## PHoM

 swMATH ID: 4645 Software Authors: Gunji, Takayuki; Kim, Sunyoung; Kojima, Masakazu; Takeda, Akiko; Fujisawa, Katsuki; Mizutani, Tomohiko Description: 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. Homepage: http://www.is.titech.ac.jp/~kojima/PHoM/index.html Keywords: homotopy continuation method; polyhedral homotopy; numerical experiment; software package; PHoM; polynomial systems Related Software: PHCpack; HOM4PS; Bertini; MixedVol; PoSSo; SINGULAR; POLSYS_PLP; DEMiCs; Macaulay2; HOMPACK90; Hom4PS-3; alphaCertified; CoCoA; HOMPACK; POLSYS_GLP; PHoMpara; Gfan; STRINGVACUA; Magma; TenEig Cited in: 30 Publications
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### Cited by 59 Authors

 5 Li, Tien-Yien 5 Mehta, Dhagash 4 Chen, Tianran 4 Kojima, Masakazu 4 Lee, Tsung-Lin 3 Hauenstein, Jonathan D. 3 Leykin, Anton 3 Verschelde, Jan 2 Fujisawa, Katsuki 2 Gunji, Takayuki 2 He, Yang-Hui Evariste 2 Kim, Sunyoung 2 Mizutani, Tomohiko 2 Sommese, Andrew John 2 Su, Hai-Jun 2 Takeda, Akiko 1 Bates, Daniel J. 1 Castañeda-Sheissa, Roberto 1 Chen, Liping 1 Di Rocco, Sandra 1 Dickenstein, Alicia M. 1 Eklund, David 1 Emiris, Ioannis Z. 1 Feng, Erbao 1 Filobello-Nino, Uriel A. 1 Galeani, Sergio 1 Han, Lixing 1 Henrion, Didier 1 Hughes, Ciaran 1 Jacquemard, Alain 1 Jimenez-Fernandez, Victor Manuel 1 Khan, Yasir 1 Luo, Zhongxuan 1 Mahmoud, Abdrhaman 1 Marin-Hernandez, A. 1 Martínez-Pedrera, Danny 1 McCarthy, J. Michael 1 Mevissen, Martin 1 Nie, Jiawang 1 Peterson, Chris 1 Rummel, Markus 1 Santoprete, Manuele 1 Skullerud, Jon-Ivar 1 Sosonkina, Masha 1 Takayama, Nobuki 1 Tari, Hafez 1 Vázquez-Leal, Héctor 1 Wampler, Charles W. II 1 Watson, Layne Terry 1 Westphal, Alexander 1 Yıldırım, Ahmet 1 Yu, Bo 1 Yu, Josephine 1 Zaccarian, Luca 1 Zhang, Jiejin 1 Zhang, Xuping 1 Zhao, Ailing 1 Zhou, Liangmin 1 Zhuang, Yan
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### Cited in 19 Serials

 3 Computing 3 Journal of Symbolic Computation 3 Journal of High Energy Physics 1 ACM Transactions on Mathematical Software 1 Applied Mathematics and Computation 1 Automatica 1 Journal of the Operations Research Society of Japan 1 Theoretical Computer Science 1 Discrete & Computational Geometry 1 SIAM Journal on Matrix Analysis and Applications 1 Annals of Physics 1 Celestial Mechanics and Dynamical Astronomy 1 Applied Mathematics. Series B (English Edition) 1 ETNA. Electronic Transactions on Numerical Analysis 1 Mechanism and Machine Theory 1 Communications in Information and Systems 1 Pacific Journal of Optimization 1 Algorithms and Computation in Mathematics 1 Advances in High Energy Physics
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### Cited in 19 Fields

 18 Numerical analysis (65-XX) 9 Algebraic geometry (14-XX) 5 Field theory and polynomials (12-XX) 5 Computer science (68-XX) 4 Commutative algebra (13-XX) 4 Quantum theory (81-XX) 3 Real functions (26-XX) 3 Functions of a complex variable (30-XX) 3 Mechanics of particles and systems (70-XX) 2 Convex and discrete geometry (52-XX) 2 Relativity and gravitational theory (83-XX) 2 Operations research, mathematical programming (90-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Group theory and generalizations (20-XX) 1 Special functions (33-XX) 1 Ordinary differential equations (34-XX) 1 Systems theory; control (93-XX)