Abstract
Homotopy continuation methods have been proved to be an efficient and reliable class of numerical methods for solving systems of polynomial equations which occur frequently in various fields of mathematics, science, and engineering. Based on the successful software package Hom4PS-2.0 for solving such polynomial systems, Hom4PS-3 has a new fully modular design which allows it to be easily extended. It implements many different numerical homotopy methods including the Polyhedral Homotopy continuation method. Furthermore, it is capable of carrying out computation in parallel on a wide range of hardware architectures including multi-core systems, computer clusters, distributed environments, and GPUs with great efficiency and scalability. Designed to be user-friendly, it includes interfaces to a variety of existing mathematical software and programming languages such as Python, Ruby, Octave, Sage and Matlab.
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References
Allgower, E., Georg, K.: Introduction to numerical continuation methods, vol. 45. Society for Industrial and Applied Mathematics (2003)
Attardi, G., Traverso, C.: The PoSSo library for polynomial system solving. In: Proc. of AIHENP 1995 (1995)
Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically Solving Polynomial Systems with Bertini. Society for Industrial and Applied Mathematics (2013)
Bernshtein, D.N.: The number of roots of a system of equations. Functional Analysis and its Applications 9(3), 183–185 (1975)
Chen, T.R., Lee, T.L., Li, T.Y.: Hom4PS-3: an numerical solver for polynomial systems using homotopy continuation methods, http://www.hom4ps3.org
Drexler, F.-J.: Eine methode zur berechnung sämtlicher lösungen von polynomgleichungssystemen. Numerische Mathematik 29(1), 45–58 (1977)
Garcia, C.B., Zangwill, W.I.: Finding all solutions to polynomial systems and other systems of equations. Mathematical Programming 16(1), 159–176 (1979)
Gunji, T., Kim, S., Kojima, M., Takeda, A., Fujisawa, K., Mizutani, T.: PHoM–a polyhedral homotopy continuation method for polynomial systems. Computing 73(1), 57–77 (2004)
Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Mathematics of Computation 64(212), 1541–1555 (1995)
Lee, T.L., Li, T.Y., Tsai, C.H.: HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2), 109–133 (2008)
Lee, T.L., Li, T.Y., Tsai, C.H.: HOM4PS-2.0 para: Parallelization of HOM4PS-2.0 for solving polynomial systems. Parallel Computing 35(4), 226–238 (2009)
Leykin, A.: NAG4M2: Numerical algebraic geometry for Macaulay2, http://people.math.gatech.edu/~aleykin3/NAG4M2/
Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Ciarlet, P.G. (ed.) Handbook of Numerical Analysis, vol. 11, pp. 209–304. North-Holland (2003)
Li, T.Y., Sauer, T., Yorke, J.: The cheater’s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM Journal on Numerical Analysis, 1241–1251 (1989)
Li, T.Y., Wang, X.: The BKK root count in ℂn. Mathematics of Computation of the American Mathematical Society 65(216), 1477–1484 (1996)
Morgan, A.P.: Solving polynomial systems using continuation for engineering and scientific problems. Classics in Applied Mathematics, vol. 57. Society for Industrial and Applied Mathematics (2009)
Morgan, A.P., Sommese, A.J.: Coefficient-parameter polynomial continuation. Applied Mathematics and Computation 29(2), 123–160 (1989)
Sommese, A.J., Wampler, C.W.: The Numerical solution of systems of polynomials arising in engineering and science. World Scientific Pub. Co. Inc. (2005)
Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software (TOMS) 25(2), 251–276 (1999)
Watson, L.T., Billups, S.C., Morgan, A.P.: Algorithm 652: Hompack: A suite of codes for globally convergent homotopy algorithms. ACM Transactions on Mathematical Software (TOMS) 13(3), 281–310 (1987)
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Chen, T., Lee, TL., Li, TY. (2014). Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods. In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_30
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DOI: https://doi.org/10.1007/978-3-662-44199-2_30
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