Tsung-Lin Lee Department of Mathematics Michigan State University Solving polynomial systems by...

Tsung-Lin Lee

Department of MathematicsMichigan State University

Solving polynomial systems by homotopy continuation method

0t 1t

)(ty

1. (Generalized) eigenvalue problem

2. Computation of equilibrium states

3. Optimal control problem

4. Kinematic synthesis problem

……

xBxA

)(xFx

Polynomial systems appear in

01

32

2

11),(

222

xy

yxt

y

xttXH

The homotopy continuation method

01

032)(

22

xy

yxXP

02

01)(

2

y

xXQ

2,1

2,1

51,52

51,52

P 和 Q的同倫 (the homotopy of P and Q)

01

322.0

2

18.0)2.0,(

222

xy

yx

y

xXH

04.12.0

02.04.0 22

xy

yx

01

32

2

11),(

222

xy

yxt

y

xttXH

01

324.0

2

16.0)4.0,(

222

xy

yx

y

xXH

08.04.0

06.08.0 22

xy

yx

0t 1t

0)( XP0)( XQ

2.0 4.0 6.0 8.0

01

32

2

11),(

222

xy

yxt

y

xttXH

0)()()1(),( XPtXQttXHstarting system target system

total degree homotopy

),,(

),,(

),,(

1

12

11

nn

n

n

xxp

xxp

xxp

T.Y. Li et al (1980’s)

1

32 22

xy

yx

ndnn

d

d

bxa

bxa

bxa

n

222

111

2

1

21

2

12

1

bya

bxa

0

0 t1 t

0

0

t t1

Theorem : For almost all choice of , this homotopy “works”. nn bbaa ,,,,, 11

0)()()1(),( XPtXQttXHstarting system target system

total degree homotopy

),,(

),,(

),,(

1

12

11

nn

n

n

xxp

xxp

xxp

ndnn

d

d

bxa

bxa

bxa

n

222

111

2

1

0

0

t t1

Case: the number of solutions << the total degree

The number of curves is(total degree)

nddd 21

For an n x n matrix A, the eigenvalue problem

xAx

It has at most n isolated solutions

Example

0 Axx

011111 nnxaxax 021212 nnxaxax

011 nnnnn xaxax

002211 cxcxcxc nn

nxxx ,,,, 21

011111 nnxaxax 021212 nnxaxax

011 nnnnn xaxax

002211 cxcxcxc nn

1211 bxa

2

222 bxa

nnn bxa 2

11 nn ba

The total degree is n2

For n = 100, 30100 102

t1 t

0t1t

may have a big waste

210

3010

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)

0542)( 345 xxxxxP

0)( 013

34

45

5 cxcxcxcxcxQ

0)()()1(),( XPtXQttXH

)()1,( XPXH )()0,( XQXH

Theorem : For almost all choice of , this homotopy “works”. 510 ,,, ccc

How to solve

0)( 013

34

45

5 cxcxcxcxcxQ

?

