The Geometry of Polynomial Division and Elimination

Transcription

1 The Geometry of Polynomial Division and Elimination Kim Batselier, Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven Department of Electrical Engineering ESAT/SCD/SISTA/SMC May / 26

2 Outline 1 Introduction 2 Multivariate Polynomial Division 3 Elimination 4 Conclusions 2 / 26

3 Symbolic Methods Computational Algebraic Geometry Emphasis on symbolic methods Computer algebra Huge body of literature in Algebraic Geometry Wolfgang Gröbner ( ) Bruno Buchberger 3 / 26

4 Changing the Point of View Richard Feynman Seeing things from a Linear Algebra perspective Is it possible to use Linear Algebra instead? New insights/interpretations? New methods? Numerical Algebraic Geometry 4 / 26

5 Research on Three Levels Conceptual/Geometric Level Polynomial system solving is an eigenvalue problem! Row and Column Spaces: Ideal/Variety Row space/kernel of M, ranks and dimensions, nullspaces and orthogonality Geometrical: intersection of subspaces, angles between subspaces, Grassmann s theorem,... Numerical Linear Algebra Level Eigenvalue decompositions, SVDs,... Solving systems of equations (consistency, nb sols) QR decomposition and Gram-Schmidt algorithm Numerical Algorithms Level Modified Gram-Schmidt (numerical stability), GS from back to front Exploiting sparsity and Toeplitz structure (computational complexity O(n 2 ) vs O(n 3 )), FFT-like computations and convolutions,... Power method to find smallest eigenvalue (= minimizer of polynomial optimization problem) 5 / 26

6 Polynomials as Vectors Graded Xel Ordering Let a and b N n 0. We say a > grxel b if a = n a i > b = i=1 n b i, or a = b and a > xel b i=1 where a > xel b if, in the vector difference a b Z n, the leftmost nonzero entry is negative. Examples (2, 0, 0) > grxel (0, 0, 1) because (2, 0, 0) > (0, 0, 1) which implies x 2 1 > grxel x 3 (0, 1, 1) > grxel (2, 0, 0) because (0, 1, 1) > xel (2, 0, 0) which implies x 2 x 3 > grxel x / 26

7 Polynomials as Vectors Vector Representation Defining a monomial ordering allows a vector representation Each column of the vector corresponds with a monomial, graded xel ordered and ascending from left to right LM(p) Leading Monomial of polynomial p according to monomial ordering Example: the polynomial 2 + 3x 1 4x 2 + x 1 x 2 7x 2 2 is represented by ( 1 x 1 x 2 x 2 1 x 1 x 2 x ) Cd n : vector space of all polynomials in n indeterminates with complex coefficients up to a degree d 7 / 26

8 Outline 1 Introduction 2 Multivariate Polynomial Division 3 Elimination 4 Conclusions 8 / 26

9 Definition Divison Definition Fix any monomial order > on Cd n and let F = (f 1,..., f s ) be a s-tuple of polynomials in Cd n. Then every p Cn d can be written as p = h 1 f h s f s + r where h i, r C n d. For each i, h if i = 0 or LM(p) LM(h i f i ), and either r = 0, or r is a linear combination of monomials, none of which is divisible by any of LM(f 1 ),..., LM(f s ). 9 / 26

10 Divisor Matrix Divisor Matrix D in C n d Given a set of polynomials f 1,..., f s Cd n, each of degree d i (i = 1... s) and a polynomial p Cd n of degree d then the Divisor matrix D is given by D = f 1 x 1f 1 x 2f 1. x k 1 n f 1 f 2 x 1f 2. x ks n f s where each polynomial f i is multiplied with all monomials x α i from degree 0 up to degree k i = deg(p) deg(f i) such that x α i LM(f i) LM(p). 10 / 26

11 Divisor Matrix Example Let p = 4 + 5x 1 3x 2 9x x 1x 2 and F = { 2 + x 1 + x 2, 3 x 1 }. The Divisor Matrix is then D = 1 x 1 x 2 x 2 1 x 1 x 2 f x 1 f f x 1 f x 2 f / 26

