PHoM -- a polyhedral homotopy continuation method for polynomial systems. PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations $f(x)= 0$. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of $f(x)= 0$. The third module Verify checks whether all isolated solutions of $f(x)= 0$ have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

References in zbMATH (referenced in 30 articles , 1 standard article )

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  1. Chen, Tianran: Unmixing the mixed volume computation (2019)
  2. Leykin, Anton; Yu, Josephine: Beyond polyhedral homotopies (2019)
  3. Mahmoud, Abdrhaman; Yu, Bo; Zhang, Xuping: Eigenfunction expansion method for multiple solutions of fourth-order ordinary differential equations with cubic polynomial nonlinearity (2018)
  4. Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien: Mixed cell computation in HOM4ps (2017)
  5. Chen, Liping; Han, Lixing; Zhou, Liangmin: Computing tensor eigenvalues via homotopy methods (2016)
  6. Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien: Hom4PS-3: a parallel numerical solver for systems of polynomial equations based on polyhedral homotopy continuation methods (2014)
  7. Galeani, Sergio; Henrion, Didier; Jacquemard, Alain; Zaccarian, Luca: Design of Marx generators as a structured eigenvalue assignment (2014)
  8. Luo, Zhongxuan; Feng, Erbao; Zhang, Jiejin: A numerical realization of the conditions of Max Nöther’s residual intersection theorem (2014)
  9. Hauenstein, Jonathan; He, Yang-Hui; Mehta, Dhagash: Numerical elimination and moduli space of vacua (2013)
  10. Hughes, Ciaran; Mehta, Dhagash; Skullerud, Jon-Ivar: Enumerating Gribov copies on the lattice (2013)
  11. Martínez-Pedrera, Danny; Mehta, Dhagash; Rummel, Markus; Westphal, Alexander: Finding all flux vacua in an explicit example (2013)
  12. Vazquez-Leal, H.; Marin-Hernandez, A.; Khan, Y.; Yıldırım, A.; Filobello-Nino, U.; Castaneda-Sheissa, R.; Jimenez-Fernandez, V. M.: Exploring collision-free path planning by using homotopy continuation methods (2013)
  13. Chen, Tianran; Li, Tien-Yien: Spherical projective path tracking for homotopy continuation methods (2012)
  14. Mehta, Dhagash; He, Yang-Hui; Hauenstein, Jonathan D.: Numerical algebraic geometry: a new perspective on gauge and string theories (2012)
  15. Di Rocco, Sandra; Eklund, David; Peterson, Chris; Sommese, Andrew J.: Chern numbers of smooth varieties via homotopy continuation and intersection theory (2011)
  16. Mehta, Dhagash: Numerical polynomial homotopy continuation method and string vacua (2011)
  17. Tari, Hafez; Su, Hai-Jun; Li, Tien-Yien: A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems (2010)
  18. Lee, Tsung-Lin; Santoprete, Manuele: Central configurations of the five-body problem with equal masses (2009)
  19. Verschelde, Jan: Polyhedral methods in numerical algebraic geometry (2009)
  20. Bates, Daniel J.; Hauenstein, Jonathan D.; Sommese, Andrew J.; Wampler, Charles W. II: Software for numerical algebraic geometry: a paradigm and progress towards its implementation (2008)

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