Extended companion matrix for approximate GCD

Transcription

1 Extended companion matrix for approximate GD Paola oito XLIM - DMI Université de Limoges - NRS 23 avenue lbert Thomas 876 Limoges EDEX paolaboitounilimfr Olivier Ruatta XLIM - DMI Université de Limoges - NRS 23 avenue lbert Thomas 876 Limoges EDEX olivierruattaunilimfr STRT We study a variant of the univariate approximate GD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring [x]/(f) In particular, the structure of the null space of the multiplication matrix contains all the essential information about GD(f, g) Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GD computation with quadratic complexity wrt polynomial degrees ategories and Subject Descriptors G [Mathematics of omputing]: General ; G2 [Mathematics of omputing]: Numerical nalysis pproximation; G3 [Mathematics of omputing]: Numerical nalysis Numerical Linear lgebra; I2 [omputing Methodologies]: Symbolic and lgebraic Manipulations lgorithms General Terms Theory; lgorithms Keywords pproximate greatest common divisor; pproximate geometry; Symbolic-numeric computation; Structured matrices INTRODUTION The approximate polynomial greatest common divisor (denoted as GD) is a central object of symbolic-numeric computation The main difficulty of the problem comes from the fact that is no universal notion of GD One can find different approaches and different notions for GD We will not give a review of all the existing work on this subject, but we will recall one of the most popular approaches Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee SN 2, June 7-9, 2, San Jose, alifornia opyright 2 M $ to show how our work brings a different point of view on the problem The main approach to the computation of an GD consists in considering two univariate polynomials whose coefficients are known with uncertainty This uncertainty can be the result of the fact that the polynomials have floating point coefficients coming from previous computation (and so are subject to round-off errors) The most frequently adopted formulation is related to semi-algebraic optimization : given f and g two approximate polynomials, find two polynomials f and g such that f f and g g are small (lower than a given tolerance for instance) and such that the degree of gcd(f, g) is maximal That is, one looks for the most singular system close to the input ( f, g) n ε-gcd is obtained if the conditions f f < ε and g g < ε are satisfied One can try to compute the tolerance on the perturbation of the input polynomial thanks to direct computation (for instance from a jump on singular values of particular matrix for instance) This last approach has received a great interest following the work of Zeng using Sylvester like matrices ([22]) following several previous work [3], [4], [7], [8], [6], [7] and [8] Here, we consider a slightly different problem One of the polynomials, say f, is known exactly (it is the result of an exact model) and the second one, say g, is an approximate polynomial (result of measures or previous approximation for instance) This case occurs in applications such as model checking (to compare results of an exact model and measures) There are many other instances of such a problem, such as simplification of fractions when one of the polynomial is known exactly but the other one is not We give a example of such a situation When modeling an electromagnetic filter, one might want to parametrize its behavior with respect to the frequency ut one may need to do so even if there are singularities and to do so one may use Padé approximations of the electromagnetic signal at each point as a function of the frequency In some cases of interest, one can know all the singularities and so compute an exact polynomial called characteristic Padé approximations are computed independently for each point by a numerical process and denominators may have a non trivial gcd with the characteristic polynomial The denominators are not known exactly So, in order to identify unwanted common factors in denominators one has to compute approximate gcds between an exact and non exact polynomials This GD problem can also be interpreted as an optimization problem Given f exactly and g approximately, compute a polynomial g close to g such that g has a maxi- 74

