Users of PHCpack

The papers listed below report the use of PHCpack in the fields of algebraic statistics, communication networks, geometric constraint solving, real algebraic geometry, computation of Nash equilibria, signal processing, magnetism, mechanical design, computational geometry, computer vision, optimal control, image processing, pattern recognition, global optimization, and computational physics:
  1. M. Abdullahi, B.I. Mshelia, and S. Hamma: Solution of polynomial system using PHCpack. Journal of Physical Sciences and Innovation, 4:44-53, 2012.
  2. Min-Ho Ahn, Dong-Oh Nam and Chung-Nim Lee: Self-Calibration with Varying Focal Lengths Using the Infinity Homography. In Proceedings of the 4th Asian Conference on Computer Vision (ACCV2000), pages 140-145, 2000.
  3. Gianni Amisano and Oreste Tristani: Exact likelihood computation for nonlinear DSGE models with heteroskedastic innovations. Journal of Economic Dynamics and Control 35:2167-2185, 2011.
  4. D. Arzelier, C. Louembet, A. Rondepierre, and M. Kara-Zaitri: A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem. Journal of Optimization Theory and Applications, 2013.
  5. Bassi, I.G., Abdullahi Mohammed, and Okechukwu C.E.: Analysis Of Solving Polynomial Equations Using Homotopy Continuation Method. International Journal of Engineering Research & Technology (IJERT) 2(8):1401-1411, 2013.
  6. Daniel J. Bates and Frank Sottile: Khovanskii-Rolle Continuation for Real Solutions. Foundations of Computational Mathematics 11:563-587, 2011.
  7. Jahan Bayat and Carl D. Crane III: Closed-Form Equilibrium Analysis of Planar Tensegrity Mechanisms. In 2006 Florida Conference on Recent Advances in Robotics, FCRAR 2006.
  8. Genevieve Belanger, Kristjan Kannike, Alexander Pukhov, and Martti Raidal: Minimal semi-annihilating Z_n scalar dark matter. Journal of Cosmology and Astroparticle Physics, June 2014 (Open Access).
  9. Ivo W.M. Bleylevens, Michiel E. Hostenbach, and Ralf L.M. Peeters: Polynomial Optimization and a Jacobi-Davidson type method for commuting matrices. Applied Mathematics and Computation 224(1): 564-580, 2013.
  10. Guy Bresler, Dustin Cartwright, David Tse: Feasibility of Interference Alignment for the MIMO interference channel. IEEE Transactions on Information Theory 60(9):5573-5586, 2014.
  11. M.-L. G. Buot and D. St. P. Richards: Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations I. Journal of Symbolic Computation 41(2): 234-244, 2005.
  12. Max-Louis G. Buot, Serkan Hosten and Donald St. P. Richards: Counting and locating the solutions of polynomial systems of maximum likelihood equations, II: The Behrens-Fisher problem. Statistica Sinica 17(4):1343-1354, 2007.
  13. Enric Celaya, Tom Creemers, Lluis Ros: Exact interval propagation for the efficient solution of position analysis problems on planar linkages. Mechanism and Machine Theory 54: 116-131, 2012.
  14. Tom Creemers, Josep M. Porta, Lluis Ros, and Federico Thomas: Fast Multiresolutive Approximations of Planar Linkage Configuration Spaces. IEEE 2006 International Conference on Robotics and Automation.
  15. R.S. Datta: Using Computer Algebra To Compute Nash Equilibria. In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC 2003), pages 74-79, ACM 2003.
  16. R.S. Datta: Finding all Nash equilibria of a finite game using polynomial algebra. Economic Theory 42(1):55-96, 2009.
  17. B.H. Dayton: Numerical Local Rings and Local Solution of Nonlinear Systems. In Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation (SNC'07), pages 79-86, ACM 2007.
  18. Max Demenkov: Estimating region of attraction for polynomial vector fields by homotopy methods. ACM Communications in Computer Algebra 46(3):84-85, 2012.
  19. Max Demenkov: A Matlab Tool for Regions of Attraction Estimation via Numerical Algebraic Geometry. In the 2015 International Conference on Mechanics - Seventh Polyakhov's Reading, February 2-6, 2015, Russia, Saint Petersburg State University, Proceedings Edited by A.A. Tikhonov. IEEE 2015.
  20. Ian H. Dinwoodie, Emily Gamundi, and Ed Mosteig: Multiple Solutions for Blocking Probabilities in Asymmetric Networks. Open Systems and Information Dynamics 12(3):273-288, 2005.
  21. Csaba Domokos and Zoltan Kato: Parametric Estimation of Affine Deformations of Planar Shapes. Pattern Recognition, 2009. In press.
