Computing Approximate GCD of Univariate Polynomials by Structure Total Least Norm 1)

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1 MM Research Preprints, MMRC, AMSS, Academia Sinica No. 24, December Computing Approximate GCD of Univariate Polynomials by Structure Total Least Norm 1) Lihong Zhi and Zhengfeng Yang Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSS, Academia Sinica Beijing , China Abstract. The problem of approximating the greatest common divisor(gcd) for polynomials with inexact coefficients can be formulated as a low rank approximation problem with Sylvester matrix. This paper presents a method based on Structured Total Least Norm(STLN) for constructing the nearest Sylvester matrix of given lower rank. We present algorithms for computing the nearest GCD and a certified ɛ-gcd for a given tolerance ɛ. The running time of our algorithm is polynomial in the degrees of polynomials. We also show the performance of the algorithms on a set of univariate polynomials. 1. Introduction Approximate GCD computation of univariate polynomials has been studied by many authors [23, 18, 7, 9, 3, 15, 14, 19, 25, 8, 29]. Particularly, in [7, 9, 25, 8, 29], Singular Value Decomposition(SVD) of the Sylvester matrix has been used to obtain an upper bound of the approximate GCD and compute it. Suppose the SVD of Sylvester matrix S is S = UΣV T, where U and V are orthogonal and Σ = diag(σ 1, σ 2,..., σ n ), then σ k is the 2-norm distance to the nearest matrix of rank strictly less than k. However, SVD computation is not appropriate for computing the distance to the nearest STRUCTURE(for example Sylvester structure) matrix of given rank deficiency. In [6], they mentioned structured low rank approximation to compute approximate GCD. The basic idea of their method is to alternate projections between the sets of the rank-k matrices and structure matrices so that the rank constrain and the structural constraint are satisfied alternatively while the distance in between is being reduced. Although the algorithm is linear convergent, the speed may be very slow as shown by the tests in the last section. We present a method based on STLN[20] for structure preserving low rank approximation of a Sylvester matrix. STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the minimal perturbation [E F ] such that (A + E)X = B + F. An algorithm in [20] has been described for Hankel structure preserving low rank approximation using STLN with L p norm. 1) Partially supported by a National Key Basic Research Project of China and Chinese National Science Foundation under Grant

2 376 L. Zhi and Z.Yang In this paper, we adjust their algorithm for computing structure preserving rank reduction of a Sylvester matrix. Furthermore, we also derive a stable and efficient way to compute the nearest GCD and ɛ GCD of univariate polynomials. The organization of this paper is as follows. In Section 2, we introduce some notations and discuss the equivalence between the GCD problems and the low rank approximation of a Sylvester matrix. In Section 3, we consider solving an overdetermined system with Sylvester structure based on STLN. Finally, in Section 4, we present algorithms based on STLN for computing the nearest GCD and ɛ-gcd. We illustrate the experiments of our algorithms on a set of univariate polynomials. 2. Preliminaries In this paper we are concerned with the following task. PROBLEM 2.1 Given two polynomials f, g C[x] with deg(f) = m, deg(g) = n, and k is a given positive integer k min(m, n). We are considering to find the minimal perturbation f g 2 2, such that deg( f) m, deg( g) n and deg(gcd(f + f, g + g)) k. When k = 1, the above problem is the same as finding the nearest GCD defined in [16]. If we are given a tolerance ɛ, then the above problem is close to computing ɛ-gcd as [7]. Suppose S(f, g) is the Sylvester matrix of polynomials f and g. We know that the degree of GCD of f and g is equal to the rank deficiency of S. The above problem can be formulated as: min f f 2 deg(gcd( f, g)) k 2 + g g 2 2 min f f 2 rank( S) n+m k 2 + g g 2 2 where S is the Sylvester matrix generated by f and g. If we solve the overdetermined system using STLN [22, 21, 20] A X B (1) for S = [B A] where B are the first k columns of S and A are the remainder columns of S, then we obtain a minimal Sylvester structured pertubation [F E] such that B + F Range(A + E) Therefore, S = [B + F, A + E] is a solution that has Sylvester structure and rank( S) n + m k. The reason why we choose the first k columns to form the overdetermined system (1) can be seen from the following example and theorem. EXAMPLE 2.1 Suppose f = x 2 + x = x (x + 1), g = x 2 + 4x + 3 = (x + 3) (x + 1), S is the Sylvester matrix by f and g S =

