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Tsung-Lin Lee Department of Mathematics Michigan State University Solving polynomial systems by homotopy continuation method

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1. (Generalized) eigenvalue problem 2.Computation of equilibrium states 3.Optimal control problem 4.Kinematic synthesis problem …… Polynomial systems appear in

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The homotopy continuation method P 和 Q 的同倫 (the homotopy of P and Q)

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starting systemtarget system

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total degree homotopy T.Y. Li et al (1980’s) Theorem : For almost all choice of, this homotopy “works”.

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starting systemtarget system

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total degree homotopy Case: the number of solutions << the total degree The number of curves is (total degree)

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For an n x n matrix A, the eigenvalue problem It has at most n isolated solutions Example

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The total degree is For n = 100,

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may have a big waste

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polyhedral homotopy 1. The number of curves is the “ mixed volume ”. Huber and Sturmfels (1995) 2. mixed volume < total degree

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Cheater’s homotopy (Li, Sauer, Yorke 1989) Theorem : For almost all choice of, this homotopy “works”.

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How to solve ?

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polyhedral homotopy Pick random powers of t, Problem : can not identify the starting system

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Binomial equation

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Note: when t = 1 Change variable with + terms with positive powers of t

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The new starting system

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Note: when t = 1 Change variable with + terms with positive powers of t

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The new starting system

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lifting => lower edges

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Altogether, we get 4+1=5 solutions of Q(x)=0

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In general

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Binomial system When the coefficients are randomly chosen, it has nonzero solutions.

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lifting => lower edges

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Choose so that is attained exactly two for each. “Mixed volume Computation”

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Note: when t = 1 Change variable Change variables Each term of looks like

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where …

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+ terms with positive powers of t Theorem (Huber and Sturmfels): For almost all choice of the (complex) coefficients and the (rational) powers of t, the polyhedral homotopy “works”.

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PHCpack : Jan Verschelde HOM4PS : T. Gao, T.Y. Li, and M. Wu (MixedVol) HOM4PS-2.0 : T.L. Lee, T.Y. Li, C.H. Tsai (MixedVol-2.0) Software : homotopy methods for polynomial systems PHoM : T. Gunji, S. Kim, M. Kojima, A. Takeda, K. Fujisawa, T. Mizutani Bertini : D. Bates, J. Hauenstein, A. Sommese, C. Wampler (1999) (2004) (2006) (2008) (2003)

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n Total degreeMixed Volume# of isolated zero eco- 153,188,6468, ,565,93816,384 noon- 919,68319, ,04959,029 cyclic- 9362,88011,0166, ,628,80035,94034,940

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n PHomPHCpackHOM4PSHOM4PS-2.0 eco- 15>12h3h55m33m15s2m25s 16>12h 2h55m6m36s noon- 95h01m33m28s21m41s22s 10>12h2h33m3h20m1m27s cyclic- 9>12h3h50m8m37s44s 10>12h11h00m57m44s2m47s Running on a Dell PC with Pentium4 CPU of 2.2GHz

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Parallel Computation

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# of CPUs CPU time (second) ratio # of CPUs CPU time (second) ratio eco-16 cyclic-11 The scalability of solving systems by polyhedral homotopy method

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Newton’s law of motion: N-body problem

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central configuration

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From Marshall Hampton central configuration

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From Marshall Hampton central configuration

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Albouy-Chenciner equation (1998) : the distance between particle i and j: slack variable

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3-body Albouy-Chenciner equation (1) 6 equations, 6 variables (2) Total degree = 1728 (3) Mixed volume = 99

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4-body Albouy-Chenciner equation (1) 12 equations, 12 variables (2) Total degree = 2,985,984 (12^6) (3) Mixed volume = 81,864

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PHCpackHOM4PS-2.0 # of complex solutions 26,53327,235 # of real solutions # of physical solutions 3132 running time 11h51m7m20s 4-body Albouy-Chenciner equation with m1=m2=m3=m4 Running on a Dell PC with Pentium D CPU of 3.2GHz

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5-body Albouy-Chenciner equation (1) 20 equations in 20 variables (2) total degree = 61,917,364,224 (3) mixed volume = 439,631,712

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5-body Albouy-Chenciner equation with equal masses (1)HOM4PS-2.0 (2)Call subroutines in MPI library (3)32 CPUs from SHARCNET (4)6 days (1)101,062,826 solutions. (2) 8775 are real solutions. (3) 258 are physical solutions (rij>0)

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(1)60 collinear c.c. (2) = 147 planar c.c. (3) = 50 spatial c.c. (4) one 4-dimensional simplex c.c.

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Conclusion: 1. Fewer curves in polyhedral homotopy. 2. HOM4PS-2.0 is very efficient. 3. 4, 5-body A.C. equations can be solved in reasonable hours.

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Thank you

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