From Isolated Points to Positive Dimensions & Back Again: A Brief History of Numerical Algebraic Geometry

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From Isolated Points to Positive
Dimensions & Back Again:

A Brief History of

Numerical Algebraic Geometry

Charles Wampler



General Motors R&D Center



(Adjunct, Univ. Notre Dame)


Including joint work with

Andrew Sommese,

University of Notre Dame

Jon Hauenstein,

University of Notre Dame

2

Sommese Co-authoring
0
5
10
15
20
25
30
35
40
45
50
1
4
7
10
13
16
19
22
25
28
31
34
37
40
Colleagues
Co-authorships
Top Ten List

43

Beltrametti

28

Wampler

14

Morgan

13

Verschelde

10

Lanteri

8

Fania

7

Carrell

7

Schneider

7

Palleschi

7

Bates

Books

2

Beltrametti

1

Shiffman

1

Schneider

1

Wampler

By Type

129

Classical

39

Numerical

1

Library Science

169

Total

42 distinct co
-
authors

3

Outline


Introduction to continuation


Historical timeline


Algorithms for zero
-
dimensional solution sets


Algorithms for positive
-
dimensional solution sets


Creation of
numerical algebraic geometry


Built on methods for zero
-
dimensional solving


Numerical irreducible decomposition


Intersecting algebraic sets


Diagonal homotopy


New:

Regeneration method


Extensions of numerical algebraic geometry


Extracting real points


Computing the genus of a curve


Exceptional sets via fiber products


Using positive
-
dimensional methods to find isolated
solution sets


Equation
-
by
-
equation solving using regeneration

4

Introduction to Continuation


Basic idea: to solve F(x)=0 (N equations, N unknowns)


Define a homotopy H(x,t)=0 such that


H(x,1) = 0 has a known isolated solutions, S
1


H(x,0) = F(x)


Track solution paths as t goes from 1 to 0


Paths satisfy the Davidenko o.d.e.


(dH/dx)(dx/dt) + dH/dt = 0


Endpoints of the paths are solutions of F(x)=0


Let S
0

be the set of path endpoints


Concerns


Points where dH/dx is singular


Turning points


Path crossings


Multiple roots


Path divergence


Will S
0

contain all isolated solutions of F(x)=0?


What happens if V(F) has positive dimensional components?


What if we have N equations, n unknowns, N

n?


Numerical accuracy, Computational efficiency

t=1

t

t=0

S
0

S
1

5

Basic Total
-
degree Homotopy

To find all isolated solutions to the polynomial
system F:
C
N


C
N
, i.e.,




form the linear homotopy

H(x,t) = (1
-
t)F(x) + tG(x)=0,

where



i
i
N
N
N
d
x
x
x
x












f
deg

,
0
)
,...,
(
f
)
,...,
(
f
1
1
1

complex

random,

,

,
)
(
i
i
i
d
i
i
i
b
a
b
x
a
x
g
i


6

Brief Timeline: Part 1 (Before Sommese)


Pre
-
computer age


Homotopy & continuity for existence proofs


1960s: early computational methods


Real number field, heuristic methods


“Bootstrap method” in kinematics


Roth 1963


1970s, early 80s: Algorithmic methods


Complex number field, Geometric arguments


Fixed
-
point continuation (T.Y.Li 1976)


Specialization to polynomial systems


All isolated solutions, 1977 (Garcia & Zangwill; Drexler)


Total degree homotopy, 1978 (Chow, Mallet
-
Paret & Yorke)


Numerical path tracking (Allgower & Georg, book ’80)


Use of projective space (Wright, 1985)


Application to 6R robot kinematics (Tsai & Morgan, 1985)


Morgan’s book on polynomial continuation 1987

7

Addressing the concerns

Concerns


Points where dH/dx is singular


Turning points


Path crossings


Multiple roots


Path divergence


Will S
0

contain all isolated solutions of F(x)=0?


Computational efficiency


Numerical accuracy


What happens if V(F) has positive dimensional
components?


What if we have N equations, n unknowns, N

n?

t=1

t

t=0

S
0

S
1

complex homotopy 1977

projective space 1985

1977

Total degree 1978

8

Kinematic Milestone


6R Serial
-
Link Inverse
Kinematics


Formulation


Pieper, 1968 & earlier


32
nd

degree polynomial


Duffy & Crane, 1980


16 solutions shown by
continuation


Tsai & Morgan, 1985


Total degree homotopy
with 256 paths


16
th

degree polynomial


Li & Liang, 1988

9

Brief Timeline: Part 2 (After Sommese)


Late 80s, early 90s: Use of algebraic geometry


Still finding all isolated solutions


Focus on reducing the number of paths


Homotopies adapted to the structure of the target system


Multihomogeneity & the “
γ

trick” (Morgan & Sommese, 1987)


Parameterized systems: f(x;p)=0


Cheater’s homotopy (Li, Sauer & Yorke, 1988)


Coefficient parameter homotopy (Morgan & Sommese, 1989)


Polytope homotopies (a.k.a., polyhedral, mixed volume, or BKK)


