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A numerical Approach toward Approximate Algebraic Computatition. Zhonggang Zeng Northeastern Illinois University, USA. Oct. 18, 2006, Institute of Mathematics and its Applications. What would happen when we try numerical computation on algebraic problems?.

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A numerical Approachtoward

Approximate Algebraic Computatition

Zhonggang Zeng

Northeastern Illinois University, USA

Oct. 18, 2006, Institute of Mathematicsand its Applications


What would happen

when we try numerical computation

on algebraic problems?

A numerical analyst got a surprise 50 years ago on a deceptively simple problem.

1


Start of project: 1948

Completed: 1950

Add time: 1.8 microseconds

Input/output: cards

Memory size: 352 32-digit words

Memory type: delay lines

Technology: 800 vacuum tubes

Floor space: 12 square feet

Project leader: J. H. Wilkinson

Britain’s Pilot Ace

James H. Wilkinson (1919-1986)

2


The Wilkinson polynomial

p(x) = (x-1)(x-2)...(x-20)

= x20 - 210 x19 + 20615 x18 + ...

Wilkinson wrote in 1984:

Speaking for myself I regard it as the most traumatic

experience in my career as a numerical analyst.

3



approximation

A factorable polynomial

irreducible

Factoring a multivariate polynomial:

6


Translation: There are two 1-dimensional solution set:

Solving polynomial systems:

Example: A distorted cyclic four system:

7


approximation

Isolated solutions

1-dimensional solution set

Distorted Cyclic Four system in floating point form:

8


tiny perturbation

in data

(< 0.00000001)

huge error

In solution

( >105 )

9


What could happen in approximate algebraic computation?

  • “traumatic” error

  • dramatic deformation of solution structure

  • complete loss of solutions

  • miserable failure of classical algorithms

  • Polynomial division

  • Euclidean Algorithm

  • Gaussian elimination

  • determinants

  • … …

10


Either

or

So, why bother with approximation in algebra?

  • You may have no choice (e.g. Abel’s Impossibility Theorem)

All subsequent computations become approximate

11


blurred image

blurred image

true image

Application: Image restoration (Pillai & Liang)

So, why bother with approximation solutions?

  • You may have no choice

  • Approximate solutions are better!


blurred image

blurred image

restored image

Application: Image restoration (Pillai & Liang)

true image

Approximate solution is better than exact solution!

13


Approximate solutions

by Bertini

(Courtesy of Bates, Hauenstein,

Sommese, Wampler)

Perturbed Cyclic 4

Exact solutions by Maple: 16 isolated (codim 4) solutions


Perturbed Cyclic 4

Exact solutions by Maple: 16 isolated (codim 4) solutions

Or, by an experimental approximate elimination combined with approximate GCD

Approximate solutions are better than exact ones

, arguably

15


Special case:

Example: JCF computation

Maple takes 2 hours

On a similar 8x8 matrix, Maple and Mathematica run out of memory

So, why bother with approximation solutions?

  • You may have no choice

  • Approximate solutions are better

  • Approximate solutions (usually) cost less

  • You may have no choice

  • Approximate solutions are better

  • Approximate solutions (usually) cost less

16


Pioneer works in numerical algebraic computation (incomplete list)

  • Homotopy method for solving polynomial systems

  • (Li, Sommese, Wampler, Verschelde, …)

  • Numerical Algebraic Geometry

  • (Sommese, Wampler, Verschelde, …)

  • Numerical Polynomial Algerba

  • (Stetter)

17


exact computation

approximate solution

using 8-digits precision

forward error

backward error

axact solution

-

-

-

=

2

4

2

(

x

1

)

(

10

)

0

What is an “approximate solution”?

To solve

with 8 digits precision:

backward error: 0.00000001 -- method is good

forward error: 0.0001 -- problem is bad

18


From problem

From numerical method

The condition number

[Forward error] < [Condition number] [Backward error]

A large condition number

<=> The problem is sensitive or, ill-conditioned

An infinite condition number <=> The problem is ill-posed

19


Wilkinson’s Turing Award contribution:

Backward error analysis

  • A numerical algorithm solves a “nearby” problem

  • A “good” algorithm may still get a “bad” answer, if the problem is ill-conditioned (bad)

20


Ill-posed problems are common in applications

  • image restoration - deconvolution

  • IVP for stiction damped oscillator - inverse heat conduction

  • some optimal control problems - electromagnetic inverse scatering

  • air-sea heat fluxes estimation - the Cauchy prob. for Laplace eq.

