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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.

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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
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
introduction to continuation

S0

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
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
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
addressing the concerns

S0

S1

t=0

t

t=1

Addressing the concerns

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?
  • 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

kinematic milestone
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 Milestone
brief timeline part 2 after sommese
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)
the trick
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
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

kinematic milestone1
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 Milestone
nine point four bar summary
Nine-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
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)

addressing the concerns1

S0

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?
brief timeline part 3 numerical alg geom

Sommese & Wampler

(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 & 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
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
Further Reading

World Scientific 2005

example griffis duffy platform
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
Real Points on a Complex Curve
  • Go to Griffis-Duffy movie…
addressing the concerns2

S0

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

equation by equation solving

Intersect

Intersect

Intersect

  • Special case:
    • N=n
    • nonsingular solutions only
    • results are very promising
Equation-by-Equation Solving

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

regeneration step 1
Regeneration: Step 1

Union of sets

move slice dk times

regeneration step 2
Regeneration: Step 2

Linear homotopy

software for polynomial continuation
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 3 lotka volterra systems
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

lotka volterra systems cont
Lotka-Volterra Systems (cont.)
  • Time Summary

xx = did not finish

All runs on a single processor

summary
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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