0)( 013

34

45

5 cxcxcxcxcxQ

polyhedral homotopy

Pick random powers of t,

1.10

2.11

9.133

8.044

3.155),( tcxtctxctxctxctxH

)()1,( XQXH 0)0,( XH

Problem : can not identify the starting system

Binomial equation

073 315 xx

315 73 xx

3/7315 x

3/712 x

0 nm bxax

nm bxax

abx nm /

1.10

2.11

9.133

8.044

3.155),( tcxtctxctxctxctxH 3.1,5 8.0,4 9.1,3 2.1,1 1.1,0

5 4 3 1 0

0 1 2 3 4 5 6

1

2

x

t

5

4

3

10

1,075.0ˆ

675.1ˆ,5 1.1ˆ,4

1.1ˆ,0 275.1ˆ,1 125.2ˆ,3

Note: when t = 1 yx

Change variable with ytx 075.0

1.10

2.11

9.13

38.04

43.15

5),( tctytctytctytctytctyH

3.1555

tyc

)ˆ,00

)ˆ,111

)ˆ,333

)ˆ,444

)ˆ,555

tctyctyctyctyc

1.10

275.111

125.233

1.144

675.155 tctyctyctyctyc

0175.01

1025.13

34

4575.05

51.1 ctyctycyctyct

0175.01

1025.13

34

4575.05

5),( ctyctycyctyctyH

04

4 cyc + terms with positive powers of t

04

4)0,( cycyH

)1,(),3.1,5(55

tyc )ˆ,555

tyc

1.10

2.11

9.133

8.044


0t1t

0)0,( yH

The new starting system

004

4 cyc

404 / ccy

0 1 2 3 4 5 6

1

2

x

t

5

4

3

10

1,5.0ˆ

2.1ˆ,5 2.1ˆ,4

1.1ˆ,0 7.0ˆ,1 4.0ˆ,3

Note: when t = 1 yx

Change variable with ytx 5.0

1.10

7.011

4.033

2.144

2.155),( tctyctyctyctyctytH

3.20

9.111

6.133

44

55

2.1 tctyctycycyct

44

55)0,( ycycyH

44

55 ycyc + terms with positive powers of t

3.20

9.111

6.133

44

55),( tctyctycycyctyH

1.10

2.11

9.133

8.044


0t1t

0)0,( yH

The new starting system

54 / ccy

044

55 ycyc

0 1 2 3 4 5 6

1

2

x

t

5

4

3

10

lifting => lower edges

0)( 013

34

45

5 cxcxcxcxcxQ

0t 1t

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

0t 1t

075.0

5.0

nSa

aann

Sa

aa

xcxp

xcxp

xP

,

,11

)(

)(

)(1

542)( 345 xxxxxP

In general

013

34

45

5)( cxcxcxcxcxQ

nSa

aann

Sa

aa

xcxq

xcxq

xQ*

,

*,11

)(

)(

)(1

0)()()1(),( xtPxQttxH 0)()()1(),( xtPxQttxH

013

34

45

5)( cxcxcxcxcxQ

nSa

aann

Sa

aa

xcxq

xcxq

xQ*

,

*,11

)(

)(

)(1

1.10

2.11

9.133

8.044

3.155),(

tctxctxc

txctxctxH

n

an

a

Sa

waann

Sa

waa

txctxh

txctxh

txH,

1

,1

*,

*,11

),(

),(

),(

0 nm byay

0'

0'

'

'11

11

nn an

an

aa

ycyc

ycyc

ytx ytx

0'

0'

0'

'

'22

'11

22

11

nn an

an

aa

aa

ycyc

ycyc

ycyc

Binomial system

When the coefficients are randomly chosen,

it has nonzero solutions.

'

'22

'11

det

nn aa

aa

aa

0 1 2 3 4 5 6

1

2

x

t

0)( 013

34

45

5 cxcxcxcxcxQ

0 1 2 3 4 5 6

1

2

x

t

5

4

3

10

lifting => lower edges

1.10

2.11

9.133

8.044


)3.1,5( )8.0,4( )9.1,3( )2.1,1( )1.1,0(

t

xycxycyxcyxcyxq 43

33

233

11 ),(

xycxycxcyxcyxcyxcyxq 63

53

433

34

224

12 ),(

)3,3( )1,3( )3,1( )1,1(

)3,3( )0,3( )3,1( )1,1()2,4( )1,4(

t

xx

t t

8.04

3.133

5.032

7.03311 ),,( txyctxyctyxctyxctyxq

2.16

1.135

9.034

6.1333

2.142

6.02412 ),,( xytctxyctxctyxcytxctyxctyxq

xx

)7.0,3,3( )5.0,1,3( )3.1,3,1( )8.0,1,1(

)6.1,3,3( )9.0,0,3( )1.1,3,1( )2.1,1,1()6.0,2,4( )2.1,1,4(

t t

Choose so that )1,(ˆ

1

4,3,2,11 ˆ,ˆmin j

ja

2

6,,2,12 ˆ,ˆmin j

ja

is attained exactly two for each.

