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Numerical solution of multivariate polynomial systems by homotopy continuation methods

  • T. Y. Li (a1)


Let P(x) = 0 be a system of n polynomial equations in n unknowns. Denoting P = (p1,…, pn), we want to find all isolated solutions of

for x = (x1,…,xn). This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc. Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Gröbner bases, are the classical approach to solving (1.1), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.



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Allgower, E. L. (1984), Bifurcation arising in the calculation of critical points via homotopy methods, in Numerical Methods for Bifurcation Problems (Kupper, T., Mittelman, H. D. and Weber, H., eds), Birkhäuser, Basel, pp. 1528.
Allgower, E. L. and Georg, K. (1990), Numerical Continuation Methods, an Introduction, Springer, Berlin. Springer Series in Computational Mathematics, Vol. 13.
Allgower, E. L. and Georg, K. (1993), Continuation and path following, in Acta Numerica, Vol. 2, Cambridge University Press, pp. 164.
Allison, D. C. S., Chakraborty, A. and Watson, L. T. (1989), ‘Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube’, J. Supercomputing 3, 520.
Bernshteín, D. N. (1975), ‘The number of roots of a system of equations’, Functional Anal. Appl. 9, 183185. Translated from Funktsional. Anal. i Prilozhen., 9, 1–4.
Brunovský, P. and Meravý, P. (1984), ‘Solving systems of polynomial equations by bounded and real homotopy’, Numer. Math. 43, 397418.
Buchberger, B. (1985), Gröbner basis: An algorithmic method in polynomial ideal theory, in Multidimensional System Theory (Bose, N., ed.), D. Reidel, Dordrecht, pp. 184232.
Canny, J. and Rojas, J. M. (1991), An optimal condition for determining the exact number of roots of a polynomial system, in Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ACM, pp. 96101.
Chow, S. N., Mallet-Paret, J. and Yorke, J. A. (1979), Homotopy method for locating all zeros of a system of polynomials, in Functional Differential Equations and Approximation of Fixed Points (Peitgen, H. O. and Walther, H. O., eds), Lecture Notes in Mathematics, Vol. 730, Springer, Berlin, pp. 7788.
Drexler, F. J. (1977), ‘Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen’, Numer. Math. 29, 4558.
Emiris, I. (1994), Sparse Elimination and Applications in Kinematics, PhD thesis, University of California at Berkeley.
Emiris, I. and Canny, J. (1995), ‘Efficient incremental algorithms for the sparse resultant and the mixed volume’, J. Symb. Computation 20, 117149.
Fulton, W. (1984), Intersection Theory, Springer, Berlin.
Garcia, C. B. and Zangwill, W. I. (1979), ‘Finding all solutions to polynomial systems and other systems of equations’, Mathematical Programming 16, 159176.
Gel'fand, I. M., Kapranov, M. M. and Zelevinskií, A. V. (1994), Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston.
Harimoto, S. and Watson, L. T. (1989), The granularity of homotopy algorithms for polynomial systems of equations, in Parallel Processing for Scientific Computing (Rodrigue, G., ed.), SIAM, Philadelphia.
Henderson, M. E. and Keller, H. B. (1990), ‘Complex bifurcation from real paths’, SIAM J. Appl. Math. 50, 460482.
Huber, B. (software), Pelican Manual. Available via the author's web page,
Huber, B. and Sturmfels, B. (1995), ‘A polyhedral method for solving sparse polynomial systems’, Math. Comp. 64, 15411555.
Huber, B. and Sturmfels, B. (1997), ‘Bernstein's theorem in affine space’, Discrete Comput. Geom. To appear.
Khovanskií, A. G. (1978), ‘Newton polyhedra and the genus of complete intersections’, Functional Anal. Appl. 12, 3846. Translated from Funktsional. Anal. i Prilozhen., 12, 51–61.
Kushnirenko, A. G. (1976), ‘Newton polytopes and the Bézout theorem’, Functional Anal. Appl. 10, 233235. Translated from Funktsional. Anal. i Prilozhen., 10, 82–83.
Lee, C. W. (1991), Regular triangulations of convex polytopes, in Applied Geometry and Discrete Mathematics – The Victor Klee Festschrift, DIMACS Series Vol. 4 (Gritzmann, P. and Sturmfels, B., eds), American Mathematical Society, Providence, RI, pp. 443456.
Li, T. Y. (1983), ‘On Chow, Mallet-Paret and Yorke homotopy for solving systems of polynomials’, Bulletin of the Institute of Mathematics, Acad. Sin. 11, 433437.
Li, T. Y. and Sauer, T. (1989), ‘A simple homotopy for solving deficient polynomial systems’, Japan J. Appl. Math. 6, 409419.
Li, T. Y. and Wang, X. (1990), A homotopy for solving the kinematics of the most general six- and-five-degree of freedom manipulators, in Proc. of ASME Conference on Mechanisms, Dl- Vol. 25, pp. 249252.
