a numerical approach toward approximate algebraic computatition zhonggang zeng northeastern illinois...
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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?
A numerical analyst got a surprise 50 years ago on a deceptively simple problem.
1
James H. Wilkinson (1919-1986)
Britain’s Pilot Ace
Start of project: 1948Completed: 1950Add time: 1.8 microsecondsInput/output: cardsMemory size: 352 32-digit wordsMemory type: delay linesTechnology: 800 vacuum tubesFloor space: 12 square feetProject leader: J. H. Wilkinson
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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.
57521 )379.18()98.11()7222.5()3145.2()99651.0( xxxxx
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Solving polynomial systems:
Example: A distorted cyclic four system:
Translation: There are two 1-dimensional solution set:
33,,6
4321tztztztz
7
Distorted Cyclic Four system in floating point form:
1-dimensional solution set Isolated solutionsapproximation
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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• … …
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So, why bother with approximation in algebra?
1. You may have no choice (e.g. Abel’s Impossibility Theorem)
All subsequent computations become approximate
Either
or
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So, why bother with approximation solutions?
1. You may have no choice
2. Approximate solutions are better!
1)),(),,(( yxgyxfGCD
true image
Application: Image restoration (Pillai & Liang)
blurred image blurred image
),(),(),( yxyxyxp
),( yxf
),(),(),( yxyxyxp
),( yxg
),( yxp
Application: Image restoration (Pillai & Liang)
),(),(),( yxyxyxp
),( yxf
),(),(),( yxyxyxp
),( yxg
),(~)),(),,(( yxpyxgyxfAGCD
true image
blurred image blurred image
restored image
),( yxp
Approximate solution is better than exact solution! 13
Perturbed Cyclic 4
Exact solutions by Maple: 16 isolated (codim 4) solutions
Approximate solutions
by Bertini
(Courtesy of Bates, Hauenstein, Sommese, Wampler)
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 , arguably15
So, why bother with approximation solutions?
1. You may have no choice
2. Approximate solutions are better
3. Approximate solutions (usually) cost lessExample: JCF computation
t
s
s
r
r
r
1
1
1
,2r
Special case:
,3s 5t
Maple takes 2 hours
On a similar 8x8 matrix, Maple and Mathematica run out of memory
1. You may have no choice
2. Approximate solutions are better
3. Approximate solutions (usually) cost less
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Pioneer works in numerical algebraic computation (incomplete list)
• Homotopy method for solving polynomial systems (Li, Sommese, Wampler, Verschelde, …)
• Numerical Polynomial Algerba (Stetter)
• Numerical Algebraic Geometry (Sommese, Wampler, Verschelde, …)
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What is an “approximate solution”?
To solve 0122 xx with 8 digits precision:
backward error: 0.00000001 -- method is good
forward error: 0.0001 -- problem is bad
00000001.010 8
bac
kwar
d e
rro
r
0001.010 4
forw
ard erro
r
0122 xx 1xexact computation
,9999.0x
approximate solution
using 8-digits precision
,0001.1axact solution
0)0001.1)(9999.0( xx
0)10()1( 242 x
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The condition number
[Forward error] < [Condition number] [Backward error]
A large condition number <=> The problem is sensitive or, ill-conditioned
From numerical method
From problem
An infinite condition number <=> The problem is ill-posed19
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)
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A well-posed problem: (Hadamard, 1923) the solution satisfies
• existence• uniqueness• continuity w.r.t data
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. … …
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An ill-posed problem is infinitely sensitive to perturbation
tiny perturbation huge error
Ill-posed problems are common in algebraic computing
- Multiple roots
- Polynomial GCD
- Factorization of multivariate polynomials
- The Jordan Canonical Form
- Multiplicity structure/zeros of polynomial systems
- Matrix rank
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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
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Does that mean:
(Most) algebraic problems are wrong problems?
A numerical algorithm seeks the exact solution of a nearby problem
Ill-posed problems are infinitely sensitive to data perturbation
Conclusion: Numerical computation is incompatible
with ill-posed problems.
Solution: Formulate the right problem.
