APAtools: A MatlabMaple toolbox for

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APAtools A Matlab/Maple toolbox for
Approximate Polynomial Algebra
Zhonggang Zeng Northeastern Illinois University
Isograph (1937) by Bell Labs
Algebraic Machine (1895) by Leonardo Torres
Quevedo
Oct. 23, 2006, Institute of Mathematics and its
Applications
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For polynomial
with coefficients in hardware precision
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For polynomial
with (inexact ) coefficients in machine precision
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Exact gcd is discontinuous
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(Zeng-Dayton 2004, Gao-Kaltofen-May-Yang-Zhi,
2004)
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Factoring a multivariate polynomial
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An approximate factorization using APAtools
on
a modified Gaos algorithm
(Also see, Sommese-Verschelde-Wampler 2004)
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Example A distorted cyclic four system
There are two 1-dimensional solution set
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Challenge in solving algebraic problems
Problems are often ill-conditioned, or even
ill-posed.
The key remove ill-posedness
curb sensitivity
P Data ? Solution
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Geometry of ill-posed algebraic problems

• 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

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Geometry of ill-posed algebraic problems
Similar manifold stratification exists for
problems like factorization, JCF,
multiple roots
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Illustration of pejorative manifolds
The nearest manifold may not be the answer
The right manifold is of highest codimension
within a certain distance
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A three-strike 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.

Ill-posedness is successfully struck out in
problems such as GCD, univariate
complete factorization,
multivariate squares-free factorization,
irreducible factorization,
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Formulation of the approximate 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-neighborhood
of 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)

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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
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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 solution z s.t .
G(z)-aminz 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)
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APAtools in a nutshell
Objective Building an
expanding toolbox, starting from basic
operations, for computing
approximate solutions in polynomial algebra
Methodology Three-strike
formulation of problems to remove ill-posedness
Two-staged algorithms for finding
approximate solutions
Main building blocks Numerical
Linear Algebra (approximate
rank/kernal, least squares, )
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The base level tools Matrix
builder, rank/kernel finder, least squares solver
To convert
a linear transformation L A
/ B
to a matrix T C m / C n
Tools like
Generate T columnwise
For j from 1 to m do p YA (ej) q L(p)
Tj YB-1(q) end do
make matrix building easy
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The base level tools Matrix
builder, rank/kernel finder, least squares solver

• Standard SVD may not be the best choice for
  rank/kernel because
• large matrices in polynomial algebra
• recursive rank computation
• only a few singular values are needed

• App-rank/kernel tools
• MinimumSingularValue
• ApproximateKernel

(Li-Zeng, SIMAX 2005, Lee-Li-Zeng 2006)
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The base level tools Matrix
builder, rank/kernel finder, least squares solver
Solving G(z) a
a
Highest codim. manifold u G( z )
zk1 zk - J(zk) G(zk) - a
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The base level tool (under consideration)
Approximate Jordan Canonical Form
(Zeng-Li, 2006)
gtgt A jormat(1 1 2 2,3 1 2 1) A 3
3 -1 1 0 -1 0 -4 -2
2 -1 -1 -2 1 0 1 1
1 -1 -2 0 0 -1 0 0
1 2 0 5 2 -2 1 3
5 -2 0 0 0 0 0 2
0 -2 0 1 -2 1 0 3
gtgt J jcf(A) gtgt J J 2.0000 1.2309
0 0 0 0 0
0 2.0000 0 0 0
0 0 0 0 2.0000
0 0 0 0 0
0 0 1.0000 1.0433 0
0 0 0 0
0 1.0000 1.7845 0 0
0 0 0 0 1.0000
0 0 0 0 0
0 0 1.0000
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Building up Univariate approximate GCD tool
Stage I Find the highest codimension manifold
(Zeng, Approximate GCD of inexact polynomials I,
2004)
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Bulding up Multivariate approximate GCD tool
Stage I Find the max-codimension manifold by
univariate AGCD algorithm on each
variable xj
(Zeng-Dayton, Approximate GCD of inexact
polynomials I, 2004)
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Matlab demo
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Building up univariate factorization tool
(Zeng, Math. Comp. 2005)
Stage I Find the max-codimension manifold by
univariate AGCD algorithm on (f, f
)
Stage II solve the (overdetermined) polynomial
system F(z1 ,,zk )f
(in the form of coefficient vectors)
for a least squares solution (z1 ,,zk ) by G-N
iteration
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For polynomial
with (inexact ) coefficients in machine precision
Approximate factorization results The backward
error 6.16 x 10-16 Approximate factors
multiplicities x-1.000000000000000 20
x-1.999999999999997 15 x-3.000000000000011 10 x
-3.999999999999985 5
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Approximate Squarefree Factorization (ASFF)
Input polynomial p, tolerance e, random vector
a
Example p (p1)5(p2)3(p3)3(p4)1
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Example approximate squarefree factorization
Approximate squarefree factorization requires a
simple sequence of
applying multivariate AGCD tool
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The multiplicity structure of polynomial systems
(Joint work with Barry H. Dayton, 2005)
Example
Whats the multiplicity of (0,0)?

• Multiplicity 12

• Hilbert function 1,2,3,2,2,1,1,0,

Multiplicity is more than a number!
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For a univariate polynomial p(x) at zero x0
Duality approach (Lasker, Macaulay,
Groebner) Multiplicity is the dimension of the
dual space, spanned by the differential
functionals that vanish on the ideal
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For a polynomial system, or ideal
at zero z
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Multiplicity matrices
and kernel
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If the polynomials are inexact, or the zeros are
approximate,
determining the multiplicity structure is an
application of the matrix builder and the
rank/kernel tool.
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Example
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Building up Approximate elimination
For
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For
find
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Find
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Find
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Start
If Mj is rank deficient, then compute
its kernel convert it to (p,q)
exit end if
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The elimination matrix Mj
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Symbolic or numerical elimination
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By elimination, Cyclic 4 system leads to
triangular system
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APAtools at a glance
Semi-completed tools

• Matrix builders
• Rank/kernel finders
• Least squares tools

• univariate GCD
• multivariate GCD
• univariate factorization

Tools in development

• dual basis and multiplicity structure
• approximate elimination
• squarefree factorizations

Tools in planning

• approximate irreducible factorization

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Conclusion
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He who wants works well done
(on algebraic geometry)
needs to sharpen his (APA)tools first
Confucius 551-479 (?) B.C
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