Li, T. Y. and Wang, X. (1991), ‘Solving deficient polynomial systems with homotopies which keep the subschemes at infinity invariant’, Math. Comp. 56, 693710.
Li, T. Y. and Wang, X. (1992), ‘Nonlinear homotopies for solving deficient polynomial systems with parameters’, SIAM J. Numer. Anal. 29, 11041118.
Li, T. Y. and Wang, X. (1993), ‘Solving real polynomial systems with real homotopies’, Math. Comp. 60, 669680.
Li, T. Y. and Wang, X. (1994), ‘Higher order turning points’, Appl. Math. Comput. 64, 155166.
Li, T. Y. and Wang, X. (1996), ‘The BKK root count in ℂn’, Math. Comp. 65, 14771484.
Li, T. Y., Sauer, T. and Yorke, J. A. (1987 a), ‘Numerical solution of a class of deficient polynomial systems’, SIAM J. Numer. Anal. 24, 435451.
Li, T. Y., Sauer, T. and Yorke, J. A. (1987 b), ‘The random product homotopy and deficient polynomial systems’, Numer. Math. 51, 481500.
Li, T. Y., Sauer, T. and Yorke, J. A. (1988), ‘Numerically determining solutions of systems of polynomial equations’, Bull. Amer. Math. Soc. 18, 173177.
Li, T. Y., Sauer, T. and Yorke, J. A. (1989), ‘The cheater's homotopy: an efficient procedure for solving systems of polynomial equations’, SIAM J. Numer. Anal. 26, 12411251.
Li, T. Y., Wang, T. and Wang, X. (1996), Random product homotopy with minimal BKK bound, in Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics on Mathematics of Numerical Analysis: Real Number Algorithms, Park City, Utah, pp. 503512.
Malajovich, G. (software), pss 2.alpha, polynomial system solver, version 2.alpha, README file. Distributed by the author through gopher, http:;~gregorio.
Morgan, A. P. (1986), ‘A homotopy for solving polynomial systems’, Appl. Math. Comput. 18, 173177.
Morgan, A. P. (1987), Solving Polynomial Systems using Continuation for Engineering and Scientific Problems, Prentice Hall, Englewood Cliffs, NJ.
Morgan, A. P. and Sommese, A. J. (1987 a), ‘Computing all solutions to polynomial systems using homotopy continuation’, Appl. Math. Comput. 24, 115138.
Morgan, A. P. and Sommese, A. J. (1987 b), ‘A homotopy for solving general polynomial systems that respect m-homogeneous structures’, Appl. Math. Comput. 24, 101113.
Morgan, A. P. and Sommese, A. J. (1989), ‘Coefficient-parameter polynomial continuation’, Appl. Math. Comput. 29, 123160. Errata: Appl. Math. Comput. 51, 207 (1992).
Morgan, A. P., Sommese, A. J. and Watson, L. T. (1989), ‘Finding all isolated solutions to polynomial systems using HOMPACK’, ACM Trans. Math. Software 15, 93122.
Rojas, J. M. (1994), ‘A convex geometric approach to counting the roots of a polynomial system’, Theoret. Comput. Sci. 133, 105140.
Rojas, J. M. and Wang, X. (1996), ‘Counting affine roots of polynomial systems via pointed Newton polytopes’, J. Complexity 12, 116133.
Shafarevich, I. R. (1977), Basic Algebraic Geometry, Springer, New York.
Shub, M. and Smale, S. (1993), ‘Complexity of Bézout's theorem I: Geometric aspects’, J. Amer. Math. Soc. 6, 459501.
Tsai, L. W. and Morgan, A. P. (1985), ‘Solving the kinematics of the most general six-and five-degree-of-freedom manipulators by continuation methods’, ASME Journal of Mechanics, Transmissions, and Automation in Design 107, 189200.
Verschelde, J. (1995), PHC and MVC: two programs for solving polynomial systems by homotopy continuation, Technical report. Presented at the PoSSo workshop on software, Paris. Available by anonymous ftp to in the directory /pub/NumAnal-ApplMath/PHC.
Verschelde, J. (1996), Homotopy continuation methods for solving polynomial systems, PhD thesis, Katholieke Universiteit Leuven, Belgium.
Verschelde, J. and Cools, R. (1993), ‘Symbolic homotopy construction’, Applicable Algebra in Engineering, Communication and Computing 4, 169183.
Verschelde, J. and Cools, R. (1994), ‘Symmetric homotopy construction’, J. Comput. Appl. Math. 50, 575592.
Verschelde, J. and Gatermann, K. (1995), ‘Symmetric Newton polytopes for solving sparse polynomial systems’, Adv. Appl. Math. 16, 95127.
Verschelde, J., Gatermann, K. and Cools, R. (1996), ‘Mixed volume computation by dynamic lifting applied to polynomial system solving’, Discrete Comput. Geom. 16, 69112.
Wampler, C. W. (1992), ‘Bézout number calculations for multi-homogeneous polynomial systems’, Appl. Math. Comput. 51, 143157.
Wampler, C. W. (1994), ‘An efficient start system for multi-homogeneous polynomial continuation’, Numer. Math. 66, 517523.
Wright, A. H. (1985), ‘Finding all solutions to a system of polynomial equations’, Math. Comp. 44, 125133.
Zulener, W. (1988), ‘A simple homotopy method for determining all isolated solutions to polynomial systems’, Math. Comp. 50, 167177.


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