  22. C. Durand and C.M. Hoffmann: Variational Constraints in 3D. In Proceedings of the International Conference on Shape Modeling and Applications, Aizu-Wakamatsu, Japan, pages 90-98, IEEE Computer Society, 1999.
  23. C. Durand and C.M. Hoffmann: A systematic framework for solving geometric constraints analytically. J. Symbolic Computation 30(5):493-520, 2000.
  24. I.Z. Emiris, E. Tsigaridas, G. Tzoumas: The predicates for the Voronoi diagram of ellipses. Proc. ACM Symp. Comput. Geom. 2006.
  25. Jonathan P. Epperlein and Bassam Bamieh: A Frequency Domain Method for Optimal Periodic Control. 2012 American Control Conference (ACC), pages 5501-5506, IEEE 2012.
  26. F. Ferrari: On the geometry of super Yang-Mills theories: phases and irreducible polynomials. Journal of High Energy Physics 1, paper 26, 2009.
  27. Jaime Gallardo-Alvarado: A simple method to solve the forward displacement analysis of the general six-legged parallel manipulator. Robotics and Computer-Integrated Manufacturing 30:55-61, 2014.
  28. Jaime Gallardo-Alvarado and Juan-de-Dios Posadas-Garcia: Mobility analysis and kinematics of the semi-general 2(3-RPS) series-parallel manipulator. Robotics and Computer-Integrated Manufactoring 29(6): 463-472, 2013.
  29. Bertrand Haas: A Simple Counterexample to Kouchnirenko's Conjecture. Beitraege zur Algebra und Geometrie/Contributions to Algebra and Geometry, 43(1):1-8, 2002.
  30. Adlane Habed and Boubakeur Boufama: Camera self-calibration from bivariate polynomial equations and the coplanarity constraint. Image and Vision Computing, 24(5):498-514, 2006.
  31. Marshall Hampton and Richard Moeckel: Finiteness of stationary configurations of the four-vortex problem. Transactions of the American Mathematical Society, 361(3): 1317-1332, 2009.
  32. Jonathan Hauenstein, Jose Israel Rodriguez, and Bernd Sturmfels: Maximum Likelihood for Matrices with Rank Constraints. Journal of Algebraic Statistics 5(1): 18-38, 2014.
  33. Christoph Hellings, David A. Schmidt, and Wolfgang Utschick: Optimized beamforming for the two stream MIMO interference channel at high SNR. In 2009 Internatial ITG workshop on smart antennas - WSA 2009, February 16-19, Berlin, Germany, pages 88-95.
  34. Gabor Horvath: Moment Matching-Based Distribution Fitting with Generalized Hyper-Erlang Distributions In Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, Volume 7984, pages 232-246, 2013.
  35. X.G. Huang: Forward Kinematics for a Parallel Platform Robot. Communications in Computer and Information Sciences 86:529-532, 2011.
  36. Xiguang Huang, Qizheng Liao, Shimin Wei, and Qiang Xu: Five precision point-path synthesis of planar four-bar linkage using algebraic method. Frontiers of Electrical and Electronic Engineering in China 3(4):470-474, 2008.
  37. Xiguang Huang, Qizheng Liao, Shimin Wei, Qiang Xu, and Shuguang Huang: The 4SPS-2CCS generalized Stewart-Gough Platform mechanisms and its direct kinematics. In Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, August 5-8, 2007, Harbin, China. Pages 2472-2477, 2007.
  38. Hamadi Jamali, Tokunbo Ogunfunmi: Stationary points of the finite length constant modulus optimization. Signal Processing 82(4): 625-641, 2002.
  39. Libin Jiao, Bo Dong, Jintao Zhang, and Bo Yu: Polynomial Homotopy Methods for the Sparse Interpolation Problem Part I: Equally Spaced Sampling. SIAM J. Numer. Anal. 54(1): 462-480, 2016.
  40. Bjorn Johansson, Magnus Oskarsson, and Kalle Astrom: Structure and motion estimation from complex features in three views. In the Online ICVGIP-2002 Proceedings (Indian Conference on Computer Vision, Graphics and Image Processing).
  41. M. Kara-Zaitri, D. Arzelier, and C. Louembet: Mixed iterative algorithm for solving optimal implusive time-fixed rendezvous problem. American Institute of Aeronautics and Astronautics Guidance, Navigation, and Control Conference, Toronto, Canada, 02-05 August 2010.
  42. P.U. Lamalle, A. Messiaen, P. Dumortier, F. Durodie, M. Evrard, F. Louche: Study of mutual coupling effects in the antenna array of the ICRH plug-in for ITER. Fusion Engineering and Design 74:359-365, 2005.