P : Data SolutionP
P
Data
Solution
P
Challenge in solving ill-posed problems:
Can we recover the lost solution when the problem is inexact?
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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.
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Why are ill-posed problems infinitely sensitive?
Plot of pejorative manifolds of degree 3 polynomials with multiple roots
• 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
• Problems with certain solution structure form a “pejorative manifold”
W. Kahan’s observation (1972)
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)( | matrices Rank * rArankCAMr nmnmr
))(( codim -- rnrmM nmr
)),((deg |),( pairs al Polynomi* , rqpGCDqpP nmr rP nm
r codim --
Geometry of ill-posed algebraic problems
nmnmn
nmn PPP
01
nmn
nmnm MMM 10
Similar manifold stratification exists for problems like factorization, JCF, multiple roots …
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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
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1 codimsion
2 ncodimensio
3 codimsion
B
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Illustration of pejorative manifolds
0 codimsion
A?
?
Problem A Problem Bperturbation
The “nearest” manifold may not be the answer
The right manifold is of highest codimension within a certain distance
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.
Finding approximate solution is (likely) a well-posed problem
Approximate solution is a generalization of exact solution.
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Formulation of the approximate rank /kernel:
)(min)( BrankArankAB
0 and nmCA
The approximate rank of A within Backward nearness: app-rank of A is the exact rank of certain matrix B within .
Maximum codimension: That matrix B is on the pejorative manifold possessing thehighest co-dimension and intersecting theneighborhood of A.
)()( BKerAKer with
2)()(2min ACAB
ArankCrank
The approximate kernel of A within
Minimum distance: That B is the nearestmatrix on the pejorative manifold .
• An exact rank is the app-rank within sufficiently small .
• App-rank is continuous (or well-posed)
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Rank
= 4nullity = 2
+ E = 6nullity = 0
kernel
basis
+ E = 4nullity = 2
98.40
1
26.61))()(( EAKerAKerdist
Rank
= 4nullity = 2
After reformulating the rank:
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Ill-posedness is removed successfully.
App-rank/kernel can be computed by SVD and other rank-revealing algorithms (e.g. Li-Zeng, SIMAX, 2005)
Formulation of the approximate GCD ngmfxxCgf l )deg( ,)deg( ,0 ],,,[),( 1
),(deg,)deg(,)deg(
],,[),( 1,
jqpGCDnqmp
xxCqpP lnm
j
nmjj Pqpqpgfgf ,),( ),(),(inf),(
),(),(),(min),(),(,),(
gfvugfqpgf kPvu nm
k
)codim(max ,
),(
nmj
gfPk
j
),(),( qpEGCDgfAGCD
The AGCD within :
),( gf
nmkP ,
),( qp
nmkP ,
1
nmkP ,
1
• Finding AGCD is well-posed if (f,g) is sufficiently small
• EGCD is an special case of AGCD for sufficiently small
(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)
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after formulating the approximate solution to problem P within
P
The two-staged algorithm
Stage II: Find/solve problem Q such that
RPQPR
min
Q
Stage I: Find the pejorative manifold of the highest dimension s.t.
),(Pdist
Exact solution of Q is the approximate solution of P within
which approximates the solution of S where P is perturbed from
S
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Case study: Univariate approximate GCD:
Stage I: Find the pejorative manifold
ngmfxCgf )deg( ,)deg( ,0 ],[),(
nmgfSPgfdist knm
k ),( ),,( min,
knwkmv
vgwfv,wgfSk
)deg( ,)deg( with
)(for matrix theis ),( where
)0 and ( vgwfwugvuf
for a least squares solution (u,v,w) by Gauss-Newton iteration
1)(u
gwu
fvu
Stage II: solve the (overdetermined) quadratic system ),(),,( gfbwvuF
(key theorem: The Jacobian of F(u,v,w) is injective.) 36
Start: k = n
Is AGCD of degree kpossible?no
k := k-1
Successful?
no
k := k-1
Refine with G-N Iteration
probably
yes
Output GCD
Univariate AGCD algorithm
Max-codimension
Min-distancenearness
37
Case study: Multivariate approximate GCD:
Stage I: Find the max-codimension pejorative manifold by applying univariate AGCD algorithm on each variable xj
ngmfxxCgf l
)deg( ,)deg( ,0 ],,,[),( 1
Stage II: solve the (overdetermined) quadratic system
1)(u
gwu
fvu
for a least squares solution (u,v,w) by Gauss-Newton iteration
),(),,( gfbwvuF
(key theorem: The Jacobian of F(u,v,w) is injective.)