2 mal degree gcd with f Our approach takes advantage of the asymmetry of the problem and of the structure of the quotient algebra [x]/(f(x)) (more accurately, of the displacement rank of the multiplication operator in this algebra) So, we address the following problem : Problem Let f(x) [x] a given polynomial and g(x) another polynomial Find g(x) close to g(x) (in a sense that will be explained) such that f(x) and g(x) have a gcd of maximal degree This may be also an interesting approach when one has two polynomials, one known with high confidence and another with worse accuracy This approach may take advantage of this asymmetry which would not be possible for classical framework based on Sylvester or ézout matrices ll the previous methods allow to deform the two input polynomials They give a symmetrical role to those polynomials If one of them is known with high confidence, the previous methods cannot take into account this information Here, the polynomial used to construct the quotient algebra is supposed to be exact or known with higher confidence than the one use to compute the multiplication matrix Furthermore, just as in [7] and [8] we can translate our problem to an optimization problem This is important in order to be able to design the numerical part of our algorithm oth the approach and the algorithm seem new since we address a new problem The proposed algorithm is fast since the exponent of its complexity is better than the classical linear algebra exponent in the degree of the input polynomials Organisation of the paper: The second section is devoted to some basic result on algebra needed after, the third section gives an algebraic method for gcd based on linear algebra, the fourth section recalls the arnett s formula allowing to compute the multiplication matrix without division, the fifth gives the displacement rank structure of the multiplication matrix, the sixth describes the final algorithm and experiments before finishing with conclusions and perpectives 2 EULIDIN STRUTURE ND QUOTIENT LGER In this section, we recall basic algebraic results needed to understand the principle of our approach ll material in this section can be found (even in the non reduced case and in the multivariate setting) in [9] for instance ssume that K is an algebraically closed field (here we think about ) Let f(x) and g(x) K[x] and assume dq that f(x) = (x ζ i) and that ζ i ζ j for all i j i= in {,, d} Let = K[x]/(f) and π : K[x] be the natural projection For i {,, d}, we define L i(x) = Q (x ζ j ) j i Q j i (ζ i ζ j ), the ith Lagrange polynomial associated with {ζ,, ζ d } learly, since deg(l i) < deg(f), we have π(l i) = L i, for all i {,, d} Let = HomK(, K) be the usual dual space of For all i {,, d}, we define ζi : K by ζi (p) = p(ζ i) for all p The following lemma is obvious from the definition of the polynomials j L i if i = j that for i and j {,, d}, we have L i(ζ j) = else This implies that the set {L,, L d } is a basis of well known fact is that the set { ζ,, ζd } form a basis dual of the basis {L,, L d } of s a corollary, we have the Lagrange interpolation formula : Each p can be written p(x) = d P i= ζi (p)l i(x) direct consequence is that if we choose {L,, L d } as a basis of, for all g K[x], the remainder π(g) resulting of the Euclidean division of g by f is given by (g(ζ ),, g(ζ d )) in the basis {L,, L d }, ie P r = d g(ζ i)l i(x) In other word, divide g by f is equivalent i= to evaluate g at the roots of f The general philosophy of this last assertion will allow us to make a lot of proof in a very simple way For example, it is very easy to see the different operation in using this representation Let g and h be two elements in, then we P have g + h = d P (g(ζ i) + h(ζ i)) L i(x) and g h = d (g(ζ i) i= h(ζ i))l i(x) in This allows us to avoid the use of the section of the surjection π In fact, the Lagrange polynomials L,, L d reveal a deeper structure on the algebra : The polynomials j L,, L d are the idempotents of, ie L i Li if i = j L j = otherwise Thanks to this description of the quotient algebra, it is easy to derive algorithms for both polynomial solving and gcd computation even though the problems are of very different nature Remark that we have expressed everything in the monomial basis since it is the most widely used basis to express polynomials We can use other bases particular basis is the hebyshev basis where all results are exactly the same since it is a graded basis 3 N LGERI LGORITHM FOR GD OMPUTTION To first give an idea on how to exploit the section above in order to design algorithm for gcd, We recall a classical method for polynomial solving (see [6] for instance) Proofs are given for the sake of completeness and because very similar ideas will lead us to the GD computation 3 Roots via eigenvalues Let f(x) = d P i= i= f ix i [x] be a polynomial oegree d Then we consider the matrix of the multiplication by x in [x]/(f) Its matrix in the monomial basis,, x d is the following: ` x x 2 x d Frob(f) = x x 2 x d f f f 2 well known as the Frobenius companion matrix associated with f Proposition Let f(x) [x]be polynomial oegree d with d distinct roots Z(f) = {z,, z d }, then the eigenvalues of Frob(f) are the roots of f(x), ie Spec(Frob(f)) = {z,, z d } 75