12 Divisor Matrix 12 / 26

13 Divisor Matrix Divisor Matrix D row space of D D : all polynomials i h if i s.t. LM(p) LM(h i f i ) dim(d) = rank(d) [p] D = {r C n d : p r D} Set of all these equivalence classes (remainders) is denoted by C d /D dim(c d /D) = nullity(d) Any monomial basis of a vector space R such that R = C d /D and R Cd n = a normal set 13 / 26

14 Divisor Matrix R r p i h if i r D 14 / 26

15 Division Algorithm Algorithm: Multivariate Polynomial Division Input: polynomials f 1,..., f s, p Cd n Output: h 1,..., h s, r D Divisor matrix for p D linear independent rows of D col indices of linear dependent columns of D R canonical basis of monomials corresponding with col q = s i h if i project p along R onto D r p q h = ( ) h 1,..., h s solve hd = q 15 / 26

16 Division Algorithm Oblique Projection p = h 1 f h s f s + r with h i f i D and r R s i h if i is found by projecting p oblique along R onto D s h i f i i=1 = p/r [D/R ] D p/r, D/R orthogonal complements of p orthogonal on R and D orthogonal on R respectively r is then found as r = p hf 16 / 26

17 Non-uniqueness of quotients Non-uniqueness of quotients General case D not of full row rank Linear independent rows of D form a basis of D Definition does not provide extra constraints to pick out a certain basis Non-uniqueness of remainders General case D not of full column rank Linear dependent columns of D form a monomial basis of R Definition does provide extra constraint but still not-unique 17 / 26

18 Implementation Implementation determine: rank(d), basis for D and kernel from kernel determine the monomial basis for R compute the oblique projection (exploiting the structure) sparse multifrontal multithreaded rank-revealing QR decomposition 18 / 26

19 Outline 1 Introduction 2 Multivariate Polynomial Division 3 Elimination 4 Conclusions 19 / 26

20 Macaulay Matrix Macaulay Matrix Given a set of multivariate polynomials f 1,..., f s Cd n, each of degree d i (i = 1... s) then the Macaulay matrix of degree d is given by M(d) = f 1 x 1 f 1. x d 1 d n f 1 f 2 x 1 f 2. x ds d n where each polynomial f i is multiplied with all monomials up to degree d d i for all i = 1... s. f s 20 / 26

21 Elimination Elimination Problem Given a set of multivariate polynomials f 1,..., f s Cd n and x e {x 1,..., x n }. Find a polynomial g = s i h if i in which all monomials x e are eliminated. Solution g lies in the intersection of two vector spaces: M d = row space of M(d) E d = vector space spanned by monomials {x 1,..., x n } \ x e containing polynomials up to degree d 21 / 26

22 Elimination E d g o M d 22 / 26

23 Elimination Elimination Algorithm Input: polynomials f 1,..., f s C n d, monomial set x e Output: g M d E d d max(deg(f 1 ), deg(f 2 ),..., deg(f s )) g [ ] while g = [ ] do E(d) canonical basis for E d M(d) Macaulay matrix of degree d if M d E d ø then g element from intersection else d d + 1 end if end while 23 / 26

24 Elimination Implementation Use canonical angles between vector spaces to determine the intersection Q m, Q e orthogonal bases for M d and E d Q m Q Y e = Y CZ T with C = diag(cosθ 1,..., cosθ k ) link with Cosine-Sine decomposition Need orthogonal basis for M d : sparse rank-revealing QR Implicitly Restarted Arnoldi Iterations to determine the canonical angle and g 24 / 26

25 Conclusions Conclusions Polynomial division: vector decomposition Elimination: intersection of vector spaces Oblique projections Principal angles and CS decomposition Sparse structured matrices Applicable on many other problems: approximate GCD polynomial system solving ideal membership problem / 26

26 Conclusions Thank You 26 / 26