3 Approximate GCD of Univariate Polynomials 377 The rank deficiency of S is 1. We partition S by two ways: S = [A 1 b 1 ] = [b 2 A 2 ], where b 1 is the last column of S, whereas b 2 is the first column of S. The overdetermined system A 1 X = b 1 has no solution, while the system has an exact solution as X = [ 3, 1, 0] T. A 2 X = b 2 Theorem 2.1 Given f, g C[x] with deg(f) = m, deg(g) = n and a nonzero integer k min(m, n). Suppose S is the Sylvester matrix of f and g. Partition S = [B A], where B are the first k columns of S and A are the last n + m k columns of S. Then we have rank(s) n + m k AX = B has a solution (2) Proof of Theorem 2.1: = Let AX = B have a solution, then B Range(A). Since B are the first k columns of S. Therefore, it is obvious that the rank of S = [B A] is less than n + m k. = Suppose rank(s) n + m k. Multiplying the vector [x n+m 1,, x, 1] to the two sides of the equation AX = B, it turns out to be [x n k 1 f,, f, x m 1 g,, g]x = [x n 1 f,, x n k f] (3) The solution X of (3) corresponds to the coefficients of polynomials u i, v i, with deg(u i ) n k 1, deg(v i ) m 1, satisfy x n i f = u i f + v i g, i = 1,..., k (4) Let d = GCD(f, g), f 1 = f/d, g 1 = g/d. Since rank(s) n + m k, we have deg(d) k and deg(f 1 ) m k, deg(g 1 ) n k. For any 1 i k, dividing x n i by g 1, we have a quotient a i and a remainder b i such that x n i = a i g 1 + b i where deg(a i ) deg(d) i, deg(b i ) n k 1. Now we can check that are solutions of (4) since deg(u i ) n k 1, u i = b i, v i = a i f 1 deg(v i ) = deg(a i ) + deg(f 1 ) deg(d) i + deg(f 1 ) m 1 and v i g + u i f = f 1 a i d g 1 + b i f = f a i g 1 + f b i = f x n i. Next, we show that for any given Sylvester matrix, when all the elements are allowed to be perturbed, it is always possible to find matrices [F E] with Sylvester structure such that B + F Range(A + E) where B are the first k columns of S and A are the last n + m k columns of S.

4 378 L. Zhi and Z.Yang Theorem 2.2 Given the integers m, n and k, k min(m, n), then there exists a Sylvester matrix S C (m+n) (m+n) with rank m + n k. Proof of Theorem 2.2: For m and n, we always can construct polynomials f, g C[x] such that deg(f) = m, deg(g) = n, and the degree of GCD(f, g) is k. Hence S is the Sylvester matrix generating by f, g and its rank is m + n k. Corollary 2.3 Given the integers m, n and k, k min(m, n), for any Sylvester matrix S = [B A] C (m+n) (m+n), where A C (m+n) (m+n k) and B C (m+n) k, then it is always possible to find a sylvester structure perturbation [F E] C (m+n) (m+n) such that B + F Range(A + E). 3. STLN for Overdetermined Systems with Sylvester Structure In this section, we will illustrate how to solve the overdetermined system AX B (5) where A C (m+n) (m+n k) and B C (m+n) k, and [B A] is the Sylvester matrix generated by polynomials f, g of degrees m and n respectively. We are to find minimum Sylvester structure perturbation [F E] such that B + F Range(A + E) The case: k = 1 In this case, (5) can be transformed into the following overdetermined system: Ax b (6) where A C (m+n) (m+n 1) and b C (m+n) 1, S = [b A] is a Sylvester matrix. According to Theorem 2.2 and Corollary 2.3, there always exists Sylvester structure perturbation [f E] so that (b + f) Range(A + E). In the following, we illustrate how to find the minimum solution using STLN. First, suppose the Sylvester matrix S is generated by f = a m x m + a m 1 x m a 1 x + a 0, g = b n x n + b n 1 x n b 1 x + b 0. The Sylvester structure perturbation [f E] of S can be represented by a vector η C (m+n+2) 1 (η 1, η 2,, η m+n+1, η m+n+2 ) T, (7) when [f E] =. η η m η 2 η η m+3 η m η m+1 η m 0 0 η m+n+2 η m+n η m η m+n+2