“Mixed volume Computation”

Note: when t = 1 yx

Change variable ytx

3.15

5

3.155),(

tytc

txctxH

3.1555

tyc

)1,(),3.1,5(55

tyc

)ˆ,555

tyc ,aaa

waa tyctxc

Change variables

ytx

ntyx

tyx

nn

111

Each term of looks likeih

n n

nn

Sa Sa

aaan

aaan

Sa Sa

aaa

aaa

Sa Sa

aaa

aaa

tycttyc

tycttyc

tycttyc

tytHtxH

ˆ,ˆ*,

ˆ,ˆ*,

ˆ,ˆ*,2

ˆ,ˆ*,2

ˆ,ˆ*,1

ˆ,ˆ*,1

2 2

22

1 1

11

),(),(

where

1

,,11 ˆ,ˆmin

1j

mja

nj

mjn a

n

ˆ,ˆmin,,1

…

2

,,12 ˆ,ˆmin

2j

mja

n

n

Sa

aaan

Sa

aaa

tyc

tyc

tyH

ˆ,ˆ*,

ˆ,ˆ*,1

1

1

),(

nn an

an

aa

ycyc

ycyc

'

'11

'

' 11

+ terms with positive powers of t

+ 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 degree Mixed Volume # of isolated zero

eco-15 3,188,646 8,192 8,192

16 9,565,938 16,384 16,384

noon-9 19,683 19,665 19,665

10 59,049 59,029 59,029

cyclic-9 362,880 11,016 6,642

10 3,628,800 35,940 34,940

n PHom PHCpack HOM4PS HOM4PS-2.0

eco-15 >12h 3h55m 33m15s 2m25s

16 >12h >12h 2h55m 6m36s

noon-9 5h01m 33m28s 21m41s 22s

10 >12h 2h33m 3h20m 1m27s

cyclic-9 >12h 3h50m 8m37s 44s

10 >12h 11h00m 57m44s 2m47s

Running on a Dell PC with Pentium4 CPU of 2.2GHz

Parallel Computation

0t 1t

)(ty

# of

CPUs

CPU time

(second)

ratio

1 445.32 1.00

2 223.49 1.99

3 150.69 2.96

5 91.31 4.88

7 68.70 6.48

# of

CPUs

CPU time

(second)

ratio

1 1475.39 1.00

2 734.96 2.00

3 494.19 2.99

5 295.90 4.99

7 212.87 6.93

eco-16 cyclic-11

The scalability of solving systems by polyhedral homotopy method

1m 2m

3m

13r

12r

23r

0im3Rxi jiij xxr

nj

r

xxmmxm

ji ij

jijijj

13

Newton’s law of motion:

N-body problem

central configuration

1m 2m

3m

From Marshall Hampton


Albouy-Chenciner equation (1998)

n

kijjkikjkijikjkikk rrrSrrrSm

1

222222 0

Nji 10133 ijijij rrS

ijr : the distance between particle i and j

ijS : slack variable

3-body Albouy-Chenciner equation

(1) 6 equations, 6 variables

(2) Total degree = 1728

(3) Mixed volume = 99


(1) 12 equations, 12 variables

(2) Total degree = 2,985,984 (12^6)

(3) Mixed volume = 81,864

PHCpack HOM4PS-2.0

# of complex solutions 26,533 27,235

# of real solutions 133 135

# of physical solutions 31 32

running time 11h51m 7m20s

0ijr

4-body Albouy-Chenciner equation with m1=m2=m3=m4

Running on a Dell PC with Pentium D CPU of 3.2GHz


(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

Tsung-Lin Lee Department of Mathematics Michigan State University Solving polynomial systems by...

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