3 Then, to compute the roots of f(x) one can compute the eigenvalues of its Frobenius companion matrix This is the object of the method proposed (reintroduced) by Edelman and Murakami [9] and revisited by Fortune [] and many others trying to use the displacement structure of the companion matrix In fact, often, the author realized that the monomial basis of the quotient algebra is not the most suitable one and proposed to express the matrix of the same linear application but in other basis In the case of the hebyshev basis this algorithm was already known by arnett [2] and ardinal later [6] In the next section, we will also take advantage of the structure of the quotient algebra to design an algorithm for gcd computation mainly using linear algebra (eigenvalues are used in theory and never computed) 32 Structure of quotient and gcd Let f(x) and g(x) K[x] such that they are both monic s above, we denote = K[x]/(f) and d = deg(f) We denote {ζ,, ζ d } the set of roots of f(x) and we assume that f(x) j is squarefree, ie ζ i ζ j if i j We define M g : where π(p) denote the remainder h π(gh) of p(x) K[x] divided by f(x) We denote M g the matrix of M g in the monomial basis, x,, x d of but other bases can be used matrix representing the map M g is called an extended companion matrix Proposition 2 The eigenvalues of M g are {g(ζ ),, g(ζ d )} Proof It is a direct corollary of the proposition since if we write the matrix of this linear map in the Lagrange basis associated with {ζ,, ζ d }, then it is and gives the wanted result Trivially, we have: g(ζ ) g(ζ d ) orollary We have corank(m g) = deg(f) rank(m g) = deg(gcd(f, g)) The i-th column of the matrix M g contains the coefficients of π(x i g(x)) Let p,, p l be a basis of Ker(M g) and let P (x),, P l (x) be the corresponding polynomials First we remark that nn(g) = {P (x) P (x) g(x) = } is an ideal of Lemma The ideal nn(g) is a principal ideal Proof Let us define v(x) = Y ζ Z(f)\(Z(f) Z(g)) (x ζ) For all h nn(g) it is clear that Z(h) Z(f)\(Z(f) Z(g)) and then v divides h Furthermore v nn(g) since in the Lagrange basis v(x) g(x) = dx v(ζ i)g(ζ i) L i(x) = i= This shows that nn(g) = (v) To compute v(x), we built the matrix with columns formed by p,, p l and we make a triangulation by operating only on the columns This way we obtain the polynomial of minimal degree as a linear combination of P (x),, P l (x) and it is easily seen that this v(x) up to a multiplicative scalar factor Lemma 2 The first column of a column echelon form of the matrix K g built from a basis of Ker(M g) is the generator of nn(g), ie it is the vector of the coefficients of v(x) up to a scalar multiplication Proof Since the columns of a column echelon form of the matrix K g are linearly independent, they form a basis of nn(g) as K-vector space So v(x) is a linear combination of the polynomials associated with those columns The polynomial associated with the column echelon form of K g have all different degree (because it is an echelon form) and so v(x) is a linear combination of those polynomial ecause v(x) as the lowest degree possible, it is a scalar multiple of the polynomial associated with the first column Proposition 3 gcd(f(x), g(x)) = f(x) v(x) Proof y construction, we have v(x) g(x) = mod f(x) and so v(x) divide f(x) We also have gcd( f(x), g(x)) = v(x) gcd(f(x), g(x)) since the roots of f(x) are the root of f(x) v(x) where g(x) vanishes Since deg( f(x) ) = deg(gcd(f(x), g(x)) v(x) we have the wanted result In all this section, we did not care if the polynomials are known in monomial or hebyshev basis for instance In fact, in order to have an algebraic algorithm, we only need to be able to perform Euclidean division and this is always the case if the polynomial basis is graduated (as for monomial, hebyshev, most of the orthogonal bases) 4 EZOUTIN ND RNETT S FORMUL classical matricial formulation of resultant is given by the ézout matrix In this part, we recall the construction of the ézout matrix and a special factorization of the