  43. E. Lee and C. Mavroidis: Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Continuation. Journal of Mechanical Design, Transactions of the ASME, 124(4):652-661, 2002.
  44. E. Lee and C. Mavroidis: Four Precision Points Geometric Design of Spatial 3R Manipulators. Proceedings of the 11th World Congress in Mechanism and Machine Sciences, August 18-21, 2003, Tianjin, China. China Machinery Press, edited by Tian Huang.
  45. E. Lee and C. Mavroidis: Geometric Design of 3R Manipulators for Reaching Four End-Effector Spatial Poses. International Journal for Robotics Research, 23(3):247-254, 2004.
  46. E. Lee, C. Mavroidis, and J. Morman: Geometric Design of Spatial 3R Manipulators. Proceedings of the 2002 NSF Design, Service, and Manufacturing Grantees and Research Conference, San Juan, Puerto Rico, January 7-10, 2002.
  47. Dimitri Leggas and Oleg V. Tsodikov: Determination of small crystal structures from a minimum set of diffraction intensities by homotopy continuation. Acta Crystallographica Section A 71(3): 319-324, 2015.
  48. Dawei Leng and Weidong Sun: Finding All the Solutions of PnP Problem. In IST 2009 - International Workshop on Imaging Systems and Techniques, Shenzhen, China, May 11-12, 2009. Pages 348-352, IEEE, 2009.
  49. Anton Leykin: Numerical Primary Decomposition. In Proceedings of ISSAC 2008, edited by David Jeffrey, pages 165-164, ACM 2008.
  50. Anton Leykin and Frank Sottile: Computing Monodromy via Parallel Homotopy Continuation. In Proceedings of the 2007 International Workshop on Parallel Symbolic Computation (PASCO'07), pages 97-98, ACM 2007. (on CDROM)
  51. Anton Leykin and Frank Sottile: Galois groups of Schubert problems via homotopy computation. Mathematics of Computation 78: 1749-1765, 2009.
  52. Shaobai Li, Srinandan Dasmahapatra, and Koushik Maharatna: Dynamical System Approach for Edge Detection Using Coupled FitzHugh-Naguma Neurons. IEEE Transactions on Image Processing 24(12), 5206-5219, 2015.
  53. Ross A. Lippert: Fixing multiple eigenvalues by a minimal perturbation. Linear Algebra Appl. 432(7): 1785-1817, 2010.
  54. M. Maniatis and O. Nachtmann: Stability and symmetry breaking in the general three-Higgs-double model. Journal of High Energy Physics 2015:58, February 2015.
  55. Hyosang Moon and Nina P. Robson: Design of spatial non-anthropomorphic articulated systems based on arm joint constraint kinematic data for human interactive robotics applications. DETC2015-46530. In the Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. IDETC/CIE 2015. August 2-5, 2015, Boston Massachusetts.
  56. Marc Moreno Maza, Greg Reid, Robin Scott, and Wenyuan Wu: On Approximate Triangular Decompositions I. Dimension Zero. In SNC 2005 Proceedings. International Workshop on Symbolic-Numeric Computation. Xi'an, China, July 19-21, 2005. Edited by Dongming Wang and Lihong Zhi. Pages 250-275, 2005.
  57. Andrew J. Newell: Transition to supermagnetism in chains of magnetosome crystals. Geochemistry Geophysics Geosystems 10(11):1-19, 2009.
  58. M. Oskarsson, A. Zisserman and K. Astrom: Minimal Projective Reconstruction for combinations of Points and Lines in Three Views. Electronic Proceedings of BMVC2002 - The 13th British Machine Vision Conference 2002, pages 63 - 72.
  59. P.A. Parrilo and B. Sturmfels. Minimizing polynomial functions. In S. Basu and L. Gonzalez-Vega, editors, Algorithmic and quantitative real algebraic geometry, volume 60 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 83-99. AMS, 2003.
  60. Alba Perez and J.M. McCarthy: Dual Quaternion Synthesis of Constrained Robotic Systems. Journal of Mechanical Design 126(3): 425-435, 2004.
  61. Nina Patarinsky-Robson, J. Michael McCarthy, and Irem Y. Tumer: The algebraic synthesis of a spatial TS chain for a prescribed acceleration task. Mechanism and Machine Theory 43(10): 1268-1280, 2008.
  62. Nina Patarinsky-Robson, J. Michael McCarthy, and Irem Y. Tumer: Failure Recovery Planning for an Arm Mounted on an Exploratory Rover. IEEE Transactions on Robotics 25(6):1448-1453, 2009.