  • … …

  • A well-posed problem: (Hadamard, 1923)

  • the solution satisfies

    • existence

    • uniqueness

    • continuity w.r.t data

An ill-posed problem is infinitely sensitive to perturbation

tiny perturbation  huge error

21


- Multiple roots

- Polynomial GCD

- Factorization of multivariate polynomials

- The Jordan Canonical Form

- Multiplicity structure/zeros of polynomial systems

- Matrix rank

Ill-posed problems are common

in algebraic computing

22


Ill-posed problems

are infinitely sensitive

to data perturbation

A numerical algorithm

seeks the exact solution

of a nearby problem

If the answer is highly sensitive to perturbations, you have probably asked the wrong question.

Maxims about numerical mathematics, computers,

science and life, L. N. Trefethen. SIAM News

Does that mean:

(Most) algebraic problems are wrong problems?

Conclusion: Numerical computation is incompatible

with ill-posed problems.

Solution: Formulate the right problem.

23


Solution

P

P

Data

P

Challenge in solving ill-posed problems:

Can we recover the lost solution

when the problem is inexact?

P:Data Solution

24


William Kahan:

This is a misconception

Are ill-posed problems really sensitive to perturbations?

Kahan’s discovery in 1972:

Ill-posed problems are sensitive to arbitrary perturbation,

but insensitive to structure preserving perturbation.

25


W. Kahan’s observation (1972)

  • Problems with certain solution structure form a “pejorative manifold”

Plot of pejorative manifolds of degree 3 polynomials with multiple roots

Why are ill-posed problems infinitely sensitive?

  • The solution structure is lost when the problem leaves the manifold due to an arbitrary perturbation

  • The problem may not be sensitive at all if the problem stays on the manifold, unless it is near another pejorative manifold

26


Geometry of ill-posed algebraic problems

Similar manifold stratification exists for problems like

factorization, JCF, multiple roots …

27


Manifolds of 4x4 matrices defined by Jordan structures

(Edelman, Elmroth and Kagstrom 1997)

e.g. {2,1} {1} is the structure of 2 eigenvalues in 3 Jordan blocks of sizes 2, 1 and 1

28


?

B

A

?

perturbation

Problem A

Problem B

Illustration of pejorative manifolds

The “nearest” manifold may not be the answer

The right manifold is of highest codimension within a certain distance

29


Finding approximate solution is (likely) a well-posed problem

Approximate solution is a generalization of exact solution.

A “three-strikes” principle for formulating

an “approximate solution” to an ill-posed problem:

  • Backward nearness: The approximate solution is the exact solution of a nearby problem

  • Maximum codimension: The approximate solution is the exact solution of a problem on the nearby pejorative manifold of the highest codimension.

  • Minimum distance: The approximate solution is the exact solution of the nearest problem on the nearby pejorative manifold of the highest codimension.

30



Formulation of the problemapproximate rank /kernel:

Backward nearness:app-rank of A is

the exact rank of certain matrix B within q.

The approximate rank of A within q :

Maximum codimension:That matrix B

is on the pejorative manifold P possessing the

highest co-dimension and intersecting the

q-neighborhoodof A.

The approximate kernel of A within q :

with

Minimum distance:That B is the nearest

matrix on the pejorative manifold P.

  • An exact rank is the app-rank within sufficiently small q.

  • App-rank is continuous (or well-posed)

31


= 4 problem

nullity = 2

= 4

nullityq = 2

After reformulating the rank:

Rank

Rankq

Ill-posedness is removed successfully.

kernel

App-rank/kernel can be computed by SVD and other rank-revealing algorithms

(e.g. Li-Zeng, SIMAX, 2005)

basis

+ eE = 6

nullity = 0

+ eE = 4

nullityq = 2

32


The AGCD within probleme :

Formulation of the approximate GCD

  • Finding AGCD is well-posed if qk(f,g) is sufficiently small

  • EGCD is an special case of AGCD for sufficiently small e(Z. Zeng, Approximate GCD of inexact polynomials, part I&II)

33


Similar formulation strikes out ill-posedness in problems such as

  • Approximate rank/kernel(Li,Zeng 2005, Lee, Li, Zeng 2006)

  • Approximate multiple roots/factorization(Zeng 2005)

  • Approximate GCD(Zeng-Dayton 2004, Gao-Kaltofen-May-Yang-Zhi 2004)

  • Approximate Jordan Canonical Form(Zeng-Li 2006)

  • Approximate irreducible factorization(Sommesse-Wampler-Verschelde 2003, Gao et al 2003, 2004, in progress)

  • Approximate dual basis and multiplicity structure(Dayton-Zeng 05, Bates-Peterson-Sommese ’06)

  • Approximate elimination ideal(in progress)

34


after formulating the approximate solution to problem such asP within e

Stage I: Find the pejorative manifold

P of the highest dimension s.t.

