Download presentation

Presentation is loading. Please wait.

1
Computing the Rational Univariate Reduction by Sparse Resultants Koji Ouchi, John Keyser, J. Maurice Rojas Department of Computer Science, Mathematics Texas A&M University ACA 2004

2
Texas A&M University ACA2004 2 RUR Outline n What is Rational Univariate Reduction? n Computing RUR by Sparse Resultants n Complexity Analysis n Exact Implementation

3
Texas A&M University ACA2004 3 RUR Rational Univariate Reduction n Problem: Solve a system of n polynomials f 1, …, f n in n variables X 1, …, X n with coefficients in the field K n Reduce the system to n + 1 univariate polynomials h, h 1, …, h n with coefficients in K s.t. if is a root of h then ( h 1 ( ), …, h n ( )) is a solution to the system

4
Texas A&M University ACA2004 4 RUR RUR via Sparse Resultant n Notation l e i the i -th standard basis vector = { o, e 1, …, e n } l u 0, u 1,…, u n indeterminates l A i = Supp ( f i ) l the algebraic closure of K

5
Texas A&M University ACA2004 5 RUR Toric Perturbation n Toric Generalized Characteristic Polynomial Let f 1 *, …, f n * be n polynomials in n variables X 1, …, X n with coefficients in K and Supp ( f i * ) A i =Supp ( f i ), i = 1, …, n that have only finitely many solutions in ( \ { 0 }) n Define TGCP ( u, Y ) = Res ( , A 1, …, A n ) ( a u a X a, f 1 - Y f 1 *, …, f n - Y f n *)

6
Texas A&M University ACA2004 6 RUR Toric Perturbation n Toric Perturbation [Rojas 99] Define Pert ( u ) to be the non-zero coefficient of the lowest degree term (in Y ) of TGCP ( u, Y ) l Pert ( u ) is well-defined l A version of “projective operator technique” [Rojas 98, D’Andrea and Emiris 03]

7
Texas A&M University ACA2004 7 RUR Toric Perturbation n Toric Perturbation l If ( 1, …, n ) ( \ { 0 }) n is an isolated root of the input system f 1, …, f n then a u a a Pert ( u ) l Pert ( u ) completely splits into linear factors over ( \ { 0 }) n. For every irreducible component of the zero set of the input system, there is at least one factor of Pert ( u )

8
Texas A&M University ACA2004 8 RUR Computing RUR n Step 1: l Compute Pert ( u ) l Use Emiris’ sparse resultant algorithm [Canny and Emiris 93, 95, 00] to construct Newton matrix whose determinant is some multiple of the resultant l Evaluate resultant with distinct u and interpolate them

9
Texas A&M University ACA2004 9 RUR Computing RUR n Step 2: l Compute h ( T ) l Set h ( T ) = Pert ( T, u 1, …, u n ) for some values of u 1, …, u n l Evaluate Pert ( u ) with distinct u 0 and interpolate them

10
Texas A&M University ACA2004 10 RUR Computing RUR n Step 3: l Compute h 1 ( T ), …, h n ( T ) l Computation of h i involves - Evaluating Pert ( u ), - Interpolate them, and - Some univariate polynomial operations

11
Texas A&M University ACA2004 11 RUR Complexity Analysis n Count the number of arithmetic operations n Notation l O˜( ) the polylog factor is ignored l Gaussian elimination of m dimensional matrix requires O ( m )

12
Texas A&M University ACA2004 12 RUR Complexity Analysis n Quantities l M A The mixed volume MV ( A 1, …, A n ) of the convex hull of A 1, …, A n l R A MV ( A 1, …, A n ) + i = 1,…,n MV ( , A 1, …, A i-1, A i+1, …, A n ) l The total degree of the sparse resultant l S A The dimension of Newton matrix l Possibly exponentially bigger than R A

13
Texas A&M University ACA2004 13 RUR Complexity Analysis n [Emiris and Canny 95] l Evaluating Res ( , A 1, …, A n ) ( a u a X a, f 1, …, f n ) requires O ˜ ( n R A S A ) or O ˜ ( S A 1+ ) if char K = 0

14
Texas A&M University ACA2004 14 RUR Complexity Analysis n [Rojas 99] l Evaluating Pert ( u ) requires O ˜ ( n R A 2 S A ) or O ˜ ( S A 1+ ) if char K = 0

