1. Numerical Polynomial Algebra and Algebraic Geometry
Matrix Computation in
Numerical Polynomial Algebra and
Algebraic Geometry
Zhonggang Zeng
Northeastern Illinois University
Linear and Numerical Linear Algebra: Theory, Method, and Applications
August 14, 2009, De Kalb, (supported in part by NSF under Grant DMS )

2. Sample problem: Factor
with inexact coefficients (a.k.a. root-finding problem) 1

3. Sample problem: Factor
with inexact coefficients (a.k.a. root-finding problem)
Matlab factorization
Numerical irreducible factorization:
>> [F,res,fcnd] = uvFactor(f,1e-10,1);
THE CONDITION NUMBER:
THE BACKWARD ERROR: e-015
THE ESTIMATED FORWARD ROOT ERROR: e-011
FACTORS
( x )^20
( x )^40
( x )^60
( x )^80
Z. Zeng, TOMS 2004, Math Comp 2005, ISSAC 2009

4. Sample problem: Computing polynomial GCD2

5. Sample problem: Factor a multivariate polynomial
(Kaltofen, Problem 1 in “Challenges of symbolic computation”, 2000)
Give an example to show it is ill-posed. Use Kaltofen example.
Factorization fails 3

6. Factoring a multivariate polynomial:
A factorable polynomial
irreducible
approximation
Question: How to factor a multivariate polynomial approximately? 4

7. Sample problem: What is the multiplicity of the nonlinear system
What’s the multiplicity of the function f(x) = x3 – 6 x cos(x2) + 6 sin(x)
at the zero x* = 0 ?
Textbooks say:
0 = f(x*)
= f’(x*)
= f”(x*)
= f’’’(x*)
= f(4) (x*)
f(5) (x*) = 366  0
Multiplicity = 5
Sample problem: What is the multiplicity of the nonlinear system
at the zero 5

8. Perturbed system has 12 zeros near (0,0)
Example:
Consider the multiplicity structure of
at an isolated multiple zero (0,0)
Multiplicity = 12
Perturbed system has 12 zeros near (0,0)
Hilbert function = {1,2,3,2,2,1,1,0,…}
Dual space D(0,0) (F) spanned by 1 2 3 4 5 6
Differential orders
depth = 6
breadth=2
Multiplicity is not just a number!
Question: How to identify the multiplicity structure? 6

9. Symbolic computation has thrived for decades
in polynomial algebra and algebraic geometry
Gröbner basis
Polynomial factorization
Computer Algebra Systems (Maple, Mathematica, etc)
Bruno Buchberger
with limitations. 7

10. Pioneer works in numerical polynomial algebra and algebraic geometry
(incomplete list)
Homotopy method for solving polynomial systems
(Li, Sommese, Wampler, Verschelde, …)
Numerical Algebraic Geometry
(Sommese, Wampler, Verschelde, …)
Numerical Polynomial Algerba
(Stetter)
A frontier in scientific computing, and may be the beginning of
Numerical Nonlinear Algebra 8

11. Recent progress in numerical polynomial algebra
Solving polynomial systems (Li, Sommese, Verschelde, …)
Approximate multiple roots/factorization (Zeng 2005, 2009)
Approximate GCD (Zeng-Dayton 2004)
Approximate irreducible factorization (Sommesse-Wampler-Verschelde 2003, Gao et al 2003, 2004, Zeng in progress)
Computing multiplicity structure (Dayton-Zeng 05, Bates-Peterson-Sommese ’06)
Polynomial elimination (Zeng, 2008)
Feedback from num. polyn. algebra to numerical linear algebra:
Approximate rank/kernel (Li,Zeng 2005, Lee, Li, Zeng 2009)
Approximate Jordan Canonical Form (Zeng-Li in progress) 9

12. Among many challenges in numerical polynomial algebra:
Regularizing ill-posed problems for approximate solutions
numerical polynomial algebra via matrix computations
Handling huge matrices with subspace strategies 10

13. The two-staged strategy
after formulating the approximate solution to problem P within e P
Stage I: Find the pejorative manifold
P of the highest codimension s.t. P
Stage II: Find/solve problem Q such that Q S
by numerical matrix computations
by the Gauss-Newton iteration
Exact solution of Q is the approximate solution of P within e
which approximates the solution of S where P is perturbed from 11

14. Example: For polynomial
with (inexact ) coefficients in hardware precision
Matlab factorization
Approximate irreducible factorization:
>> [F,res,fcnd] = uvFactor(f,1e-10,1);
THE CONDITION NUMBER:
THE BACKWARD ERROR: e-015
THE ESTIMATED FORWARD ROOT ERROR: e-011
FACTORS
( x )^20
( x )^40
( x )^60
( x )^80
Zeng, 2005, 2009 12

