Tutorial: Numerical Algebraic Geometry · Homotopy continuation Singular isolated solutions Positive dimension Certiﬁed homotopy tracking Tutorial: Numerical Algebraic Geometry

Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking

Tutorial: Numerical Algebraic GeometryBack to classical algebraic geometry...

with more computational power andhybrid symbolic-numerical algorithms

Anton Leykin

Georgia Tech

Waterloo, November 2011


Outline

Homotopy continuationpredictor-corrector numerical methods, Newton’s method, (global)homotopy continuation scenarios

Singular isolated solutionsregularization of singular solutions, deflation, dual spaces/inversesystems

Positive dimensionwitness sets, numerical irreducible decomposition, numericalprimary decomposition

Certified homotopy trackingnumerical zeros, α-theory of Smale, heuristic vs. rigorouspath-tracking


Computational algebraic geometry

What is the game?• Level 0: Given a system of polynomial equations in K[x1, ..., xn]

with finitely many solutions,

SOLVE.

( K could be Q, Z/pZ, R, C, ... )• Level 1+: Describe positive-dimensional solutions (curves,

surfaces, ...)Classical methods “generalize” linear algebra:• Gröbner basis: a generalization of Gaussian reduction;• Resultant: a generalization of determinant.

These methods are symbolic.


Linear Algebra Numerical Linear Algebray yAlgebraic Geometry Numerical Algebraic Geometry


Applications

Robotics: Stewart-Gough platforms.

Griffis-Duffy platform: the solutioncontains a curve of degree 28.

`1

`2

`3`4

s1

s2

p��3

Enumerative algebraic geometry:solutions of Schubert problems.

... control theory, optimization, computer vision, math biology, realalgebraic geometry, algebraic curves ...


Polynomial homotopy continuation

• Target system: n equations in n variables,

F (x) = (f1(x), . . . , fn(x)) = 0,

where fi ∈ R = C[x] = C[x1, ..., xn] for i = 1, ..., n.• Start system: n equations in n variables:

G(x) = (g1(x), . . . , gn(x)) = 0,

such that it is easy to solve.• Homotopy: for γ ∈ C \ {0} consider

H(x, t) = (1− t)G(x) + γtF (x), t ∈ [0, 1].


Exampletarget start

f1 = x41x2 + 5x21x32 + x31 − 4 g1 = x51 − 1

f2 = x21 − x1x2 + x2 − 8 g2 = x22 − 1

Start solutions→ target solutions:

H(x, t) = 0 impliesdx

dt= −

(∂H

∂x

)−1∂H

∂t.


Exampletarget start

f = −x2 + 2 g = x2 − 1

The solution of the homotopy equation

H(x, t) = (1− t)g(x) + tf(x) = (1− 2t)x2 − 1 + 3t = 0

is singular for t = 1/3.


Randomization

• Note: the complement of a complex algebraic variety isconnected.

space of polynomials

f

g

• For all but finite number of γ ∈ C the homotopy

H(x, t) = (1− t)G(x) + γtF (x).

is regular for 0 ≤ t < 1.


Global picture

Optimal homotopy:

• the continuation paths are regular;• the homotopy establishes a bijection

between the start and target solutions.

Possible singular scenarios:

non-generic diverging paths multiple solutions


Numerical algebraic geometry

• Sommese, Verschelde, and Wampler, Introduction to NumericalAG (2005)

• Sommese and Wampler, The numerical solution of systems ofpolynomials (2005)

Software:• PHCpack (Verschelde);• HOM4PS (group of T.Y.Li);• Bertini (group of Sommese);• NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.).

and more, e.g.: Maple’s ROOTFINDING[HOMOTOPY].


Possible improvements• Parallel computation:

Paths are mutually independent⇒ linear speedups.

• Minimize the number of diverging paths:

• Total degree: Number of start solutions =product of degrees of equations (Bézoutbound).

• Polyhedral homotopies: Number of startsolutions = mixed volume of sparse system(BKK bound).

• Optimal homotopies:

• Cheater’s homotopy;

• Special homotopies: e.g., Pieri homotopy.


Multiple solutions

In general, with probability 1, thepicture looks like this:

Singular end games [Morgan, Sommese, Wampler (1991)]:• power-series method;• Cauchy integral method;• trace method.

Deflation:• regularizes an isolated singular solution;• restores quadratic convergence of the Newton’s method.

How to describe a singularity?


Cauchy integral endgame

• An implication of Cauchy residue theorem:Let y : U → C holomorphic on a simply connected U ⊂ C,a ∈ C, and C ⊃ C ' S1 be a contour winding I(C, a) timesaround a.Then

y(a) =1

2πiI(C, a)

∮C

y(z)

z − adt,

• H(x, t) = 0 defines (a possibly multivalued function) x = x(t) ina neighborhood of t = 1.