multiplication matrix expressed in the monomial basis This factorization is called arnett s formula (see [2]) The arnett s formula allows to build the classical extended companion matrix without using Euclidean division and only stable numerical computations Furthermore, this factorization reveals that the extended companion matrix has a special rank structure and we will use this fact later to design a fast algorithm to compute GD Definition Let f and g [x] oegree m and n respectively (with n m), we denote Θ f,g (x, y) = f(x)g(y) f(y)g(x) P θ i,jx i y j = m P κ f,g,j (x)y j The ézout matrix associated i,j j= with f and g is f,g = ` θ i,j i,j {,,m } = Remark that since Θ f,g (x, y) = Θ f,g (y, x) the matrix f,g is symmetric The polynomials κ f,g,j (x) are univariate polynomials oegree at most m One particular case of interest is when f = In this case the ézout matrix has a Hankel structure, ie θ i,j = θ i,j+ In this case we denote 76

4 H g,i(x) = κ,g,i(x) for i {,, m } which are called the Horner polynomials Proposition 4 Let i {,, m }, the polynomial H g,i(x) = c,m i + + c,mx i has degree i and since they have different degree, they form a basis of [x]/(g) Furthermore, Θ,g(x, y) = m P H g,m i(x)y i i= orollary 2 The matrix,g is the basis conversion from the Horner basis H,, M m to the monomial basis, x,, x m of [x]/(g) This leads us to the following theorem, known as arnett s formula (see [2]): Theorem Let M g be the multiplication matrix associated with g in [x]/(f) in the monomial basis, we have: M g = f,g,f Proof We have Θ f,g (x, y) = f(x) g(y) g(x) +g(x) f(x) f(y) and so g(x) f(x) f(y) Θ f,g (x, y) in [x, y]/(f(x)) So, for each i {,, deg(f) }, we have Θ f,g,i (x) g(x)θ,f,i (x) This last equality means that f,g is the matrix of the multiplication by g(x) in [x]/(f) The result follows directly from this fact The arnett s formula reveals the rank structure of the multiplication matrix Furthermore, this formula is already known if we choose hebyshev basis instead of monomial basis to express the polynomials and the matrices have exactly the same nature 5 STRUTURED MTRIES ND SYMP- TOTILLY FST LGORITHMS In this section, we briefly recall some basics on displacement structured matrices and related algorithms We are especially interested in two forms oisplacement structure: Toeplitz-like and auchy-like structure See [5] for a more detailed treatment of this topic 5 Displacement structure Given an integer n and a complex number ϑ with ϑ =, define the circulant matrix ϑ Zn ϑ = n n Next, define the Toeplitz-like displacement operator as the linear operator T : m n m n T () = Z m Z ϑ n matrix m n is said to be Toeplitz-like if T () is a small rank matrix (where small means small with respect to the matrix size) The number α = rank( ()) is called the displacement rank of If is Toeplitz-like, then there exist (non-unique) displacement generators G m α and H α n such that T () = GH Other formally different (but essentially equivalent) definitions of the Toeplitz-like displacement operator are found in the literature; as a consequence, the exact displacement rank of a given matrix may vary slightly depending on the definition that has been chosen Toeplitz-like structure is a generalization of Toeplitz structure: indeed, it is easy to see that Toeplitz matrices are also Toeplitz-like Moreover, note that displacement structure is preserved under inversion: the inverse of a Toeplitz-like matrix is again Toeplitz-like, with the same displacement rank Sums and products of Toeplitz-like matrices are also Toeplitz-like, even though the displacement rank may increase More examples of Toeplitz-like matrices include: Toeplitz-block matrices, such as the Sylvester matrix associated with two given polynomials; ézout matrices (which can be defined via sums of products of Sylvester matrices, see eg [2]), the multiplication matrix M f (which is a product of ézoutians) In particular, M f has Toeplitz-like displacement rank equal to 2, regardless of its size, with respect to the displacement operator defined above similar definition holds for auchy-like structure; here the relevant displacement operator is : m n m n () = D D 2, where D and D 2 are diagonal matrices of appropriate size with disjoint spectra matrix is said to be auchylike if () has small rank One can then define notions of auchy-like displacement generators and auchy-like displacement rank 52 Fast solution oisplacement structured linear systems Gaussian elimination with partial pivoting (GEPP) is a well-known and