Abstract
This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step, it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.
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Acknowledgements
The author wants to thank Paul Breiding, Peter Deuflhard, Michael Joswig, Simon Telen, and Marc Van Barel for helpful discussions.
Funding
Open Access funding enabled and organized by Projekt DEAL. The author was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) Graduiertenkolleg Facets of Complexity (GRK 2434).
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Communicated by: Jon Wilkening
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Timme, S. Mixed precision path tracking for polynomial homotopy continuation. Adv Comput Math 47, 75 (2021). https://doi.org/10.1007/s10444-021-09899-y
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DOI: https://doi.org/10.1007/s10444-021-09899-y