Algorithm 801: POLSYS_PLP. A partitioned linear product homotopy code for solving polynomial systems of equations. The problem of solving large systems of polynomial equations arises in many application areas and poses difficult mathematical questions and computational challenges. This paper critically reviews the literature of the past two decades, during which there has been rapid evolution of both theory and codes. Homotopy (continuation) methods are a principal tool for such problems. The code under consideration uses modules from the more general HOMPACK90 package and is compatible with it. The middle game of following a homotopy path is well understood at this point. The paper focuses principally on the end and opening games. The authors treat mainly isolated, finite, regular solutions, but they also introduce suitable transformations for coping gracefully and efficiently with more awkward situations. They exploit partitioned linear product structure to provide convenient starting points and to prune the number of paths to be followed. The discussion relies heavily on the previous literature cited throughout, but the relevant results are clearly explained – except for minor mysteries. (Review ACM)

References in zbMATH (referenced in 18 articles , 1 standard article )

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  1. Luo, Zhongxuan; Feng, Erbao; Zhang, Jiejin: A numerical realization of the conditions of Max Nöther’s residual intersection theorem (2014)
  2. Wampler, Charles W.; Sommese, Andrew J.: Numerical algebraic geometry and algebraic kinematics (2011)
  3. Di Rocco, Sandra; Eklund, David; Sommese, Andrew J.; Wampler, Charles W.: Algebraic $\Bbb C^*$-actions and the inverse kinematics of a general 6R manipulator (2010)
  4. Tari, Hafez; Su, Hai-Jun; Li, Tien-Yien: A constrained homotopy technique for excluding unwanted solutions from polynomial equations arising in kinematics problems (2010)
  5. Zhu, Xin-Guang; Alba, Rafael; de Sturler, Eric: A simple model of the Calvin cycle has only one physiologically feasible steady state under the same external conditions (2009)
  6. Kumar, M.Vasudeva; Kienle, A.; Zeyer, K.P.; Pushpavanam, S.: Nonlinear analysis of the effect of maintenance in continuous cell cultures (2008)
  7. Yu, Bo; Dong, Bo: A hybrid polynomial system solving method for mixed trigonometric polynomial systems (2008)
  8. Gunji, T.; Kim, S.; Fujisawa, K.; Kojima, M.: PHoMpara-parallel implementation of the polyhedral homotopy continuation method for polynomial systems (2006)
  9. Leykin, Anton; Verschelde, Jan; Zhuang, Yan: Parallel homotopy algorithms to solve polynomial systems (2006)
  10. Su, Hai-Jun; Mccarthy, J.Michael; Sosonkina, Masha; Watson, Layne T.: Algorithm 857: POLSYS$_-$GLP -- a parallel general linear product homotopy code for solving polynomial systems of equations. (2006)
  11. Dickenstein, Alicia (ed.); Emiris, Ioannis Z. (ed.): Solving polynomial equations. Foundations, algorithms, and applications (2005)
  12. Zwolak, Jason W.; Tyson, John J.; Watson, Layne T.: Finding all steady state solutions of chemical kinetic models (2004)
  13. Hochstenbach, Michiel E.; van der Vorst, Henk A.: Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem (2003)
  14. Zavorin, Ilya; O’Leary, Dianne P.; Elman, Howard: Complete stagnation of GMRES (2003)
  15. Tawarmalani, Mohit; Sahinidis, Nikolaos V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming. Theory, algorithms, software, and applications (2002)
  16. Watson, Layne T.: Probability-one homotopies in computational science (2002)
  17. Wise, Steven M.; Sommese, Andrew J.; Watson, Layne T.: Algorithm 801: POLSYS_PLP. A partitioned linear product homotopy code for solving polynomial systems of equations. (2000)
  18. Verschelde, Jan: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation (1999)