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
Sommese Co-authoring
50
Top Ten List
43
Beltrametti
2
Beltrametti
28
Wampler
1
Shiffman
14
Morgan
1
Schneider
35
13
Verschelde
1
Wampler
30
10
Lanteri
45
40
Co-authorships
Books
By Type
8
Fania
25
7
Carrell
20
7
Schneider
39
7
Palleschi
1
7
Bates
15
10
129
169
Classical
Numerical
Library Science
Total
42 distinct co-authors
5
0
1
4
7
10 13 16 19 22 25 28 31 34 37 40
Colleagues
2
Outline


Introduction to continuation
Historical timeline


Algorithms for zero-dimensional solution sets
Algorithms for positive-dimensional solution sets

Creation of numerical algebraic geometry



Numerical irreducible decomposition
Intersecting algebraic sets



Diagonal homotopy
New: Regeneration method
Extensions of numerical algebraic geometry




Built on methods for zero-dimensional solving
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
3
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, S1
H(x,0) = F(x)
S1
Track solution paths as t goes from 1 to 0

Paths satisfy the Davidenko o.d.e.




S0
(dH/dx)(dx/dt) + dH/dt = 0
Endpoints of the paths are solutions of F(x)=0
Let S0 be the set of path endpoints
Concerns

Points where dH/dx is singular








Turning points
Path crossings
Multiple roots
t=0
Path divergence
Will S0 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
4
t
t=1
Basic Total-degree Homotopy
To find all isolated solutions to the polynomial
system F: CN CN, i.e.,
 f1 ( x1 ,..., xN ) 




  0, degf i   d i
f N ( x1 ,..., xN )
form the linear homotopy
H(x,t) = (1-t)F(x) + tG(x)=0,
where
g i ( x)  a x  bi , ai , bi random, complex
di
i i
5
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
6
Addressing the concerns
S0
S1
Concerns

Points where dH/dx is singular









Turning points
Path crossings
Multiple roots
complex homotopy 1977
t=0
t
Path divergence projective space 1985
Will S0 contain all isolated solutions of F(x)=0? 1977
Computational efficiency Total degree 1978
Numerical accuracy
What happens if V(F) has positive dimensional
components?
What if we have N equations, n unknowns, N≠n?
7
t=1
Kinematic Milestone

6R Serial-Link Inverse
Kinematics

Formulation


32nd degree polynomial


Duffy & Crane, 1980
16 solutions shown by
continuation



Pieper, 1968 & earlier
Tsai & Morgan, 1985
Total degree homotopy
with 256 paths
16th degree polynomial

Li & Liang, 1988
8
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



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


Cheater’s homotopy (Li, Sauer & Yorke, 1988)
Coefficient parameter homotopy (Morgan & Sommese, 1989)
Morgan, Sommese, & Wampler (1991,1992)
Kinematics applications


Tutorial for mechanical engineers (MSW 1990)
Complete solution of the 9-point four-bar, (MSW 1992)
9
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) + gτg(x)

10
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
11
Kinematic Milestone

9-Point Path Generation
for Four-bars

Problem statement


Bootstrap partial solution


Alt, 1923
Roth, 1962
Complete solution



Wampler, Morgan &
Sommese, 1992
m-homogeneous continuation
1442 Robert cognate triples
12
Nine-point Four-bar summary

Symbolic reduction






Numerical reduction (Parameter continuation)



Initial total degree
≈1010
Roth & Freudenstein, tot.deg.=5,764,801
Our reformulation,
tot.deg.=1,048,576
Multihomogenization
286,720
2-way symmetry
143,360
Nondegenerate solutions
Roberts cognate symmetry
4326
1442
Synthesis program follows 1442 paths
13
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)
14
Addressing the concerns





Turning points
Path crossings
Multiple roots: Singular endgames 1991-92



t
Ab initio:


t=0
Path divergence
Will S0 contain all isolated solutions of F(x)=0?
Computational efficiency


S1
Points where dH/dx is singular


S0
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?
15
t=1
Brief Timeline: Part 3 (Numerical Alg.Geom.)

