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

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1 MM Reseach Pepints, MMRC, AMSS, Academia Sinica No 23, Decembe 24 Computing Appoximate GCD of Multivaiate Polynomials by Stuctue Total Least Nom 1) Lihong Zhi and Zhengfeng Yang Key Laboatoy of Mathematics Mechanization Institute of Systems Science, AMSS, Academia Sinica Beijing 18, China (lzhi, Abstact The poblem of appoximating the geatest common diviso(gcd) fo multivaiate polynomials with inexact coefficients can be fomulated as a low ank appoximation poblem with Sylveste matix This pape pesents a method based on Stuctued Total Least Nom(STLN) fo constucting the neaest Sylveste matix of given lowe ank We pesent algoithms fo computing the neaest GCD and a cetified ɛ-gcd fo a given toleance ɛ The unning time of ou algoithm is polynomial in the degees of polynomials We also show the pefomance of the algoithms on a set of multivaiate polynomials 1 Intoduction Appoximate GCD computation of multivaiate polynomials has been studied by many authos [5, 11, 15, 16, 17] Paticulaly, in [5, 16, 17], Singula Value Decomposition(SVD) of the Sylveste matix has been used to obtain an uppe degee bound of the appoximate GCD Suppose the SVD decomposition of Sylveste matix is S = UΣV T, whee U and V ae othogonal and Σ = diag(σ 1, σ 2,, σ n ) Then σ k is the 2-nom distance to the neaest matix of ank stictly less than k Howeve, SVD computation is not appopiate fo computing the distance to the neaest STRUCTURE(fo example Sylveste stuctue) matix of given ank deficiency In [4], they mentioned stuctued low ank appoximation to compute appoximate GCD The basic idea of thei method is to altenate pojections between the sets of the ank k matices and stuctue matices so that the ank constain and the stuctual constaint ae satisfied altenatively while the distance in between is being educed Although the algoithm is linea convegent, the speed may be vey slow We pesent a method based on STLN[12] fo stuctue peseving low ank appoximation of a Sylveste matix STLN is an efficient method fo obtaining an appoximate solution to an ovedetemined linea system AX B, peseving the given linea stuctue in the minimal petubation [E F ] such that (A + E)X = B + F An algoithm in [12] has been descibed fo Hankel stuctue peseving low ank appoximation using STLN with L p nom In this pape, we adjust thei algoithm fo computing stuctue peseving ank eduction 1) Patially suppoted by a National Key Basic Reseach Poject of China and Chinese National Science Foundation unde Gant 14135

2 Appoximate GCD of Multivaiate Polynomials 389 of Sylveste matices Futhemoe, we also deive a stable and efficient way to compute the neaest GCD and ɛ GCD of multivaiate polynomials The oganization of this pape is as follows In Section 2, we intoduce some notations and discuss the equivalence between the GCD poblems and the low ank appoximation of Sylveste matices In Section 3, we conside solving ovedetemined system with Sylveste stuctue based on STLN Finally, in Section 4, we pesent algoithms based on STLN fo computing the neaest GCD and ɛ-gcd We also illustate the expeiments of ou algoithms on a set of andom multivaiate polynomials 2 Peliminaies In this pape we ae concened with the following task PROBLEM 21 Given two polynomials f, g C[x 1, x 2 x ] with deg(f) = m, deg(g) = n, deg denotes the total degees of the multivaiate polynomials Fo a given positive intege k, we ae going to find the minimal petubation f g 2 2, such that deg( f) m, deg( g) n and deg(gcd(f + f, g + g)) k Suppose S(f, g) is the Sylveste matix that is geneated by the polynomials f and g Fist, we quote the binomial numbe β(k, ): ( ) k + β(k, ) = (1) We show how to constuct the Sylveste matix S of multivaiate polynomials intoduce a tivial lemma[6] Lemma 21 Let f, g C[x 1, x ] be nonzeo polynomials, f 1 = f/gcd(f, g) and g 1 = g/gcd(f, g) Then all the solutions u, v C[x 1, x ] to the equation must be of the fom whee q C[x 1, x ] We uf + vg = (2) u = g 1 q, v = f 1 q (3) Fom the equation (2), we can know that it is a linea system fo the coefficients of u and v We equie that deg(u) deg(g) 1, deg(v) deg(f) 1 (4) Suppose deg(gcd(f, g)) = k, by(3) and (4), all the solutions fo u and v ae detemined by q C[x 1,, x ] with deg(q) k 1 The dimension of the solution space fo u and v is β(k 1, ), ie the ank deficiency of M is β(k 1, ) Hence, PROBLEM 21 can be fomulated as : min f f 2 deg(gcd( f, g)) k 2 + g g 2 2 min f f g g 2 2 dim Nullspace( S) β(k 1,) whee S is the Sylveste matix geneated by f and g