• Idea: as the homotopy tracker approaches a singular x∗ = x(1)use Cauchy integral to compute x∗ staying away from x∗.


Winding numbers

(1 2 3 4 5)(6 7 8)(9 10)

|1− t| = ε


Cauchy integral endgame

1. Pick a point on x = x(t), a solution to H(x, t) = 0 for t = t ∈ R;let ε = 1− t.

2. Track the path

C = {x(1− εeiθI) | θ ∈ [0, 2π]},

where I > 0 is such that

x = x(1− εeiθI) ⇒ θ ∈ {0, 2π}.

3. Let y(z) = x(1− zI), then y(z) is holonomic for |z| < ε (if ε� 1).4. Find numerically the integral

x(1) = y(0) =1

2π

∫|z|=ε

y(z)

zdz =

1

2π

∫[0,2π]

x(1− εeiθI) dθ.

(Note: one may use samples made when tracking the path C.)


Newton’s method: x(n+1) = x(n) −f(x(n))

f ′(x(n))

Example 1: f(x) = x(x− 1)3, x(0) = 0.1

x(1) = −0.05000000000000000000000000000000000000000000000000x(2) = −0.00625000000000000000000000000000000000000000000000x(3) = −0.00011432926829268292682926829268292682926829268293x(4) = −0.00000003919561993882928315798471103711494222972094x(5) = −0.00000000000000460888914457438597268761599543603706x(6) = −0.00000000000000000000000000006372557744092567103642

Example 2: f(x) = x2(x− 1)3, x(0) = 0.1

x(1) = 0.04000000000000000000000000000000000000000000000000x(2) = 0.01866666666666666666666666666666666666666666666667x(3) = 0.00905920745920745920745920745920745920745920745921x(4) = 0.00446662546689373374865785737653016156369492056043x(5) = 0.00221818070337351048684295922675846246257988477728x(6) = 0.00110537952927547542499858913840929687677679537995


Deflation method

Let f(x) = (f1(x), ..., fN (x)), N ≥ n, fi(x) ∈ C[x] = C[x1, ..., xn].

Let A(x) =

(∂fi∂xj

)∈ CN×n be the Jacobian matrix.

Given: an approximation x(0) of an exact isolated solution x∗, whichis singular, i.e., corankA(x∗) = n− rankA(x∗) > 0.

Newton’s method in homotopy continuationloses quadratic convergence around x∗. Isthere a way to restore the convergence?

• Want: a symbolic procedure that “makes” x∗ regular.• Rules:

• New variables are allowed.• Assume that the numerical rank of A(x(0)) equals A(x∗).


Deflation step: create an augmented system in C[x,a]1. Introduce n new variables a;2. Add equations coming from A(x)a = 0;

Example. Let f1 = x31 + x1x22, f2 = x1x

22 + x32, f3 = x21x2 + x1x

22 and

x∗ = 0. ∂1 ∂2

f1 3x21 + x22 2x1x2

f2 x22 2x1x2 + 3x22

f3 2x1x2 + x22 x21 + 2x1x2

[a1

a2

]= 0.

3. Compute the rank r of A(x∗); (r = 0 for our example)

4. Add n− r random linear equations.

5. Find the solution (x∗,a∗) of the augmented system; (8 equations)

6. Repeat if (x∗,a∗) is singular. (2 steps for the example)

Theorem (L., Verschelde, Zhao)The multiplicity of (x∗,a∗) in the augmented system is smaller thanthat of x∗ in the original system.


Multiplicity

Staircases for I = 〈f1, f2, f3〉, wheref1 = x31 + x1x

22, f2 = x1x

22 + x32, f3 = x21x2 + x1x

22.

6

-hh

hh

hh

x

xx

x1x22 + x32

x21x2 + x1x22

x31 + x1x22

xhx42

ω = (−1,−2)�

6

-hh

hh

hh

xx

xx21x2 + x1x

22

x1x22 + x31

x32 + x1x22

xh x41

ω = (−2,−1)�

Multiplicity is the number of integer points under the staircase.


Dual space (local inverse system)

• For x ∈ Cn, let ∆βx : R→ C be a linear functional,

∆βx(f) = (∂β · f)(x) =

∂|β|f

∂β(x), f ∈ R.

• For an ideal I, the dual space Dx[I] is the subspace ofSpanC{∆β

x} of the functionals that annihilate I.• Filter by order:

D(0)x [I] ⊂ D(1)

x [I] ⊂ D(2)x [I] ⊂ . . .

where D(d)x [I] is the set of functionals of order at most d.