5 Approximate GCD of Univariate Polynomials 379 f is the first column of the above matrix. Thus, we can find a matrix P 1 R (m+n) (m+n+2) such that f = P 1 η. ( ) Im+1 0 P 1 = (8) 0 0 where I m+1 is a (m + 1) (m + 1) identity matrix. We define the structured residual r = r(η, x) = b + f (A + E)x. (9) We are going to solve the equality-constrained least squares problems: min η,x η 2, subject to r = 0. (10) Formulation (10) can be solved by using the penalty method [1], (10) is transformed into: min η,x wr(η, x) η w 1, (11) 2 where w is a large penalty value such as As in [21, 20], we use a linear approximation to r(η, x) to compute the minimization of (11). Let η and x represent a small change in η and x respectively, and E represents the corresponding change in E. The first order approximation to r(η + η, x + x) is r(η + η, x + x) = b + P 1 (η + η) (A + E + E)(x + x) b + P 1 η (A + E)x + P 1 η (A + E) x Ex Introducing a Sylvester structure matrix Y 1 C (m+n) (m+n+2) satisfies that Y 1 η = E x, (12) where x = (x 1, x 2 x m+n 1 ) T. Y 1 has the form as Y 1 = 0 x n x 1... xn xn x n 1 x 1 x m+n 1 x n x n 1 x m+n 1 m+1 n+1 (11) can be written as min x η ( w(y1 P 1 ) w(a + E) I m+n+2 0 ) ( η x ) + ( wr η ) 2 (13)

6 380 L. Zhi and Z.Yang EXAMPLE 3.1 Suppose m = n = 3, k = 1, A = 0 0 b a 3 0 b 2 b 3 0 a 2 a 3 b 1 b 2 b 3 a 1 a 2 b 0 b 1 b 2 a 0 a 1 0 b 0 b 1 0 a b 0, b = a 3 a 2 a 1 a Then and Y 1 = P 1 = ( I x x x 4 x x 2 x x 5 x 4 x x 2 x x 5 x 4 x x 2 x x 5 x x x 5 ) It is easy to see that the coefficient matrix of the vector [ η x] T is also of block Toeplitz structure. We could apply fast least squares method[12] to solve it quickly The case: k > 1 In this case, (5) can be transformed into the following overdetermined system: AX B (14) where A C (m+n) (m+n k) and B C (m+n) k, [B A] has Sylvester structure. By stacking the columns of X = [x 1,..., x k ] and B = [b 1,..., b k ] into long vectors, (14) can be written as the following equivalent structured overdetermined system with a single right hand side vector A x 1 b 1 0 A... 0 x b A x k b k However, since the dimension of the diagonal matrix diag(a,... A) is k(m+n) k(m+n k). The size of the linear overdetermined system (15) increases quadratically along with k. The algorithm we proposed in the above subsection may suffer from efficiency and stability problems. In the following, rather than working on the big system (15), we introduce the k-th Sylvester matrix S k which is a (m + n k + 1) (m + n 2k + 2) submatrix of S = [B A] by deleting the last k 1 rows of S and the last k 1 columns of coefficients of f and g (15)

