Abstract
Numerical algebraic geometry provides tools for approximating solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently used for solving a parameterized family of polynomial systems at multiple parameter values. This article describes Paramotopy, a parallel, optimized implementation of this technique, making use of the Bertini software package. The novel features of this implementation include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on an automatically generated or manually provided mesh in the parameter space of coefficients, front ends and back ends that are easily specialized to particular classes of problems, and adaptive techniques for solving polynomial systems near singular points in the parameter space.
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Notes
- 1.
In fact, this technique applies when the coefficients are holomorphic functions of the parameters [1], but we restrict to the case of polynomials for simplicity.
- 2.
Bertini provides this functionality as well.
- 3.
Bertini and PHCpack both have parallel versions, but not for multiple parameter homotopy runs.
- 4.
Handling non-convex parameter spaces is significantly more difficult and is described later.
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Acknowledgements
The authors appreciate the useful comments from several anonymous referees and Andrew Sommese as these have greatly contributed to the quality of this paper. The first author would also like to recognize the hospitality of Institut Mittag-Leffler and the Mathematical Biosciences Institute, as well as partial support from the NSF via award DMS-1719658.
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Bates, D., Brake, D., Niemerg, M. (2018). Paramotopy: Parameter Homotopies in Parallel. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_4
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