15
Texas A&M University ACA2004 15 RUR Complexity Analysis l Computing h ( T ) requires O ˜ ( n M A R A 2 S A ) or O ˜ ( M A S A 1+ ) if char K = 0

16
Texas A&M University ACA2004 16 RUR Complexity Analysis l Computing every h i ( T ) requires O ˜ ( n M A R A 2 S A ) or O ˜ ( M A S A 1+ ) if char K = 0

17
Texas A&M University ACA2004 17 RUR Complexity Analysis l Computing RUR h ( T ), h 1 ( T ), …, h n ( T ) for fixed u 1, …, u n requires O ˜ ( n 2 M A R A 2 S A ) or O ˜ ( n M A S A 1+ ) if char K = 0

18
Texas A&M University ACA2004 18 RUR Complexity Analysis n Derandomize the choice of u 1, …, u n l Computing RUR h ( T ), h 1 ( T ), …, h n ( T ) requires O ˜ ( n 4 M A 3 R A 2 S A ) or O ˜ ( n 3 M A 3 S A 1+ ) if char K = 0

19
Texas A&M University ACA2004 19 RUR Complexity Analysis Emiris DivisionEmiris GCD char K = 0 “Small” Newton Matrix Evaluating Res n R A S A S A 1+ RARA Evaluating Pert n R A 2 S A S A 1+ R A 1+ RUR for fixed u n 2 M A R A 2 S A n M A S A 1+ n M A R A 1+ RUR n 4 M A 3 R A 2 S A n 3 M A 3 S A 1+ n 3 M A 3 R A 1+

20
Texas A&M University ACA2004 20 RUR Complexity Analysis l A great speed up is achieved if we could compute “small” Newton matrix whose determinant is the resultant No such method is known

21
Texas A&M University ACA2004 21 RUR Khetan’s Method l Khetan’s method gives Newton matrix whose determinant is the resultant of unmixed systems when n = 2 or 3 [Kehtan 03, 04] l Let B = A 1 A n Then, computing RUR requires n 3 M A 3 R B 1+ arithmetic operations

22
Texas A&M University ACA2004 22 RUR ERUR:Implementation n Current implementation l The coefficients are rational numbers l Use the sparse resultant algorithm [Emiris and Canny 93, 95, 00] to construct Newton matrix l All the coefficients of RUR h, h 1,…, h n are exact

23
Texas A&M University ACA2004 23 RUR ERUR l Non square system is converted to some square system l Solutions in ( ) n are computed by adding the origin o to supports.

24
Texas A&M University ACA2004 24 RUR ERUR n Exact Sign l Given an expression e, tell whether or not e ( h 1 ( ), …, h n ( )) = 0 l Use (extended) root bound approach. l Use Aberth’s method [Aberth 73] to numerically compute an approximation for a rootof h to any precision.

25
Texas A&M University ACA2004 25 RUR Applications by ERUR n Real Root l Given a system of polynomial equations, list all the real roots of the system n Positive Dimensional Component l Given a system of polynomial equations, tell whether or not the zero set of the system has a positive dimensional component

26
Texas A&M University ACA2004 26 RUR Applications by ERUR l Presented today’s last talk in Session 3 “Applying Computer Algebra Techniques for Exact Boundary Evaluation” 4:30 – 5:00 pm

27
Texas A&M University ACA2004 27 RUR The Other RUR n GB+RS [Rouillier 99, 04] l Compute the exact RUR for real solutions of a 0-dimensional system l GB computes the Gröebner basis n [Giusti, Lecerf and Salvy01] l An iterative method

28
Texas A&M University ACA2004 28 RUR Conclusion n ERUR l Strong for handling degeneracies l Need more optimizations and faster algorithms

29
Texas A&M University ACA2004 29 RUR Future Work n RUR l Faster sparse resultant algorithms l Take advantages of sparseness of matrices [Emiris and Pan 97] l Faster univariate polynomial operations

30
Texas A&M University ACA2004 30 RUR Thank you for listening! n Contact l Koji Ouchi, kouchi@cs.tamu.edukouchi@cs.tamu.edu l John Keyser, keyser@cs.tamu.edukeyser@cs.tamu.edu l Maurice Rojas, rojas@math.tamu.edurojas@math.tamu.edu n Visit Our Web l http://research.cs.tamu.edu/keyser/geom/erur/ http://research.cs.tamu.edu/keyser/geom/erur/ Thank you

Similar presentations

Presentation is loading. Please wait....

OK

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google