7 Approximate GCD of Univariate Polynomials 381 separately in S, S k = a m b n a m 1 a m 0 0 b n 1 b n a b 0 n k+1 For k = 1, S 1 = S is the Sylvester matrix. m k+1. (16) Theorem 3.1 Given f, g C[x], deg(f) = m, deg(g) = n. 1 k min(m, n). S(f, g) is the Sylvester matrix of f and g, S k is the k th Sylvester matrix of f and g. Then the following statements are equivalent: (a) rank(s) m + n k. (b) deg(gcd(f, g)) k. (c) rank(s k ) m + n + 1 2k. (d) Rank deficiency of S k is greater than one. Proof of Theorem 3.1: The equivalence between (a) and (b) is a classic result. Since the column dimension of S k is m + n + 2 2k, it is obvious (c) and (d) are equivalent. We only need to prove that (a) and (d) are equivalent. Suppose rank(s) m + n k. Denote p = GCD(f, g), f = p u and g = p v, where GCD(u, v) = 1. Then d = deg(p) k, deg(u) m k and deg(v) n k. We have It can be certified that v f u g = 0. [0 n k deg(v), v, 0 m k deg(u), u] T is a non-zero null vector of S k, where u and v are coefficient vectors of the polynomials u and v respectively. So the rank deficiency of S k is at least one. On the other hand, suppose the rank deficiency of S k is greater than one, then there exists a vector w C (m+n+2 2k) 1 such that S k w = 0. We can form polynomials v and u from the first n k + 1 rows and the last m k + 1 rows of w. It is easy to check that v f + u g = 0. Suppose p = GCD(f, g), since f/p is a factor of u, we have deg(f/p) deg(u) m k = deg(p) k. By the above theorem, the overdetermined system (14) can be reformulated as single right hand side vector A k x b k (17)

8 382 L. Zhi and Z.Yang where S k = [b k A k ]. The Sylvester structure perturbation [f k E k ] of S k can still be represented by the vector η = (η 1, η 2,, η m+n+1, η m+n+2 ) T as (7), and η η m η 2 η η m+3 η m [f k E k ] = 0 0 η m+1 η m 0 0 η m+n+2 η m+n η m η m+n+2 n k+1 m k+1 As in the subsection 3.1, we can find a matrix P k C (m+n k+1) (m+n+2) has the similar form as (8) such that f k = P k η, and a Sylvester structure matrix Y k C (m+n k+1) (m+n+2) satisfies Y k η = E k x, Y k has the Sylvester structure 0 x n+1 k. x... 1 xn+2 k xn+1 k Y k = x n k x 1 x m+n+1 2k x n+2 k x n k m+1 x m+n+1 2k n+1 We can use STLN to solve the following minimization problem ( ) ( ) ( min w(yk P k ) w(a k + E k ) η wr x η + I m+n+2 0 x η ) (18) 2 (19) EXAMPLE 3.2 Suppose m = n = 3, k = 2, then the submatrix S 2 = [b 2 A 2 ] where 0 b 3 0 a a 3 b 2 b 3 a A 2 = a 2 b 1 b 2, b 2 = a 1, P 2 = a 1 b 0 b 1 a a 0 0 b Y 2 = x x x 3 x x x 3 x x x 3 x x x 3

9 Approximate GCD of Univariate Polynomials Approximate GCD Algorithm and Experiments The following algorithm is designed for solving the problem (2.1). Algorithm AppSylv-k Input - A Sylvester matrix S generated by two polynomials f and g of degrees m n respectively. An integer 1 k n and a tolerance tol. Output- Two polynomials f and g with rank(s( f, g)) m+n k. and the Euclidean distance f f g g 2 2 is reduced to a minimum. 1. Compute the k-th Sylvester matrix S k as (16), set b k the first column of S k and A k be the last m + n 2k + 1 columns of S k. Let E k = 0, f k = Compute x from min A k x b k 2 and r = b k A k x. Form P k and Y k as showed in the above section. 3. Repeat (a) min x η ( w(yk P k ) w(a k + E k ) I n+m+2 0 (b) Set x = x + x, η = η + η. ) ( η x ) + ( wr η (c) Construct the matrix E k and f k from η, and Y k from x. Set A k = A k + E k, b k = b k + f k, r = b k A k x. until ( x 2 tol and η 2 tol) 4. Output the polynomials f and g formed from [b k A k ]. Given a tolerance ɛ, the algorithm AppSylv-k can be used to compute an ɛ-gcd of polynomials f and g with degrees m n respectively. The method starts with k = n m, using AppSylv-k to compute the minimum N = f f 2 + g g 2 with rank(s( f, g)) m + n k. If N < ɛ, then we can compute the ɛ-gcd from the matrix S k ( f, g) [10, 27]; Otherwise, we reduce k by one and repeat the AppSylv-k algorithm. Another method tests the degree of ɛ-gcd by computing the singular value decomposition of Sylvester matrix S(f, g), find the upper bound degree r of the ɛ-gcd as shown in [7, 9]. So we can start with k = r rather than k = n to compute the certified ɛ-gcd of the highest degree. EXAMPLE 4.1 Given polynomials p = x 2 + 2x + 4, f = p (5 + 2x) x x , g = p (5 + x) x x , and ɛ = 10 3, we are going to compute the ɛ-gcd of f and g. Set tol = 10 3, k = 2, and S is the Sylvester matrix of f and g. The singular values of S are: , , , 2.472, , ) 2