Verschelde, Verlinden & Cools, ’94; Huber & Sturmfels, ’95


T.Y. Li with various co
-
authors, 1997
-
present


Computing singular endpoints


Morgan, Sommese, & Wampler (1991,1992)


Kinematics applications


Tutorial for mechanical engineers (MSW 1990)


Complete solution of the 9
-
point four
-
bar, (MSW 1992)

10

The “
γ

trick”


Simple but useful consequence of algebraicity


Instead of the homotopy

H(x,t) = (1
-
t)f(x) + tg(x)


use



H(x,
τ
) = (1
-
τ
)f(x) +
g
τ
g(x)

11

Parameter Continuation

initial
parameter
space

target
parameter
space


Start system easy in initial parameter space


Root count may be much lower in target parameter space


Initial run is 1
-
time investment for cheaper target runs

12

Kinematic Milestone


9
-
Point Path Generation
for Four
-
bars


Problem statement


Alt, 1923


Bootstrap partial solution


Roth, 1962


Complete solution


Wampler, Morgan &
Sommese, 1992


m
-
homogeneous continuation


1442 Robert cognate triples

13

Nine
-
point Four
-
bar summary


Symbolic reduction


Initial total degree



≈10
10


Roth & Freudenstein, tot.deg.=5,764,801


Our reformulation, tot.deg.=1,048,576


Multihomogenization 286,720


2
-
way symmetry 143,360


Numerical reduction (Parameter continuation)


Nondegenerate solutions


4326


Roberts cognate symmetry


1442


Synthesis program follows 1442 paths

14

Parameter Continuation: 9
-
point problem

2
-
homogeneous
systems with
symmetry:

143,360 solution
pairs

9
-
point
problems*:

1442 groups of
2x6 solutions

*Parameter space of 9
-
point problems is 18 dimensional (complex)

15

Addressing the concerns

t=1

t

t=0

S
0

S
1


Points where dH/dx is singular


Turning points


Path crossings


Multiple roots: Singular endgames 1991
-
92


Path divergence


Will S
0

contain all isolated solutions of F(x)=0?


Computational efficiency


Ab initio:


Multihomogeneous 1987


Polytopes 1994
-
present


Parameter homotopy 1989


Numerical accuracy


What happens if V(F) has positive dimensional
components?


What if we have N equations, n unknowns, N≠n?

16

Brief Timeline: Part 3 (Numerical Alg.Geom.)


“Numerical Algebraic Geometry” founded


Treatment of positive dimensional sets by slicing


Treatment of N
≠n by randomization


Cascade algorithm for slicing


Sommese & Verschelde, 2000


Numerical irreducible decomposition


Factoring by sampling & fitting (Sommese, Verschelde, & Wampler, 2001)


Coinage of “witness set”


Use of monodromy (SVW 2001 )


Use of symmetric functions (SVW 2002)


Intersection of components


Diagonal homotopy (SVW 2004)


Book by Sommese & Wampler, 2005


Local methods


Deflation of
μ
>1 i
solated points (Leykin, Verschelde, & Zhao, 2006)


Multiplicity structure (Dayton & Zeng 2005, Bates, Sommese, Peterson 2006)


Deflation of nonreduced positive dimensional components (S&W 2005)


Curves


Real curves (Lu, Bates, S&W, 2007), Curve genus (Bates, Peterson, S&W, preprint)


Equation
-
by
-
equation solving


Using diagonal homotopy (SVW 2008)


Using regeneration (Hauenstein, S, & W, in preparation)


Sommese & Wampler

(FoCM 1995 , publ. 1996)

Up to

positive

dimensions

And back
again!

17

Slicing & Witness Sets


Slicing theorem


An

degree
N
reduced algebraic
set of dimension
m

in
n
variables hits a general (
n
-
m
)
-
dimensional linear space in
N

isolated points


Witness generation


Slice at every dimension


Use continuation to get sets
that contain all isolated
solutions at each dimension


“Witness supersets”


Irreducible decomposition


Remove “junk”


Monodromy on slices finds
irreducible components


Linear traces verify
completeness

18

Membership Test

19

Linear Traces


Track witness paths as
slice translates parallel to
itself.


Centroid of witness
points for an algebraic set
must move on a line.

20

Further Reading

World Scientific
2005

21

Example: Griffis
-
Duffy Platform

Special Stewart
-
Gough platform

Studied by:


Husty & Karger, 2000

Degree 28 motion curve (in Study
coordinates)



if legs are equal & plates congruent:


factors as 6+(6+6+6)+4

22

Real Points on a Complex Curve


Go to Griffis
-
Duffy movie…

23

Addressing the concerns

t=1

t

t=0

S
0

S
1


Points where dH/dx is singular


Turning points


Path crossings


Multiple roots


Path divergence


Will S
0

contain all isolated solutions of F(x)=0?


What happens if V(F) has positive dimensional
components?


What if we have N equations, n unknowns, N≠n?