15. How to identify the solution structure?
Answer: Matrix computations 13

16. Matrix computation is not always good.
Is this matrix nonsingular?
(polynomial long division)
The condition number ≥ 10n
Distance to singularity  1/10n
The error in data can be magnified by 10n times into the solution z.
“Life is too short for long division” anyway. 14

17. Matrix for L is rank-deficient
Example: How to determine the GCD structure If
then
The linear transformation
has a nontrivial kernel:
Matrix for L is rank-deficient 15

18. xj xj [ ] xj Moreover The linear transformation On the vector space
Has the kernel
Linear transformation L  Sylverster matrix S(f,g)
James J. Sylvester
Numerical rank-deficiency = degree of the approx. GCD 16

19. Case study: How to solve
Find
such that
Namely, (p, q) satisfy linear equation
--- leading to matrix rank/kernel problem(Zeng, TCS 2008) 17

20. # of factors = # of solutions to
Multivariate factorization structure: Matrix computations! =
For any polynomial f(x,y)
Assume f = f1 f2 f3 with distinct factors f1, f2, and f3 =
The equation
has three solutions
# of factors = # of solutions to 18

21.  a random nulvector of matrix Rf corresponds to
Irreducibility condition (Ruppert ‘99, and Gao ‘03, Kaltofen-May ’03,Gao-Kaltofen-May-Yang-Zhi’04)
A squarefree polynomial f  C[x,y] of bidegree [m,n] has k distince factors
 the homogeneous linear equation
has k linearly independent solutions (g,h) of bidegrees
deg(g)  [m-1,n], deg(h)  [m,n-1].
is a linear transformation corresponding to a matrix Rf
Rank-deficiency = # of irreducible factors
 a random nulvector of matrix Rf corresponds to
g = l1 f1,x f2 … fk + … + lk f1 f2 … fk,x
h = l1 f1,y f2 … fk + … + lk f1 f2 … fk,y 19

22. S( f(, y*), g (, y*) - li fx (, y*) ) is of nullity mi,
Taking g = l1 f1,x f2 f3 + l2 f1, f2x f3 + l3 f1 f2 f3,x ( f = f1 f2 f3 )
g - l1 fx = (l2 - l1) f1 f2,x f3 + (l3 - l1) f1 f2 f3,x
GCD( f, g - li fx ) = fi for i = 1, 2, …, k
Namely,
Let deg( fi ) = [mi,, ni ]. Then, for fixed random ( x*, y* ),
the Sylvester matrices
S( f(, y*), g (, y*) - li fx (, y*) ) is of nullity mi,
S( f(x* ,), g (x* ,) - li fx (x* ,) ) is of nullity ni,
li is an eigenvalue of geometric multiplicity mi the matrix pencil
S( f(, y*), g (, y*) ) - l S( 0, fx (, y*) )
A - l B
Numerical polynomial factorization can be done by matrix computation! 20

23. Perturbed system has 12 zeros near (0,0)
Example:
Consider the multiplicity structure of
at an isolated multiple zero (0,0)
Multiplicity = 12
Perturbed system has 12 zeros near (0,0)
Hilbert function = {1,2,3,2,2,1,1,0,…}
Dual space D(0,0) (F) spanned by 1 2 3 4 5 6
Differential orders
depth = 6
breadth=2
Case study: How to identify the multiplicity structure of a nonlinear system? 21

24. = the # of linearly Independent functionals vanishing on ideal I
Univariate case: x* is m-fold if = f(x*) = f’(x*) = … = f(m-1)(x*) and f(m)(x*)≠0
Multivariate case:
A (differential) functional at zero
linear combination of
evaluated at
Dual space
defines the multiplicity structure at
= the # of linearly Independent functionals vanishing on ideal I 22

25. Fundamental Theorems of the multiplicity:
Local Finiteness Theorem: The zero is isolated iff the multiplicity is finite.
Perturbation Invariance Theorem: The number of zeros is invariant counting
multiplicities under small perturbation to the system.
Multiplicity Consistence Theorem: The intersection multiplicity = dual multiplicity.
Depth Deflation Theorem: Multiple zeros can be accurately computed by a deflation method, with the number of deflation steps bounded by the depth of the zero.
(Dayton, Li, and Zeng, Multiple zeros of nonlinear systems, 2009)
Joint work with 23

26. Multiplicity matrices and nullspaces24