10 384 L. Zhi and Z.Yang Construct the submatrix S 2 of S S 2 = Apply the algorithm AppSylv-k to S 2, after three iterations of the step 2, we obtain the minimal perturbed S 2 of rank deficiency 1 S 2 = The singular values of S 2 are: We can form the polynomials f, g from S 2 The minimum perturbation is , , , f = x x x g = x x x N = f f g g 2 2 = It is easy to run the algorithm again for k = 3, we will find the least perturbation for f and g have a degree three GCD is So for ɛ = 10 3, we can say that the highest degree of ɛ-gcd of f and g is 2. Moreover, this GCD can be computed from S 2 as shown in [10] p = x x The forward error of this ɛ-gcd is: p p 2 / p 2 = EXAMPLE 4.2 The following example is given in Karmarkar and Lakshman s paper [16]. f = x 2 6x + 5 = (x 1)(x 5), g = x 2 6.3x = (x 1.1)(x 5.2). We are going to find the minimal perturbation that f and g have a common root. We consider this problem in two cases: the leading coefficients can be perturbed and the leading coefficients can not be perturbed.

11 Approximate GCD of Univariate Polynomials Case 1 : the leading coefficients can be perturbed. Apply the algorithm AppSylv-k to f, g with k = 1 and tol = 10 3, after three iterations, we obtain polynomials f and g : with minimum distance as f = x x , g = x x , N = f f g g 2 2 = The common root of the f and g is Case 2: the leading coefficients can not be perturbed. This case corresponds to that the first and the fourth term of η are fixed as one. Running the algorithm AppSylv-k for k = 1 and tol = 10 3, after three iterations, we get the polynomials f and g : with minimum distance as f = x x , g = x x , N = f f g g 2 2 = The common root of the f and g is In the paper[16], they restricted to the second case where the perturbed polynomials remained to be monic. The minimum perturbation they obtained is which corresponding to the perturbed common root Ex deg(f), coeff. iterative. iterative backward backward deg(g) k error num.(chu) num. error(zeng) error σ k σ k 1 2, e e e e , e e e e , e e e , e e e e , e e e e , e e e , e e e e , e e e e Table 1. Algorithm performance on benchmarks(univariate case) In Table 1, we show the performance of our new algorithms for computing ɛ-gcd of univariate polynomials randomly generated in Maple. deg(f) and deg(g) denote the degree of the polynomials f and g. k is a given integer. coeff. error is the range of perturbations we add to f and g. iterative num.(chu) is the number of the iteration needed by Chu s[6] method; whereas iterative num. denotes the number of iterations by AppSylv-k algorithm. σ k and σ k are the last k-th singular value of S(f, g) and S( f, g) respectively. We show that the backward error computation. Suppose the polynomial p is GCD(f,g), u and v are the primary part of f and g respectively. Then the backward error in Table 1 is f p u g p v 2 2. backward error(zeng) denotes the backward error by Zeng s algorithm[27]; whereas backward error is the backward error by our algorithm.

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