Macaulay array• For I = 〈f1, . . . , fm〉, the deflation matrix A(d)

I (x) is a part of theMacaulay array, the infinite matrix with entries(

∂

∂xβ(xαfi)

)((i,α),β)

where |α| < d and |β| ≤ d.• For example, for I = 〈f1, f2〉 ⊂ C[x, y],

A(2)I =

id ∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ ∗ ∂

∂x2 (xf1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗ ∗


Dual spaces and deflation• For x ∈ V (I), get D(d)

x [I] by computing kerA(d)I (x).

• For the running example,

D0[I] = Span{ ∆(3,0) −∆(2,1) −∆(1,2) + ∆(0,3),

∆(2,0), ∆(1,1), ∆(0,2),

∆(1,0), ∆(0,1), ∆(0,0) }.

The leading terms with respect toω = (2, 1) correspond to themonomials under the staircase forthe standard basis forω = (−2,−1).

6

-hh

hh

hh

xx

xx21x2 + x1x

22

x1x22 + x31

x32 + x1x22

xh x41

ω = (−2,−1)�


Deflation continued...

Idea of proof of termination for deflation:• Deflation "deflates the staircase";• the multiplicity of becomes 1 after a finite number of steps.

Related work:• Dual spaces: Macaulay (1916), ..., Stetter (1993), Mourrain

(1997), Dayton and Zeng (2005), Krone (2011).• Deflation: Lyapunov (1900?) , ..., Ojika et al (1987), Lecerf

(2002), L. et al (2006), ..., Lihong Zhi et al (2010).


Higher-dimensional solution sets• Let I = (f1, . . . , fN ) be an ideal of C[x1, . . . , xn].• Goal: Understand the variety

X = V(I) = {x ∈ Cn | ∀f ∈ I, f(x) = 0}.

• A witness set for an equidimensional component Y of X:• a generic “slicing” plane L with dimL = codimY• witness points wY,L = Y ∩ L• (generators of I)


Numerical irreducible decomposition• Homotopy mapping wY,L → wY,L′ :

HL,L′,γ(x, t) =

(f(x)

(1− t)L(x) + γtL′(x)

), t ∈ [0, 1].

L

L’

• Monodromy action: a permutation on wY,Lis produced by homotopy HL,L′,γ followedby HL′,L,γ′ for random γ, γ′ ∈ C.

• Irreducible decomposition: a partition of thewitness set wY,L stable under this action.

• Linear trace test: the average of thepoints in a witness set of an irreduciblecomponent behaves linearly.


Example with an embedded component

ExampleI = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1).

y + 1 = 0

x = 0

��u(0, 0)

��solution set of

{x · x(y + 1) = 0;y · x(y + 1) = 0.

Numerical irreducible decomposition*sees two 1-dimensional components ...

... but does not discover the embedded point.

*NID reference: Sommese, Verschelde, Wampler “Numerical decompositionof the solution sets...” (2001)




y + 1 = 0

x = 0��

u(0, 0)

��solution set of

{x · x(y + 1) = 0;y · x(y + 1) = 0.







y + 1 = 0

x = 0��u(0, 0) ��

solution set of{x · x(y + 1) = 0;y · x(y + 1) = 0.





Deflation matrix, system, and ideal

Example (Deflation ideal of order d = 2)original system f(x, y) → deflation matrix A(2)

I (x, y)

→→ deflation system D(2)f(x, y,a) →→ deflation ideal I(2)

∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ 6x(y + 1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗

axayaxx

axyayy

=: D(2)f(x, y,a)

6

∂2(xf1)∂x2

6

C[x, y,a]6

Deflation idealI(2) = 〈f , D(2)f〉 ⊂ C[x, y,a]




I (x, y) →→ deflation system D(2)f(x, y,a)

→→ deflation ideal I(2)


ax

ay

axx

axy

ayy

=: D(2)f(x, y,a)6

∂2(xf1)∂x2

6

C[x, y,a]6





I (x, y) →→ deflation system D(2)f(x, y,a) →→ deflation ideal I(2)


ax

ay

axx

axy

ayy

=: D(2)f(x, y,a)6

∂2(xf1)∂x2

6

C[x, y,a]6



Deflated variety

• For the given variety X = V (I) = {x | f(x) = 0 for all f ∈ I}define deflated variety of order d:X(d) = V (I(d)) ⊂ CB(n,d), where B(n, d) = n− 1 +

(n+d+1

d

).

• Projection: πd : CB(n,d) → Cn, πd(x,a) 7→ x; πdX(d) = X.

• A component Y ⊂ X is called visible at order d if Y = πdZ for anisolated component Z ⊂ X(d).

s(0, 0) Theorem (L.)Every component is visible at some order d.

ExampleFor the running example d = 1 is sufficient.


Example: I = (f1, f2), f1 = x2(y + 1), f2 = xy(y + 1)

• Isolated components of X = V (I):

V (y + 1), V (x).