3 39 LZhi and ZYang 3 STLN fo Ovedetemined Systems with Sylveste Stuctue 31 The case: k = 1 In this case, β(k 1, ) = 1, Hence, PROBLEM 21 can be fomulated as : min f f 2 deg(gcd( f, g)) g g 2 2 min f f g g 2 2 dim Nullspace( S) 1 whee S is the Sylveste matix geneated by f and g If we solve the ovedetemined system using STLN [12, 13, 14] A x b (5) fo Sylveste stuctued S = [A 1 b A 2 ], whee b is a column of S and A = [A 1 A 2 ] ae the emainde columns of S, we obtain a minimum stuctued petubation f and E = [E 1 E 2 ] such that b + f Range(A 1 + E 1 A 2 + E 2 ), ie, S = [A 1 + E 1 b + f A 2 + E 2 ] is a Sylveste matix and dim Nullspace( S) 1 EXAMPLE 31 Suppose f = x 2 + xy = x (x + y), g = x 2 + 2xy + 3x + y 2 + 3y = (x + y + 3) (x + y), and S is the Sylveste matix of f and g S = The ank deficiency of S is β(, 2) = 1 We patition S by two ways: S = [A 1 b 1 ] = [b 2 A 2 ], whee b 1 is the last column of S and b 2 is the fist column of S The ovedetemined system A 1 x = b 1 has no solution, while the system A 2 x = b 2 has an exact solution as x = [ 1, 3,, 1, ] T Theoem 31 Given f, g C[x 1, x ] with deg(f) = m, deg(g) = n Suppose S is the Sylveste matix of f and gthen the following statements ae equivalent: (a) dim Nullspace(S) 1 (b) We can choose a column b of S and patition S as S = [A 1 b A 2 ] such that Ax = b has a solution, whee A = [A 1 A 2 ]

4 Appoximate GCD of Multivaiate Polynomials 391 Poof of Theoem 31: (b) = (a) Suppose Ax = b has a nonzeo solution, then b Range(A) Theefoe, it is obvious that the dimension of the null space of S = [A 1 b A 2 ] is at least one (a) = (b) Suppose d = GCD(f, g), f 1 = f/d, g 1 = g/d Then deg(d) = k 1, deg(f 1 ) = m k m 1 and deg(g 1 ) = n k n 1 Choosing a vaiable ode: x 1 x 2 x, polynomials can be aanged by gaded lexicogaphic ode If we multiply the vecto v = [x n+m 1 to the matix S, then we have x n k, x 1 x n+m 2,, x n+m 1 1, x n+m 2,, x n+m+2 1,, x 2, x 1, 1] vs = [x n 1 f, x 1 x n 2 f, f, x m 1 g, x 1 x m 2 g, g] Suppose the leading tem of the g 1 is x t 1 1 x t 2 2 x t, hee i=1 t i = n k We can find a monomial x s 1 1 xs 2 2 xs such that i=1 s i n 1 and s i t i, i = 1, 2, Then x t 1 1 x t 2 2 x t is a facto of x s 1 1 xs 2 2 xs Thus we can choose a column b of S such that vb = x s 1 1 xs 2 2 xs f Without loss of geneality, we can assume the leading tem of g 1 is In this case, we can choose the fist column of S as b, because vb = x n 1 f and x n k is a facto of x n 1 Then A ae the emainde columns of S and S = [b A] Next, we need to pove that Ax = b has a solution Multiplying v to two sides of the equation Ax = b, it tuns out to be [x 1 x n 2 f, x 2 x n 2 f,, x n 1 1 f, x 1 f, f, x m 1 g,, x 1 g, g]x = [x n 1 f] (6) The solution x of (6) coesponds to the coefficients of polynomials u 1, v 1, with deg(v 1 ) m 1, deg(u 1 ) n 1 and u 1 has no x n 1 tem, satisfy Dividing x n 1 x n 1 f = u 1 f + v 1 g (7) by g 1, we have a quotient a 1 and a emainde b 1 such that x n 1 = a 1 g 1 + b 1 whee deg(a 1 ) = k 1, deg(b 1 ) n 1 and b 1 has no x n 1 of g 1 is a facto of x n 1 Now we can check that u 1 = b 1, v 1 = a 1 f 1 tem because the leading tem ae solutions of (7) since deg(u 1 ) n 1 and has no x n 1 tem, and deg(v 1 ) = deg(a 1 ) + deg(f 1 ) = k 1 + deg(f 1 ) m 1 v 1 g + u 1 f = f 1 a 1 d g 1 + b 1 f = f a 1 g 1 + f b 1 = x n 1 f Hence, in this case, Ax = b has a solution, and S = [b A] Next, we show that fo any given Sylveste matix, when all the elements ae allowed to be petubed, it is always possible to find f and E with Sylveste stuctue such that b + f Range(A + E), whee b is some column of S, A ae the emainde columns of S