“Numerical Algebraic Geometry” founded









Diagonal homotopy (SVW 2004)
Deflation of μ>1 isolated points (Leykin, Verschelde, & Zhao, 2006)
Multiplicity structure (Dayton & Zeng 2005, Bates, Sommese, Peterson 2006)
Deflation of nonreduced positive dimensional components (S&W 2005)
Curves


Use of monodromy (SVW 2001 )
Use of symmetric functions (SVW 2002)
Book by Sommese & Wampler, 2005
Local methods


Coinage of “witness set”
Intersection of components


And back
again!
Factoring by sampling & fitting (Sommese, Verschelde, & Wampler, 2001)

positive
dimensions
Sommese & Verschelde, 2000
Numerical irreducible decomposition

Up to
Sommese & Wampler
(FoCM 1995 , publ. 1996)
Cascade algorithm for slicing


Treatment of positive dimensional sets by slicing
Treatment of N≠n by randomization
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)
16
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
17
Membership Test
18
Linear Traces
Track witness paths as
slice translates parallel to
itself.
Centroid of witness
points for an algebraic set
must move on a line.
19
Further Reading
World Scientific
2005
20
Example: Griffis-Duffy Platform
Special StewartGough 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
21
Real Points on a Complex Curve

Go to Griffis-Duffy movie…
22
Addressing the concerns







Turning points
Path crossings
Multiple roots
t=0
t
Path divergence
Will S0 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



S1
Points where dH/dx is singular


S0
Using traditional homotopies
Cascade goes like total degree
Working equation-by-equation helps! (esp., regeneration)
Numerical accuracy – a talk for another day
23
t=1
Equation-by-Equation Solving
N equations, n variables
f1(x)=0  Co-dim 1
Intersect
f2(x)=0  Co-dim 1
•Special case:
Co-dim 1,2
f3(x)=0  Co-dim 1
•N=n
•nonsingular solutions only
•results are very promising
Intersect
Co-dim 1,2,3
Theory is in place for μ>1
isolated and for full witness
set generation.
Similar intersections
Final Result
Co-dim 1,2,...,N-1
Co-dim 1,2,...,min(n,N)
Intersect
fN(x)=0  Co-dim 1
24
Working Equation-by-Equation

Basic step
  f1 ( x)  



  
  f ( x)  
  k 1  
V0   Lk ( x)  



  Lk 1 ( x) 
  
 


  LN ( x )  
  f1 ( x)  



  
  f ( x)  
  k 1  
V0   f k ( x)  



  Lk 1 ( x) 
  
 


  LN ( x )  
25
Regeneration: Step 1
  f1 ( x)  



  
  f ( x)  
  k 1  
V0   Lk ( x)  



L
(
x
)
  k 1  
  
 


  LN ( x )  
  f1 ( x)  



  
  f ( x)  
  k 1  
V0   Lk ,1 ( x)  



L
(
x
)
  k 1  
  
 


  LN ( x )  
Union of
sets

move
slice dk
times
  f1 ( x)  



  
  f ( x)  
  k 1  
V0   Lk ,d k ( x) 



L
(
x
)
  k 1  
  
 
 
L
(
x
)

 N
26

f1 ( x)









f k 1 ( x)


V0   Lk ,1 ( x)  Lk ,d k ( x) 



L
(
x
)
k 1





 


LN ( x )


Regeneration: Step 2

f1 ( x)









f k 1 ( x)


V0   Lk ,1 ( x)  Lk ,d k ( x) 



Lk 1 ( x)





 


LN ( x )


Linear
homotopy
27
  f1 ( x)  






  f ( x)  
  k 1  
V0   f k ( x)  



L
(
x
)
k

1


  
 


  LN ( x )  
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


Positive dimensional sets


Basics, diagonal homotopy
Hom4PS-2.0, Hom4PS-2.0para (recent release)


T.Y. Li
Isolated points:


Multihomogeneous & polytope method
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
28
Test Run 1: 6R Robot Inverse Kinematics
Method*
Work
Time
Total-degree 1024 paths
traditional
54 s
Diagonal
eqn-by-eqn
649 paths
23 s
Regeneration 628 paths
eqn-by-eqn
313 slice
moves
9s
*All runs in Bertini
29
Test Run 2: 9-point Four-bar Problem
Method
Work
Time
Polytope
Mixed volume
(Hom4PS-2.0) 87,639 paths
Regeneration
(Bertini)
136,296 paths
66,888 slice
moves
30
11.7 hrs
8.1 hrs
1442 Roberts
cognates
Test Run 3: Lotka-Volterra Systems

Discretized (finite differences) population model

Order n system has 8n sparse bilinear equations
Work Summary
+ mixed
volume
Total degree = 28n
Mixed volume = 24n is exact
31
Lotka-Volterra Systems (cont.)

Time Summary
xx = did not finish
All runs on a single processor
32
Algebraic vs. Geometric Approaches
Pros
Cons
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
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
Algebraic/Symbolic
Geometric/
Numerical Methods
33
Summary

Polynomials arise in applications


Continuation methods for isolated solutions




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,

Most efficient method yet for large, sparse systems


Highly developed in 1980’s, 1990’s
Numerical algebraic geometry


Especially kinematics
without a mixed volume computation
Thanks, Andrew, for 2 decades of fun collaboration!
34