reliable algorithm that computes the solution of a linear system Its arithmetic complexity for an n n matrix is asymptotically O(n 3 ) ut if the system matrix exhibits displacement structure, it is possible to apply a variant of GEPP with complexity O(n 2 ) The main idea consists in operating on displacement generators rather than on the whole matrix; see [] for a detailed description of the algorithm (which will be denoted as GKO in the following) Strictly speaking, the GKO algorithm performs GEPP (or, equivalently, computes the PLU factorization) for auchylike matrices However, several authors have pointed out (see [], [3], [2]) that Toeplitz-like matrices can be stably and cheaply transformed into auchy-like matrices; the same is true for displacement generators onsider, for instance, the case of square matrices, with ϑ = Recall that the Fourier matrix F of size n n, which defines the discrete Fourier transform, is such that F jk = n e 2πi(j )(k ) We have the following result (taken from []): 77

5 Proposition 5 Let be an n n Toeplitz-like matrix with generators G and H Denote by D the matrix diag(, e πi/n,, e (n )πi/n ) and let F be the Fourier matrix of size n n Then the matrix F D F H is auchy-like, of the same displacement rank as, with respect to the displacement operator defined by D = diag(, e 2πi/n,, e 2πi(n )/n ) and D 2 = diag(e πi/n, e 3πi/n,, e (2n )πi/n ) Its auchy-like generators can be computed as Ĝ = F G and ĤH = F D H H Here M H denotes the transpose conjugate of a given matrix M Generalization to the case of m n rectangular matrices is possible In this case, the parameter ϑ should be chosen so that the spectra of D and D 2 are well separated (see [] and [5]) We also point out that the GKO algorithm can be adapted to pivoting techniques other than partial pivoting ([2], [2]) This is especially useful in case of instability due to internal growth of generator entries Matlab implementation of the GKO algorithm that takes into account several pivoting strategies is found in the package DRSolve described in [] In our implementation, we use the pivoting strategy proposed in [2] t each step, the displacement generators G and are redefined, so that the columns of the new first generator G are orthogonal This result can be achieved by computing the QR factorization G = QR, where Q has size n 2, and defining new generators G = Q and H = RH This ensures good conditioning of G One then performs pivoted Gaussian elimination on the matrix column corresponding to the column of H with maximum 2-norm Such a technique involves both row and column permutations; it is not equivalent to complete pivoting, which would be too expensive, but numerical results show that it generally allows a good choice of pivots and reduces the growth of generator entries Error analysis for this pivoting strategy shows that the backward error essentially depends on the auchy-like displacement operator and on the magnitude of the computed upper triangular factor (but not on the magnitude of the generators) 6 STRUTURED PPROH TO GD OMPUTTION We propose here an algorithm that exploits the algebraic and displacement structure of the multiplication matrix to compute the GD of two given polynomials with real coefficients (as defined in Section ) 6 Rank estimation It has been pointed out in Section 3 that the rank deficiency of the multiplication matrix equals the gcd degree In an approximate setting, a relevant notion is the approximate rank (or ε-rank): recall that a matrix Mhas ε-rank k if there exists a matrix M of exact rank k such that M M ε 2 The GD degree can then be estimated using the approximate rank of the multiplication matrix Observe that the technique of computing the GD degree via approximate rank of a resultant (eg, Sylvester) matrix is often exploited in the literature: see for instance [7], [8], [6], [5] Here we use the structured pivoted LU decomposition to estimate the approximate rank of the multiplication matrix Recall that M g has a Toeplitz-like structure with displacement rank 2; it can then be transformed into a auchy-like matrix ˆM g as described in Section 52 Fast pivoted Gauss elimination yields a factorization ˆM g = P LUP 2, where L is a square, nonsingular, lower triangular matrix with diagonal entries equal to, U is upper triangular and P,2 are permutation matrices Inspection of the diagonal entries (or of the row norms) of U allows to estimate the approximate rank of ˆMg and, therefore, of M g Developing a reliable and numerically robust factorizationbased