P

P

S

Stage II: Find/solve problem Q such that

Q

The two-staged algorithm

Exact solution of Q is the approximate solution of P within e

which approximates the solution of S where P is perturbed from

35


Stage II: solve the (overdetermined) quadratic system such as

for a least squares solution (u,v,w) by Gauss-Newton iteration

Case study: Univariate approximate GCD:

Stage I: Find the pejorative manifold

(key theorem: The Jacobian of F(u,v,w) is injective.)

36


Max-codimension such as

k := k-1

k := k-1

Is AGCD of degree k

possible?

no

no

probably

Refine with

G-N Iteration

Successful?

yes

Min-distance

nearness

Output GCD

Start: k = n

Univariate AGCD algorithm

37


Stage II: solve the (overdetermined) quadratic system such as

for a least squares solution (u,v,w) by Gauss-Newton iteration

(key theorem: The Jacobian of F(u,v,w) is injective.)

Case study: Multivariate approximate GCD:

Stage I: Find the max-codimension pejorative manifold by

applying univariate AGCD algorithm on each variable xj

38


Stage II: solve the (overdetermined) polynomial system such asF(z1 ,…,zk )=f

(in the form of coefficient vectors)

for a least squares solution (z1 ,…,zk) by Gauss-Newton iteration

(key theorem: The Jacobian is injective.)

Case study: univariate factorization:

Stage I: Find the max-codimension pejorative manifold by

applying univariate AGCD algorithm on (f, f’ )

39


A such as ~ J

l 1

l 1

l

J =

Segre characteristic = [3,1]

B

l

A

codim = -1 + 3 + 3(1) = 5

l

l

s13

s23

s14

s24

s34

Wyre characteristic = [2,1,1]

lI+S=

l

l

3

1

2

1

1

Ferrer’s diagram

Case study: Finding the nearest matrix with a Jordan structure

P

Equations determining the manifold

40


A such as ~ J

B

A

For B not on the manifold, we can still solve

for a least squares solution :

When

is minimized, so is

of

is injective.

The crucial requirement: The Jacobian

(Zeng & Li, 2006)

Case study: Finding the nearest matrix with a Jordan structure

Equations determining the manifold

P

41


Project to tangent plane such as

u1 = G(z0)+J(z0)(z1- z0)

~

tangent plane P0 :

u = G(z0)+J(z0)(z- z0)

Least squares solution

u*=G(z*)

new iterate

u1=G(z1)

initial iterate

u0=G(z0)

Solving G(z) = a

a

Pejorative manifold

u = G( z )

Solve

G( z ) = a

for nonlinear least squares solutionz=z*

Solve

G(z0)+J(z0)(z - z0 ) = a

for linear least squares solutionz =z1

G(z0)+J(z0)(z - z0 ) = a

J(z0)(z - z0 ) = - [G(z0) - a ]

z1 = z0 - [J(z0)+][G(z0) - a]

42


condition number such as

(sensitivity measure)

Stage I: Find the nearby max-codim manifold

Stage II: Find/solve the nearest problem on the manifold via solving an overdetermined system G(z)=a for a least squares solutionz* s.t . ||G(z*)-a||=minz ||G(z)-a||

by the Gauss-Newton iteration

Key requirement: Jacobian J(z*) of G(z) at z* is injective (i.e. the pseudo-inverse exists)

43


univariate/multivariate GCD such asmatrix rank/kernel

dual basis for a polynomial ideal

univariate factorization

irreducible factorizationelimination ideal

… …

(to be continued in the workshop

next week)

Summary:

  • An (ill-posed) algebraic problem can be formulated using the three-strikes principle (backward nearness, maximum-codimension, and minimum distance) to remove the ill-posedness

  • The re-formulated problem can be solved by numerical computation in two stages (finding the manifold, solving least squares)

  • The combined numerical approach leads to Matlab/Maple toolbox ApaTools for approximate polynomial algebra. The toolbox consists of

44


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