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

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

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

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

starting systemtarget system

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

starting systemtarget system

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

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

The total degree is For n = 100,

may have a big waste

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

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

How to solve ?

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

Binomial equation

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

The new starting system

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

The new starting system

lifting => lower edges

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

In general

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

lifting => lower edges

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

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

where …

+ 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”.

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)

n Total degreeMixed Volume# of isolated zero eco- 153,188,6468,192 169,565,93816,384 noon- 919,68319,665 1059,04959,029 cyclic- 9362,88011,0166,642 103,628,80035,94034,940

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

Parallel Computation

# of CPUs CPU time (second) ratio 1445.321.00 2223.491.99 3150.692.96 591.314.88 768.706.48 # of CPUs CPU time (second) ratio 11475.391.00 2734.962.00 3494.192.99 5295.904.99 7212.876.93 eco-16 cyclic-11 The scalability of solving systems by polyhedral homotopy method

Newton’s law of motion: N-body problem

central configuration

From Marshall Hampton central configuration

From Marshall Hampton central configuration

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

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

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

PHCpackHOM4PS-2.0 # of complex solutions 26,53327,235 # of real solutions 133135 # 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

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

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)

(1)60 collinear c.c. (2) 15+12+60+60 = 147 planar c.c. (3) 10+15+5+20 = 50 spatial c.c. (4) one 4-dimensional simplex c.c.

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. http://hom4ps.math.msu.edu/HOM4PS_soft.htm

Thank you

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