strategy to compute the GD degree is not an easy task s far as the LU factorization is concerned, we mention that Gauss elimination with complete pivoting is rankrevealing in the exact case, but in principle it might not detect near rank-deficiency Since we cannot expect the pivoting strategy we use to perform better than complete pivoting, we conclude that our strategy cannot ensure an a priori certification of approximate rank, nor of GD degree There is, however, a relationship between GKO pivots and distance from a rank-deficient matrix Indeed at any GKO step we have a Ũ k 3/2 ρ a, 2 where e U k k is the trailing matrix block that has not yet been factorized, a is the current pivot and ρ is a positive constant depending on the auchy-like displacement operator This bound can be derived directly using the results presented in [2]; see also [5] We propose in the future to give a more detailed theoretical and numerical analysis of the effectiveness of our degree estimation strategy; the proposed numerical experiments are a first step in this direction 62 Minimization of a quadratic functional Let us suppose that: the polynomial f(x) = P n j= fjxj is exactly known, the polynomial g(x) = P m j= gjxj is approximately known and may be perturbed, so that we consider its coefficients as variables, the GD degree is known Then we can reformulate the problem of GD computation as the minimization of a quadratic functional Indeed, recall that the cofactor v(x) with respect to f(x) is defined by the shortest vector (ie, the vector with the maximum number of trailing zeros) that belongs to the null space of M g We assume v(x) to be monic; we denote its degree as k and we have M gv = M g v v k = lso observe that the entries of M g are linear functions of the coefficients of g(x) Then the equation M gv = can be 78

6 rewritten as F(g, v)=, where the functional F is defined as F : m+ k R + F(g, v) = M gv 2 2 For a preliminary study of the problem, we have chosen to solve the equation F(g, v)= by means of Newton s method, applied so as to exploit structure Denote by z = [g,, g m, v,, v k ] T the vector of unknowns; then each Newton step has the form z (j+) = z (j) J(g (j), v (j) ) M g (j)v (j) In particular, notice that the Jacobian matrix associated with F is an n (m + k + ) Toeplitz-like matrix oisplacement rank 3 This property allows to compute a solution of the linear system J(g (j), v (j) )y = M g (j)v (j) in a fast way; therefore, the arithmetic complexity of each iteration is quadratic wrt the degree of the input polynomials We propose in the future to take into consideration other optimization methods in the quasi-newton family, such as FGS 63 omputation oisplacement generators In order to perform fast factorization of the multiplication matrix M g, we need to compute Toeplitz-like displacement generators It turns out that the range of (M g) is spanned by the first and last column of the displaced matrix, and the columns of indices from 2 to n are multiples of the first one Therefore, it suffices to compute a few rows and columns of M g in order to obtain displacement generators This can be done in a fast and stable way by using arnett s formula If we denote e j the j-th vector of the canonical basis of n, then the computation of the j-th column of M gcan be seen as M g(:, j) = (f, g) `(, f) e j, that is, it consists in solving a triangular Hankel linear system and computing a matrix-vector product For row computation, recall that the ezoutian is a symmetric matrix; we have analogously: M g(j, :) = e T j (f, g) (, f) = `(, f) (f, g)e j T, so that the computation of a row of M g amounts to performing a matrix-vector product and solving a Hankel triangular system similar approach holds for computation oisplacement generators of the Jacobian matrix J(g, v) associated with the functional F(g, v) 64 Description of the algorithm Input: coefficients of polynomials f(x) and g(x) Output: a perturbed polynomial g(x) such that f and g have a nontrivial common factor Estimate the approximate rank k of M g by computing a fast pivoted LU decomposition of the associated auchylike matrix 2 gain by using fast LU, compute a vector v = [v, v,, v k,,,, ] T in the approximate null space of M g 3 pply structured Newton with initial guess (g, v) and compute polynomials g and ṽ such that f and g have a common factor oegree deg f k and ṽ is the monic cofactor for f The algebraic complexity of step is O(deg(f) 2 ) and this is also the complexity of step 2 and of each iteration