Solving Systems of Polynomial Equations: Algebraic Geometry, Linear Algebra, and Tensors?

Transcription

1 Solving Systems of Polynomial Equations: Algebraic Geometry, Linear Algebra, and Tensors? Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven ESAT/SCD Workshop on Tensor Decompositions and Applications (TDA2010) Monopoli, Italy, September / 27

2 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 2/ 27

3 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 3/ 27

4 Polynomials, Matrices and Eigenvalue Problems Characteristic Polynomial The eigenvalues of A are the roots of p(λ) = det(a λi) = 0 Companion Matrix Solving q(x) = 7x 3 2x 2 5x + 1 = 0 leads to /7 5/7 2/7 1 x x 2 = x 1 x x 2 4/ 27

5 Sylvester Matrix Sylvester Resultant Consider two polynomials f(x) and g(x): f(x) = x 2 3x + 2 g(x) = x 3 4x 2 11x + 30 Common roots iff S(f, g) = S(f,g) = det James Joseph Sylvester 5/ 27

6 Sylvester Matrix Sylvester s construction can be understood from 1 x x 2 x 3 x 4 f(x) = x f(x) = x x 2 f(x) = x 2 g(x) = x 3 = 0 x g(x) = x 4 evaluate the vector containing the powers of x at x = 2 6/ 27

7 Sylvester Matrix Find a vector in the nullspace of the Sylvester matrix, = normalize such that the first entry equals 1: = / 27

8 Conclusion: Main Ingredients Linear Algebra turns out to be suitable framework Main Ingredients: Linearize problem by separating coefficients and monomials Solutions live in the nullspace of coefficient matrix Exploit structure in monomial basis Eigenvalue problems Multivariate case? 8/ 27

9 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 9/ 27

10 Outline of Algorithm Macaulay: multivariate Sylvester construction Linearize by separating coefficients and monomials Algorithm: 1 Build coefficient matrix M 2 Find basis for nullspace of M 3 Find solutions from eigenvalue problem Étienne Bézout James Joseph Sylvester Francis Sowerby Macaulay 10/ 27

11 Build Matrix M Consider j p(x, y) = x 2 + 3y 2 15 = 0 q(x, y) = y 3x 3 2x x 2 = 0 Construct M Write the system in matrix-vector notation: 1 x y x 2 2 xy y 2 x 3 x 2 y xy 2 y 3 3 p(x, y) q(x, y) x p(x, y) y p(x, y) / 27

12 Build Matrix M Continue to enlarge M: 1 x y x 2 xy y 2 x 3 x 2 y xy 2 y 3 x 4 x 3 yx 2 y 2 xy 3 y 4 x 5 x 4 yx 3 y 2 x 2 y 3 xy 4 y 5... p q xp yp x 2 p xyp y 2 q xq yq x 3 p x 2 yp xy 2 p y 3 p x 2 q xyq y 2 q / 27

13 Solutions in Nullspace of M Coefficient matrix M: M = " Solutions generate vectors in nullspace of M: Mv = 0 Number of solutions s follows from corank # Canonical nullspace V built from s solutions (x i, y i): x 1 x 2... x s y 1 y 2... y s x 2 1 x x 2 s x 1 y 1 x 2 y 2... x sy s y1 2 y ys 2 x 3 1 x x 3 s x 2 1 y 1 x 2 2 y 2... x 2 sy s x 1 y1 2 x 2 y x sys 2 y1 3 y ys 3 x 4 1 x x 4 4 x 3 1 y 1 x 3 2 y 2... x 3 sy s x 2 1 y2 1 x 2 2 y x 2 s y2 s x 1 y1 3 x 2 y x sys 3 y1 4 y ys / 27

14 Solutions in Nullspace of M Nullspace of M Find a basis for the nullspace of M using an SVD: [ ][ M = Σ1 0 W T ] = [ X Y ] Z T Hence, MZ = 0 14/ 27

15 Find Solutions Shift property in monomial basis [ ] x = [ ] 1 x y x 2 xy y 2 1 x y x 2 xy y 2 y = [ [ Finding the x-roots: let D = diag(x 1, x 2,..., x s ), then S 1 VD = S 2 V, where S 1 and S 2 select rows from V wrt. shift property Reminiscent of Realization Theory ] ] 1 x y x 2 xy y 2 1 x y x 2 xy y 2 15/ 27

16 Find Solutions We have S 1 VD = S 2 V However, V is not known, instead a basis Z is computed as ZT = V Which leads to or Hence, S 1 ZTD = S 2 ZT S 1 Z ( TDT 1) = S 2 Z TDT 1 = (S 1 Z) S 2 Z 16/ 27

17 Algorithm Summary Algorithm 1 Construct coefficient matrix M 2 Compute basis for nullspace of M, Z 3 Choose shift function, e.g., x 4 Write down shift relation in monomial basis v for the chosen shift function using row selection matrices S 1 and S 2 5 The construction of above gives rise to a generalized eigenvalue problem S 1 Z ( TDT 1) = S 2 Z of which the eigenvalues correspond to the, e.g., x-solutions of the system of polynomial equations. 17/ 27

18 Algorithm Summary Approach has been generalized to Multivariate Polynomials Elegant link with Linear Algebra, especially eigenvalue problems Finds all solutions (or alternatively, global solutions) (Numerical) LA: embed into well-known matrix computations Limiting computational complexity (but inherent to the problem) 18/ 27

19 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 19/ 27

20 Relevant applications are found in Polynomial Optimization Problems Structured Total Least Squares Model order reduction Analyzing identifiability nonlinear model structures Robotics: kinematic problems Computational Biology: conformation of molecules Algebraic Statistics Signal Processing... 20/ 27

21 6 3 Hankel STLS 3 min v τ 2 = v T A T D 1 v Av s.t. v T v = 1. phi STLS Hankel cost function TLS/SVD soln STSL/RiSVD/invit steps STLS/RiSVD/invit soln STLS/RiSVD/EIG global min STLS/RiSVD/EIG extrema theta method TLS/SVD STLS inv. it. STLS eig v v v τ global solution? no no yes (eigenvalue decomposition on matrix) 21/ 27

22 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 22/ 27

23 Links with Tensors A homogeneous polynomial of degree d is a d-mode tensor, e.g., a 0 + a T 1 x + x T A 2 x + A 3 (x,x,x) A d (x,...,x) +... where a 0 R a 1 R n A 2 R n n A 3 R n n n. 23/ 27

24 Is it useful to decouple along tensor directions? 24/ 27

25 Outline 1 Univariate Polynomials 2 Multivariate Polynomials 3 Applications 4 Tensors 5 Conclusions and Future Work 25/ 27

26 Conclusions Link polynomial system solving and linear algebra Polynomial system solving reduces to eigenvalue problems! Problem tackled using matrix computations Computational complexity Topic touches on the fundamentals of applied mathematics Will it be interesting to look at tensors? 26/ 27

27 Thank You! 27/ 27