Wampler: Numerical Algebraic Geometry

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Mechanism and Robot Kinematics, Part II:
Numerical Algebraic Geometry

C= harles Wampler

General Motors R&D Center

Including joint wo= rk with

Andrew Sommese, University of Notre Dame = ;

Jan Verschelde, Univ. Illinois Chicago <= /span>

Alexander Morgan, <= span style=3D'mso-tab-count:1;width:1.72%'> GM R&D

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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry

2&#= 13;

Outline

nZero-dimensional solution sets <= /div>

n= Numerical solution by polynomial continuation =

nRoot counts and homotopies

nParameter homotopies

nPositive-dimensional solution sets

n= Basic constructs

nWitness sets

nNumerical irreducible decomposition

n= Basic operations

nIntersection of algebraic sets <= /div>

nDeflation of nonreduced sets

n= Higher-level operations

nEquation-by-equation intersections

nFiber products

nExtracting real points from a complex set

nApplications

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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry

Numerical Algebraic Geometry

nPurpose

nNumerically represent & manipulate algebraic sets

nApproach

nNumerical continuation operating on witness sets

Basic Operations

<= span style=3D'font-size:86%'>n Witness generate

<= span style=3D'font-size:86%'>n Witness decomposition <= /span>

<= span style=3D'font-size:86%'>n Membership tests

<= span style=3D'font-size:86%'>n Intersection

<= span style=3D'font-size:86%'>n Deflation

Basic Constructs

n Witness sets

<= span style=3D'font-size:86%'>n Irreducible decomposition

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Why study polynomial systems?

nApplication areas

nEconomics & finance

nChemical equilibrium

nComputer-aided Geometric Design (CAGD)

nControl theory

nKinematics

nConstrained mechanical motion

nLinkages for motion constraint & transformation

nSuspensions, engines, swing panels, etc.

nComputer-controlled motion devices

nRobots, human-assist devices, etc.

------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0130.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry

Zero-Dimensional Sets

n Sol= ving by polynomial continuation

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6&#= 13;

What is Continuation?

nFor some class of parameterized problems:

n<= span style=3D'font-size:86%'>H(x;p) =3D 0

nWant solutions at pfinal

nWe have solutions xstart,i for parameters pstart

n<= span style=3D'font-size:86%'>H(xstart,i;pst= art) =3D 0

nForm a parameter path

n<= span style=3D'font-size:86%'>p(t) =3D t pstart + (1-t) pfi= nal

nThis defines a homotopy

n<= span style=3D'font-size:86%'>H(x;p(t)) =3D 0

nNumerically follow solution path

n<= span style=3D'font-size:86%'>from t=3D1 to t=3D0

------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry

Example: Ellipse & Hyperbola

nWish to solve F(x,y)

<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>na1x2<= /span>+b1xy+c1y2+d1x+e1y+f1 =3D 0

na2x2<= /span>+b2xy+c2y2+d2x+e2y+f2 =3D 0

nKnow how to solve G(x,y)

<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>na1x2<= /span>+f1 =3D 0

<= span class=3DHB1B style=3D'position:absolute;left:-5.99%;top:.74em'>nc2y2<= /span>+f2 =3D 0

nHomotopy H(x,y,t)=3D0

nt(b1xy+c1y2<= /span>+d1x+e1y)+a1x2+f1 =3D 0

nt(a2x2<= /span>+b2xy+d2x+e2y)+c2y2+f2 =3D 0

nFollow 4 solution paths

nfrom t=3D0 to t=3D1.

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Solution paths

nImplicitly defined by H(x(t);p(t))= =3D 0

Nongeneric

´=

Parameter space

pstar= t

pfina= l

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An Ill-Conceived Homotopy

nQ: How do we make sure this doesn’t happen?&= #13;

nA: Use complex space

n exceptions are

complex co-dimension 1 =3D real codimension 2 =

nGeneral 1-dim parameter path miss exceptions with probability 1

Parameters for which H(x,p) has fewer = solutions

´=

Parameter s= pace (real)

pstar= t

pfina= l

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Polynomial Structures

(A) Start system solved with <= span style=3D'position:absolute;top:80.48%;left:14.59%;width:73.72%;height:5.92= %'>linear algebra

(B) Start system solved via convex hulls, polytope theory

Landmarks

n all isolated solutions =

nGarcia & Zangwill, ‘77

nDrexler, ‘77

n total degree

nChow, Mallet-Paret & Yorke, ‘78

n projective space

nWright, ‘85

nMorgan, ‘86; book, ‘87

= Landmarks

nmulti= -homogeneous

nMorgan & Sommese, ‘87

nparam= eterized systems

nLi, Sauer & Yorke, ‘88

nMorgan & Sommese, ‘89

Landmarks

nPolyt= opes (BKK)

nVerschelde, Verlinden & Cools, ‘94

nHuber & Sturmfels, ’95

nGao & Li, ’03

nPolyn= omial products

n Morgan,Sommese & Wampler,’95

nSet structures

nVerschelde & Cools, ‘94

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Parameter Continuation

initial parameter space

target parameter space

nStart system easy in initial parameter space <= /span>

nRoot count may be much lower in target parameter space

nInitial run is 1-time investment for cheaper targe= t runs

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Positive-Dimensional Sets

nBasic Constructs

nWitness Sets

nIrreducible Decomposition

------=_NextPart_01C6DF35.194A2C70 Content-Location: file:///C:/1E95224D/Wampler_NumAlgGeom_files/slide0137.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Wampler: Numerical Algebraic Geometry

Slicing & the Witness Cascade

nFundamental theorem of algebra

nA degree N <= /i>square-free polynomial p(x,y)=3D0 hits a gener= al horizontal line y=3Dc in N isolated points

nSlicing theorem

nAn degree N <= /i>reduced algebraic set of dimension m in n = variables hits a general (n-m)-dimensional linear space in N = isolated points

nWitness generation algorithm
nWitness points at every dimension

n= Relies on traditional homotopy properties to get all isolated solutions at each dimension

= <= /span>

= <= /span>

Sommese & Wampler, ’95

Sommese & Verschelde, ’00

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Witness Set

nSuppose AÎCn<= /span> is pure-m-dimensional algebraic set that is a solution of F(x)=3D0

nWitness set for A consists of:

nF(x) ð the system

na system of polynom= ials (straight-line function)

nL(x) ð generic slicing plane

na linear space of dimension (n-m)

nW =3D {x1,..., xd<= /span>} ð “Witness points”

nsolution points of {F(x),L(x)}=3D0

nd =3D degree of A

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Decomposed Witness Set

nPure-dimensional A=3D{A1,..., Ak}

nwhere each Ai is irreducible

nDecomposed witness set for A

nSystem, F(x)

nSlice, L(x)

nDecomposed witness point set

nW=3D{W1,..., Wk},
nwhere Wi=3D{x1,..., xdi<= /span>} is witness point set for Ai </