of step 3 This asymptotic bound seems to be realistic even if the degrees used for our experimentations are not large enough to truly confirm it in practice The number of Newton iterations needed, in our tests, does not seem to grow with the degree as far as our implementation allows us to go So, in our tests, the complexity of the rank computation dominates the arithmetical cost of the algorithm 65 Numerical experiments and computational issues We have written a preliminary implementation of the proposed method in Matlab (available at the URL The results of a few numerical experiments are shown below The polynomials f and g are monic and have random coefficients uniformly distributed over [, ] They have an exact GD of prescribed degree perturbation is then added to g The perturbation vector has random entries uniformly distributed over [ η, η] and its norm is of the order of magnitude of η We show: the residual F( g, ṽ), the 2-norm distance between the exact and the computed cofactor v, the 2-norm distance between the exact and the computed perturbed polynomial g (which is expected to be roughly of the same order of magnitude as η) In the following table we have taken η =e-5 n, m, deg gcd(f, g) F( g, ṽ) v ṽ 2 g g 2 8, 7, 3 2e-5 9e-5 4e-5 5, 4, 5 5e-5 226e-5 35e-4 22, 22, 7 27e-3 4e-3 22e-4 36, 36, 9e-2 57e-4 2 Here are results for η =e-8: n, m, deg gcd(f, g) F( g, ṽ) v ṽ 2 g g 2 8, 7, 3 549e-5 63e-5 585e-8 28, 27, 3 79e-4 898e-4 65e-7 38, 37, 3 488e-2 426e-2 23e-5 58, 57, 23 23e-2 44e-2 254e-4 There are several issues in our approach that deserve further investigations Let us mention in particular: The choice of a threshold (or a more refined technique) for estimating approximate rank Normalization of polynomials: here we mostly work with monic polynomials, but other normalizations may be considered The structured implementation of the optimization step (minimizing F(g, v)) We have used for now a heuristic structured version of the Gauss-Newton algorithm Observe that each step of classical Gauss-Newton applied to our problem has the form z (j+) = z (j) y (j), where z (j) is the vector containing the coefficients of the j-th iterate polynomials g (j) and v (j), and y (j) is the least-norm solution to the underdetermined system J(g (j), v (j) )y (j) = M g (j)v (j) omputing this leastnorm solution in a structured and fast way is a difficult point that will require more work Our implementation gives a solution which is not, in general, the 79

7 least-norm one, even though it is typically quite close Further work will also include a study of other possible optimization methods that lend themselves well to a structured approach 7 ONLUSIONS We have proposed and implemented a fast structured matrixbased approach to a variant of the GD problem, namely, the problem of computing an approximate greatest common divisor of two univariate polynomials, one of which is known to be exact To our knowledge, this variant has been so far neglected in the existing literature It may be also interesting when one polynomial is known with high accuracy and the other is not Our approach is based on the structure of the multiplication matrix and on the subsequent reformulation of the problem as the minimization of a suitably defined functional Our choice of the multiplication matrix M g over other resultant matrices (eg, Sylvester, ézout) is motivated by the smaller size of M g, with respect eg to the Sylvester matrix, the strong link between the null space of M g and the gcd, and in particular the fact that the null space of M g immediately yields a gcd cofactor, the displacement structure of M g, the possibility of computing selected rows and columns of M g in a stable and cheap way, thanks to arnett s formula This is, however, a preliminary study Further work will include generalizations of the proposed problem and a more thorough analysis of the optimization part of the algorithm Furthermore, this approach can be generalized in several interesting ways: using better bases than the monomial one, it can be extended to some multivariate setting to compute the co-factor of a polynomial g in [x,, x n]/(f,, f n) when f,, f n define a complete intersection since arnett s formula still holds, to compute the GD of f with g,, g k where f is known with accuracy but g,, g k are inaccurate, one can take g as a linear combination of g,, g k with our method and succeed with a high probability 8 REFERENES [] ricò, G Rodriguez, fast solver 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