• Additional equations A(1)I a = 0:[

2x(y + 1) x2

y(y + 1) x(2y + 1)

] [axay

]= 0.

• Isolated components of X(1) = V (I(1)):

V (y + 1, ay), V (x, y), V (x, ax), V (x, y + 1)

• The first three project onto the components of X, the last one (a“pseudo-component”) projects onto a singular point.


Witness sets

For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal,a

generalized

witness set consists of• a generic “slicing” plane L with dimL = codimY

(d)

• witness points w = Y

(d)

∩ L

and their projections via πd

• generators of I

(d)

Definition: Y (d) is an isolated irreducible component of the deflated varietyX(d) mapping onto Y under projection πd.


Witness sets

For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal,a generalized witness set consists of• a generic “slicing” plane L with dimL = codimY (d)

• witness points w = Y (d) ∩ L and their projections via πd• generators of I(d)

Definition: Y (d) is an isolated irreducible component of the deflated varietyX(d) mapping onto Y under projection πd.


The algorithm and NPD concept

• Idea of the algorithm: Use numerical irreducible decompositionof a deflated variety to find (generalized) witness setsrepresenting components.

• Definition: Numerical primary decomposition is a collection ofsuch witness sets, one per component.

• Deficiencies:• There is an apriori bound on the order d of needed to make all

components visible, however it is not practical.• Pseudocomponents (projections of components of X(d) that are

not components of X) are hard to eliminate.


Local global?

Local knowledge:• Dx[I] describes the local ring (R/I)x = Rx/Ix.• A generic point on a component Y is a smooth point of Y that

does not belong to any component not containing Y properly.• A generic point x ∈ Y together with the algorithm for computingD

(d)x [I] describe Y .

Global knowledge:• Given a NPD (as a collection of witness sets), mark one generic

point on each component.• The set of marked points describe the ideal I.


Path switching

Applications in mathematics wherepath certification is desirable:• Numerical irreducible

decomposition algorithms• Galois group computation based

on monodromy• Problems where the root count is

impossible by other methods


Newton’s method and approximate zerosGiven f ∈ C[x], consider the Newton operator associated to f ,

N(f)(x) = x−Df(x)−1f(x),

where Df(x) is the n× n derivative (Jacobian) matrix of f at x ∈ Cn.

• Definition: x ∈ Cn is an approximate zero of f with associatedzero η ∈ Cn if

‖N(f)l(x)− η‖ ≤ ‖x− η‖22l−1

, l ≥ 0.

• γ-theorem(Smale): Let x ∈ Cn, η ∈ f−1(0), and

‖x− η‖ ≤ 3−√

7

2γ(f, x), where γ(f, x) = sup

k≥2

∥∥∥∥Df(x)−1Dkf(x)

k!

∥∥∥∥1

k−1

.

Then x is an approximate zero of f associated to η.• Hauenstein, Sottile: alphaCertify, software for certification of

regular solutions.


Smale’s α theory

• α-theorem: Let

β(f, x) = ‖x−N(f)(x)‖ = ‖Df(x)−1f(x)‖ .

Then α(f, x) = β(f, x)γ(f, x) < 0.15767 certifies that x is anapproximate zero of f .

• "robust" theorem: Let x ∈ Cn with α(f, x) < 0.03. If

‖x− y‖ < 1

20γ(f, x),

then y is an approximate zero associated to the same zero as x.


Newton’s operator attraction basins


Certified regions


Robust regions


Linear and segment homotopy• Let H(d) be the space of systems of homogeneous polynomials

of fixed degrees (d) = (d1, . . . , dn) (with the Bombieri-Weylnorm).

• Consider f, g ∈ S = {f ∈ H(d) : ‖f‖ = 1} ⊂ H(d).• Using α-theory we design certified homotopy tracking (CHT)

algorithm that tracks a linear homotopy on S (assuming BSSmodel of computation).

• The “robust” α-theory leads to the robust CHT algorithm (Beltrán,L.):

g

f Take input f, g with coefficients in Q[i].Use the segment homotopy:

t→ ht = (1− t)g + tf, t = [0, 1].

All computations use exact linearalgebra over Q[i].


Future

• General methods• (Numerical) local ring structure• (Numerical) primary decomposition• Real solutions, real homotopy continuation

• Certification• Generalizations to higher order methods• Certification of singular isolated solutions• Certification of (irreducible/primary) decomposition

• Upcoming events• SI(AG)2: SIAM activity group in algebraic geometry• IMA PI summer program for graduate students on

Algebraic Geometry for Applications

June 18th – July 6th at Georgia Tech

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Tutorial: Numerical Algebraic Geometry · Homotopy continuation Singular isolated solutions Positive dimension Certiﬁed homotopy tracking Tutorial: Numerical Algebraic Geometry