Computational efficiency


Isolated points


Positive dimensional components


Cascade goes like total degree


Working equation
-
by
-
equation helps! (esp.,
regeneration
)


Numerical accuracy


a talk for another day

Using traditional homotopies

24

Equation
-
by
-
Equation Solving

f
1
(x)=0


Co
-
dim 1

f
2
(x)=0


Co
-
dim 1

f
3
(x)=0


Co
-
dim 1

Intersect

Co
-
dim 1,2

Co
-
dim 1,2,3

Co
-
dim 1,2,...,N
-
1

f
N
(x)=0


Co
-
dim 1

Co
-
dim 1,2,...,min(n,N)

Final Result

Similar intersections


Special case:


N=n


nonsingular solutions only


results are very promising

N equations, n variables

Intersect

Intersect

Theory is in place for
μ
>1
isolated and for full witness
set generation.

25

Working Equation
-
by
-
Equation


Basic step

















































)
(
)
(
)
(
)
(
)
(
1
1
1
0
x
L
x
L
x
L
x
f
x
f
V
N
k
k
k


















































)
(
)
(
)
(
)
(
)
(
1
1
1
0
x
L
x
L
x
f
x
f
x
f
V
N
k
k
k


26

Regeneration: Step 1

















































)
(
)
(
)
(
)
(
)
(
1
1
1
0
x
L
x
L
x
L
x
f
x
f
V
N
k
k
k


















































)
(
)
(
)
(
)
(
)
(
)
(
1
,
1
,
1
1
0
x
L
x
L
x
L
x
L
x
f
x
f
V
N
k
d
k
k
k
k



















































)
(
)
(
)
(
)
(
)
(
1
1
,
1
1
0
x
L
x
L
x
L
x
f
x
f
V
N
k
k
k



















































)
(
)
(
)
(
)
(
)
(
1
,
1
1
0
x
L
x
L
x
L
x
f
x
f
V
N
k
d
k
k
k


move
slice
d
k

times

Union of
sets

27

Regeneration: Step 2

















































)
(
)
(
)
(
)
(
)
(
)
(
1
,
1
,
1
1
0
x
L
x
L
x
L
x
L
x
f
x
f
V
N
k
d
k
k
k
k



















































)
(
)
(
)
(
)
(
)
(
1
1
1
0
x
L
x
L
x
f
x
f
x
f
V
N
k
k
k


Linear
homotopy

28

Software for polynomial continuation


PHC (first release 1997)


J. Verschelde


First publicly available implementation of polytope method


Used in SVW series of papers


Isolated points


Multihomogeneous & polytope method


Positive dimensional sets


Basics, diagonal homotopy


Hom4PS
-
2.0, Hom4PS
-
2.0para (recent release)


T.Y. Li


Isolated points:


Multihomogeneous & polytope method


Bertini (ver1.0 released Apr.20, 2008)


D. Bates, J. Hauenstein, A. Sommese, C. Wampler


Isolated points


Multihomogeneous


Positive dimensional sets


Basics, diagonal homotopy, & regeneration

29

Test Run 1: 6R Robot Inverse Kinematics

Method*

Work

Time

Total
-
degree
traditional

1024 paths

54 s

Diagonal

eqn
-
by
-
eqn

649 paths

23 s

Regeneration
eqn
-
by
-
eqn

628 paths
313 slice
moves

9 s

*All runs in Bertini

30

Test Run 2: 9
-
point Four
-
bar Problem

1442 Roberts
cognates

Method

Work

Time

Polytope

(Hom4PS
-
2.0)

Mixed volume
87,639 paths

11.7 hrs

Regeneration

(Bertini)

136,296 paths

66,888 slice
moves

8.1 hrs

31

Test Run 3: Lotka
-
Volterra Systems


Discretized (finite differences) population model


Order n system has 8n sparse bilinear equations

+ mixed
volume

Work Summary

Total degree = 2
8n

Mixed volume = 2
4n

is exact

32

Lotka
-
Volterra Systems (cont.)


Time Summary

xx = did not finish

All runs on a single processor

33

Algebraic vs. Geometric Approaches



Pros

Cons

Algebraic/Symbolic

Highly developed

Parallelizes very poorly

Can give exact answers

Unstable dependence on inputs

Not restricted to characteristic
zero

Memory requirements grow
rapidly with number of variables

Geometric/
Numerical Methods

Parallelizes almost perfectly

New
--
less developed than
algebraic/symbolic approach

Relatively stable dependence on
inputs

Certifiably exact answers are
computationally very expensive

Depending on # of digits used,
gives answers reasonable for
that accuracy

Restricted to real and complex
numbers

34

Summary


Polynomials arise in applications


Especially kinematics


Continuation methods for isolated solutions


Highly developed in 1980’s, 1990’s


Numerical algebraic geometry


Builds on the methods for isolated roots


Treats positive
-
dimensional sets


Witness sets (slices) are the key construct


Regeneration: equation
-
by
-
equation approach


Uses slice moves to regenerate each new equation


Captures much of the same structure as polytope methods,
without a mixed volume computation


Most efficient method yet for large, sparse systems


Thanks, Andrew, for 2 decades of fun collaboration!