5 392 LZhi and ZYang Theoem 32 Given the positive integes m, n,, thee exists a Sylveste matix S C (β(m+n 1,)) (β(m 1,)+β(n 1,)) with ank deficiency 1 Poof of Theoem 32: Fo m and n, we always can constuct polynomials f, g C[x 1, x 2,, x ] such that deg(f) = m, deg(g) = n, and deg(gcd(f, g)) = 1 Hence S is the Sylveste matix geneated by f, g and its ank deficiency is 1 Coollay 33 Given integes m, n and, fo any Sylveste matix S = [A 1 b A 2 ] and S C (β(m+n 1,)) (β(m 1,)+β(n 1,)), whee b is some column of S and A = [A 1 A 2 ] ae the emainde columns of S, then it is always possible to find a sylveste stuctue petubation f and E such that b + f Range(A + E) Next, we will illustate how to solve the ovedetemined system (5) Fo simplicity of discussion, we choose b as the fist column of S, and A ae the emainde columns of S The ovedetemined system(5) can be fomulated as: Ax b (8) Accoding to Theoem 32 and Coollay 33, thee always exists Sylveste stuctue petubation [f E] such that (b + f) Range(A+E) In the following, we illustate how to find the minimum solution using STLN Suppose f, g C[x 1, x 2,, x ], deg(f) = m, deg(g) = n Denotes: f = a,,,,m x m + a,,,1,m 1 x 1 x m a m 1,1,, x m 1 1 x 2 + a m,,, x m a,,,1 x + + a,1,,, x 2 + a 1,,, x 1 + a,,,, g = b,,,,n x n + b,,,1,n 1 x 1 x n b n 1,1,, x n 1 1 x 2 + b n,,, x n b,,,1 x + + b,1,,, x 2 + b 1,,, x 1 + b,,, Then the Sylveste matix S(f, g) = [b A] of f and g is: [b A] = a,,,m a,,,1,m 1 a,,,m a m,, a m 1,1,, a m,,, a,,,m 1 a,,,1,m 2 a,,,m 1 a,,, } {{ } β(n 1,) b,,,n b,,,1,n 1 b,,,n b n,,, b n 1,1,,, b n,,, b,,,n 1 b,,,1,n 2 b,,,n 1 b,, } {{ } β(m 1,) We use a vecto η C (β(m,)+β(n,)) 1 to epesent β(m, ) + β(n, ) fee elements of the Sylveste matix [f E] We defined η as (η 1, η 2,, η β(m,)+β(n,) 1, η β(m,)+β(n,) ) T