  63. Jose Israel Rodriguez: Combinatorial excess intersection. Journal of Symbolic Computation 68(2): 297-307, 2015.
  64. Roger E. Sanchez-Alonso, Jose-Joel Gonzalez-Barbosa, Eduardo Castilo-Castaneda, and Jaime Gallardo-Alvarado: Kinematic analysis of a novel 2(3-RUS) parallel manipulator. Robotica, available on CJO2015.
  65. H. Schreiber, K. Meer, and B.J. Schmitt: Dimensional synthesis of planar Stephenson mechanisms for motion generation using circlepoint search and homotopy methods. Mechanism and Machine Theory 37(7):717-737, 2002.
  66. Ben Shirt-Ediss, Ricard V. Sole, and Kepa Ruiz-Mirazo: Emergent Chemical Behavior in Variable-Volume Protocells. Life 5: 181-121, 2015.
  67. Frank Sottile: Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro. Experimental Mathematics 9(2): 161-182, 2000.
  68. H. Stewenius and K. Astrom: Structure and Motion Problems for Multiple Rigidly Moving Cameras. In Computer Vision - ECCV 2004: 8th European Conference on Computer Vision, Prague, Czech Republic, May 11-14, 2004. Proceedings, Part III. Edited by T. Pajdla and J. Matas. Lecture Notes in Computer Science 3023, pages 252-263, Springer, 2004.
  69. H.-J. Su and J.M. McCarthy: Kinematic Synthesis of RPS Serial Chains. In Proceedings of the ASME Design Engineering Technical Conferences (CDROM). Paper DETC03/DAC-48813. Chicago, IL, Sept. 02-06, 2003.
  70. H.-J. Su and J.M. McCarthy: Synthesis of Compliant Mechanisms with Specified Equilibrium Positions. In Proceedings of the ASME International Design Engineering Technical Conferences. Paper DETC 2005-85085. Long Beach, CA, Sept. 24-28 2005.
  71. H.-J. Su and J.M. McCarthy: Kinematic Synthesis of RPS Serial Chains for a Given Set of Task Positions. Mechanism and Machine Theory, 40(7):757-775, 2005
  72. H.-J. Su and J.M. McCarthy: A Polynomial Homotopy Formulation of the Inverse Static Analysis of Planar Compliant Mechanisms ASME Journal of Mechanical Design128(4): 776-786, 2006.
  73. H.-J. Su, C.W. Wampler, and J.M. McCarthy: Geometric Design of Cylindric PRS Serial Chains. ASME Design Engineering Technical Conferences, Chicago, IL, Sep 2-6, 2003.
  74. Attila Tanács and Joakim Lindblad and Nataša Sladoje and Zoltan Ka: Estimation of linear deformations of 2D and 3D fuzzy objects, Pattern Recognition 48(4):1391-1403, 2015.
  75. N. Trawny, X.S. Zhou, K.X. Zhou, S.I. Roumeliotis: 3D Relative Pose Estimation from Distance-Only Measurements. Proceedings of the 2007/IEEE/RSJ International Conference on intelligent Robots and Systems. San Diego, CA, Oct 29-Nov 2, 2007, pages 1071-1078, IEEE, 2007.
  76. T. Turocy: Towards a black-box solver for finite games: Computing all equilibria with Gambit and PHCpack. In Software for Algebraic Geometry, volume 148 of the IMA volumes in Mathematics and its Applications, edited by M.E. Stillman, N. Takayama, and J. Verschelde, pages 133-148, Springer-Verlag, 2008.
  77. Konstantin Usevich and Ivan Markovsky: Structured low-rank approximation as a rational function minimization. In 16th IFAC Symposium on System Identification Brussels, 11 - 13 Jul 2012, pages 722-727.
  78. C.W. Wampler: Isotropic coordinates, circularity and Bezout numbers: planar kinematics from a new perspective. Proceedings of the 1996 ASME Design Engineering Technical Conference. Irvine, CA, Aug 18-22, 1996. Available on CD-ROM.
  79. Wenyuan Wu and Greg Reid: Symbolic-numeric computation of implicit Riquier bases for PDE. In Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, edited by C.W. Brown, pages 377-385, ACM 2007.
  80. Jonathan Widger and Daniel Grosu: Parallel Computation of Nash Equilibria in N-Player Games. In Proc. of the 12th IEEE International Conference on Computational Science and Engineering (CSE 2009), August 29-31, 2009, Vancouver, Canada, pages 209-215.
  81. F. Xie, G. Reid, and S. Valluri: A numerical method for the one dimensional action functional for FBG structures. Can J. Phys. 76: 1-21, 2002.
