homotopy continuation. SIAM Journal on Scientific Computing 42(6):A3610-A3637, 2020.
2. J. Verschelde and K. Viswanathan. Locating the Closest Singularity in a Polynomial Homotopy. In the Proceedings of the 24th International Workshop on Computer Algebra in Scientific Computing (CASC 2022), volume 13366 of Lecture Notes in Computer Science, pages 333-352. Springer-Verlag, 2022.

CAM23, 1 September 2023, Selva di Fasano, Italy.

slides of the talk