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Charles Wampler General Motors R&D Center (Adjunct, Univ. Notre Dame) Including joint work with

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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

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

S1

t=0

t

t=1

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)
- 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 S0 be the set of path endpoints
- Concerns
- Points where dH/dx is singular
- Turning points
- Path crossings
- Multiple roots
- 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

Basic Total-degree Homotopy

To find all isolated solutions to the polynomial system F: CN CN, i.e.,

form the linear homotopy

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

where

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

S1

t=0

t

t=1

Addressing the concernsConcerns

- Points where dH/dx is singular
- Turning points
- Path crossings
- Multiple roots
- Path divergence
- Will S0 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?

complex homotopy 1977

projective space 1985

1977

Total degree 1978

6R Serial-Link Inverse Kinematics

Formulation

Pieper, 1968 & earlier

32nd degree polynomial

Duffy & Crane, 1980

16 solutions shown by continuation

Tsai & Morgan, 1985

Total degree homotopy with 256 paths

16th degree polynomial

Li & Liang, 1988

Kinematic MilestoneBrief 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)

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)

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

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

Kinematic MilestoneNine-point Four-bar summary

- Symbolic reduction
- 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
- Numerical reduction (Parameter continuation)
- Nondegenerate solutions 4326
- Roberts cognate symmetry 1442
- Synthesis program follows 1442 paths

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)

S1

t=0

t

t=1

Addressing the concerns- Points where dH/dx is singular
- Turning points
- Path crossings
- Multiple roots: Singular endgames 1991-92
- Path divergence
- Will S0 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?

(FoCM 1995 , publ. 1996)

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 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
- 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)

Up to

positive

dimensions

And back again!

Slicing & Witness Sets

- Slicing theorem
- Andegree 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

Linear Traces

Track witness paths as slice translates parallel to itself.

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

Further Reading

World Scientific 2005

Example: Griffis-Duffy Platform

Special Stewart-Gough platform

Degree 28 motion curve (in Study coordinates)

- if legs are equal & plates congruent:
- factors as 6+(6+6+6)+4

Studied by:

Husty & Karger, 2000

Real Points on a Complex Curve

- Go to Griffis-Duffy movie…

S1

t=0

t

t=1

Addressing the concerns- Points where dH/dx is singular
- Turning points
- Path crossings
- Multiple roots
- 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
- Cascade goes like total degree
- Working equation-by-equation helps! (esp., regeneration)
- Numerical accuracy – a talk for another day

Using traditional homotopies

Intersect

Intersect

- Special case:
- N=n
- nonsingular solutions only
- results are very promising

N equations, n variables

f1(x)=0 Co-dim 1

f2(x)=0 Co-dim 1

Co-dim 1,2

f3(x)=0 Co-dim 1

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)

fN(x)=0 Co-dim 1

Working Equation-by-Equation

- Basic step

Regeneration: Step 2

Linear homotopy

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

Test Run 1: 6R Robot Inverse Kinematics

*All runs in Bertini

Test Run 2: 9-point Four-bar Problem

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

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!

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