  82. Hong Bing Xin, Qiang Huang, and Yueqing Yu: Position and Orientation Analyses of Mechanism by PHCpack Solver of Homotopy Continuation. Applied Mechanics and Materials152-254: 1779-1784, 2012.
  83. K. Yang and R. Orsi: Static output feedback pole placement via a trust region approach. IEEE Transactions on Automatic Control 52(11): 2146-2150, 2007.
  84. Yan Yang, Yao Zhang, Fangxing Li, and Haoyong Chen: Computing All Nash Equilibria of Multiplayer Games in Electricity Markets by Solving Polynomial Equations. IEEE Transactions on Power Systems 27(1): 81-91, 2012.
  85. Jun Zhang and Mohan Sarovar: Identification of open quantum systems from observable time traces. Physical Review A 91, 052121, 2015.
  86. Shiqiang Zhang, Shufang Zhang, and Yan Wan: Biorthogonal Wavelet Construction Using Homotopy Method. Chinese Journal of Electronics 24(4), pages 772-775, 2015.
  87. Xun S. Zhou and Stergios I. Roumeliotis: Determining 3-D Relative Transformations for Any Combination of Range and Bearing Measurements. IEEE Transactions on Robotics 29(2):458-474, 2013.
In addition to the publications listed above, PHCpack was used as a benchmark to measure the progress of new algorithms in the following papers:
  1. T. Gao and T.Y. Li: Mixed volume computation via linear programming. Taiwanese Journal of Mathematics 4(4): 599-619, 2000.
  2. T. Gao and T.Y. Li: Mixed volume computation for semi-mixed systems. Discrete Comput. Geom. 29(2):257-277, 2003.
  3. L. Granvilliers: On the Combination of Interval Constraint Solvers. Reliable Computing 7(6): 467-483, 2001.
  4. Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler: Regeneration Homotopies for Solving Systems of Polynomials. Mathematics of Computation 80(273): 345-377, 2011.
  5. S. Kim and M. Kojima: Numerical Stability of Path Tracing in Polyhedral Homotopy Continuation Methods. Computing 73(4): 329-348, 2004.
  6. Y. Lebbah, C. Michel, M. Rueher, D. Daney, and J.P. Merlet: Efficient and safe global constraints for handling numerical constraint systems. SIAM J. Numer. Anal. 42(5):2076-2097, 2005.
  7. T.L. Lee, T.Y. Li, and C.H. Tsai: HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2-3): 109-133, 2008.
  8. Anton Leykin: Numerical Algebraic Geometry. The Journal of Software for Algebra and Geometry volume 3, pages 5-10, 2011.
  9. T.Y. Li and X. Li: Finding Mixed Cells in the Mixed Volume Computation. Foundations of Computational Mathematics 1(2): 161-181, 2001.
  10. T.Y. Li, X. Wang, and M. Wu: Numerical Schubert Calculus by the Pieri Homotopy Algorithm. SIAM J. Numer Anal. 40(2): 578-600, 2002.
  11. J.M. Porta, L. Ros, T. Creemers, and F. Thomas: Box approximations of planar linkage configuration spaces. Journal of Mechanical Design 129(4):397-405, 2007.
  12. Laurent Sorber, Marc Van Barel, and Lieven De Lathauwer: Numerical solution of bivariate and polyanalytic polynomial systems. SIAM J. Numer. Anal. 52(4):1551-1572, 2014.
  13. Yang Sun, Yu-Hui Tao, Feng-Shan Bai: Incomplete Groebner basis as a preconditioner for polynomial systems. Journal of Computational and Applied Mathematics 226(1):2-9, 2009.
PHCpack was used to develop new homotopy algorithms:
  1. Bo Dong, Bo Yu, and Yan Yu: A symmetric and hybrid polynomial system solving method for mixed trigonometric polynomial systems. Mathematics of Computation 83(288): 1847-1868, 2014.
  2. Bo Yu and Bo Dong: A hybrid polynomial system solving method for mixed trigonometric polynomial systems. SIAM J. Numer. Anal. 46(3): 1503-1518, 2008.
  3. Xuping Zhang, Jintao Zhang, and Bo Yu: Eigenfunction expansion method for multiple solutions of semilinear elliptic equations with polynomial nonlinearity. SIAM J. Numer. Anal. 51(5): 2680-2699, 2013.
Last, but certainly not least, there is the wonderful book of Bernd Sturmfels which contains a section on computing Nash equilibria with PHCpack.
  1. B. Sturmfels: Solving Systems of Polynomial Equations. CBMS Regional Conference Series of the AMS, Number 97, 2002.
If you achieved some nice results with PHCpack, please let me know!