),,( ),,(),,(
),,( ),,(),,(
and
jjj
jjj
xwxuxg
xvxuxf
wugvuf
38
Case study: univariate factorization:
Stage I: Find the max-codimension pejorative manifold by applying univariate AGCD algorithm on (f, f’ )
nfxCf )deg( ,0 ],[
1m1m1
1m1m1
mm1
k1
k1
k1
)()( )',(
)()()()('
)()()(
k
k
k
zxzxffAGCD
xqzxzxxf
zxzxxf
Stage II: solve the (overdetermined) polynomial system F(z1 ,…,zk )=f
for a least squares solution (z1 ,…,zk ) by Gauss-Newton iteration
(key theorem: The Jacobian is injective.)
)( ) () ( k1 mm1 fzz k
(in the form of coefficient vectors)
39
Case study: Finding the nearest matrix with a Jordan structure
J =
1 1
JxxxxxxxxA ,,,,,, 43214321
Segre characteristic = [3,1]
Equations determining the manifold
0
0,,,,,,
0)( ,,,,,,
1
43214321
43214321
ub
Iuuuuuuuu
SIuuuuuuuuA
T
T
0),,,,,,,,,,( 34241423134321 sssssuuuuAF
3 1
2
1
1Ferrer’s diagram
A ~ J
codim = -1 + 3 + 3(1) = 5
A
B
)( ,,,,,, 43214321 SIuuuuuuuuA
I+S=
s13 s23
s14
s24
s34
Wyre characteristic = [2,1,1]
ijji uu
0),,,( SUAF 40
Case study: Finding the nearest matrix with a Jordan structure
Equations determining the manifold
0
0,,,,,,
0)( ,,,,,,
1
43214321
43214321
ub
Iuuuuuuuu
SIuuuuuuuuA
T
T
A ~ J
A
B
2
2,,
2
2),,,(min )ˆ,ˆ,ˆ,( SUBFSUBF
SU
For B not on the manifold, we can still solve
for a least squares solution :
0),,,( SUBF
When2
)( SIUBU is minimized, so is 22)( ABUSIUB T
The crucial requirement: The Jacobian ),,,( AJ of ),,,( AF is injective.
(Zeng & Li, 2006)
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tangent plane P0 :
u = G(z0)+J(z
0)(z- z0)
initial iterate
u0 =
G(z
0 )
Least squares solution
u* =
G(z
* )
a
Project to tangent plane
u 1 = G(z 0
)+J(z 0)(z 1
- z 0)
~
new iterate
u1 =
G(z
1 )
Pejora
tive m
anifo
ld
u = G
( z )
Solve G( z ) = a for nonlinear least squares solution z=z*
Solve G(z0)+J(z0)( z - z0 ) = a for linear least squares solution z = z1
G(z0)+J(z0)( z - z0 ) = aJ(z0)( z - z0 ) = - [G(z0) - a ] z1 = z0 - [J(z0)+] [G(z0) - a]
Solving G(z) = a
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Stage II: Find/solve the nearest problem on the manifold
via solving an overdetermined system G(z)=a for a least squares solution z* s.t . ||G(z*)-a||=minz ||G(z)-a|| by the Gauss-Newton iteration
Stage I: Find the nearby max-codim manifold
,2,1,0 ,)()( 1 kazGzJzz kkkk
Key requirement: Jacobian J(z*) of G(z) at z* is injective
(i.e. the pseudo-inverse exists)
tohzGzGzJ
zz ..)ˆ()()ˆ(
1 ˆ
)ˆ(zG
)(zG
condition number(sensitivity measure)
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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
univariate/multivariate GCD
matrix rank/kernel
dual basis for a polynomial ideal
univariate factorization
irreducible factorization
elimination ideal
… …
(to be continued in the workshop next week)
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