From Isolated Points to Positive Dimensions & Back Again:
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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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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


42 distinct co-authors


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


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

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


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


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)

  • 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 Milestone


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 solutions4326

    • Roberts cognate symmetry1442

  • 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)


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?


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 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


Membership Test


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…


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


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


Working Equation-by-Equation

  • Basic step


Regeneration: Step 1

Union of sets

move slice dk times


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


Lotka-Volterra Systems (cont.)

  • Time Summary

xx = did not finish

All runs on a single processor


Algebraic vs. Geometric Approaches


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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