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

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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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

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

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
-
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

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

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

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

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

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

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

Numerical accuracy

a talk for another day

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

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

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

computationally very expensive

Depending on # of digits used,
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!