6 Appoximate GCD of Multivaiate Polynomials 393 Since f is the fist column of[f E], then we can find a matix P 1 C β(m+n 1,) (β(m,)+β(n,)) such that f = P 1 η P 1 = Q m+1 Q m Q1 ( ) Ii whee Q i =, i = 1, 2, m + 1, and I i is an i i identity matix, is a (n 1) i zeo matix We define the stuctued esidual = (η, x) = b + f (A + E)x (1) We ae going to solve the equality-constained least squaes poblems: min η,x η 2, subject to = (11) Fomulation (11) can be solved by using the penalty method [1], (11) is tansfomed into: min w(η, x) η,x η, w 1 (12) 2 whee w is a lage penalty value such as 1 1 As in [13, 12], we use a linea appoximation to (η, x) to compute the minimization of (12) Let η and x epesent a small change in η and x espectively, and E epesents the coesponding change in E The fist ode appoximation to (η + η, x + x) is (η + η, x + x) = b + P 1 (η + η) (A + E + E)(x + x) Intoducing a Sylveste stuctue matix Y 1, and b + P 1 η (A + E)x + P 1 η (A + E) x Ex x = (x 1, x 2 x β(m 1,)+β(n 1,) 1 ) T (9) (12) can be witten as ( min w(y1 P 1 ) w(a + E) x η I β(m,)+β(n,) ) ( η x ) + ( w η ) 2 (13) whee Y 1 C β(m+n 1,) (β(m,)+β(n,)) [13] satisfies that Y 1 η = E x (14) In the following, we popose a new method based on the constuction of the Sylveste matix to obtain the matix Y 1 Suppose f, g, E, η and x ae defined as above, multiplying the vecto v = [x m+n 1, x 1 x m+n 2,, x m+n 1 1, x m+n 2, x 1, 1]

7 394 LZhi and ZYang to the two sides of the equation (14), it tuns out to be Let ˆx = [, x], it is obvious that v Y 1 η = v E x v E x = V [f, E] ˆx The ight side of the above equation tuns out to be V [f, E] ˆx = ĥ1û 1 + ĥ2û 2 whee ĥ1 is the polynomial of degee m, geneated by the subvecto of η: [η 1, η 2,, η β(m,) ], ĥ 2 is the polynomial of degee n, geneated by the subvecto of η: [η β(m,)+1, η β(m,)+2,, η β(m,)+β(n,) ], û 1 is the polynomial of degee n 1, geneated by the subvecto of ˆx: [, x 1, x 2,, x β(n 1,) 1 ], û 2 is the polynomial of degee m 1, geneated by the subvecto of ˆx: [x β(n 1,), x β(n 1,)+1,, x β(m 1,)+β(n 1,) 1 ] We denote Y 1 is the matix constucted by the linea system fo the coefficients of ĥ1 and ĥ 2, ie û 1 ĥ 1 + û 2 ĥ 2 = vy 1 η We can show that the matix Y 1 satisfies (14) Hee is a simple example to illustate how to find Y 1 EXAMPLE 32 Suppose k = 1, the matix M is geneated by the bivaiate polynomials f and g, whee f = a + a 1 y + a 1 x + a 2 y 2 + a 11 xy + a 2 x 2, g = b + b 1 y + b 1 x + b 2 y 2 + b 11 xy + a 2 x 2 A = b 2 a 2 b 11 b 2 a 11 b 2 b 11 a 2 b 2 a 2 b 1 b 2 a 1 a 11 b 1 b 1 b 11 a 1 a 2 b 1 b 2 a 1 b b 1 a a 1 b b 1 a b, b = a 2 a 11 a 2 a 1 a 1 a

8 Appoximate GCD of Multivaiate Polynomials 395 Suppose x = [x 1, x 2, x 3, x 4, x 5 ] T, then x 3 x 1 x 4 x 3 x 1 x 4 x 3 x 1 x 4 Y 1 = x 2 x 5 x 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 x 5 x 4 x 2 x 5 x 3 x 2 x 1 x 5 x 4 x 2 x 5 We can see that the matix Y 1 has quasi-sylveste stuctue 32 The case: k > 1 Suppose f, g C[x 1, x ], deg(gcd(f, g)) = k > 1, then the ank deficiency of the Sylveste matix of f and g is β(k 1, ) > 1 We constuct the matix S k by equation(2): whee u, v satisfies uf + vg = deg(u) deg(g) k, deg(v) deg(f) k Accoding to the Lemma 21, suppose g 1 = g/gcd(f, g), f 1 = f/gcd(f, g), we can constuct solutions u = g 1, v = f 1 Since deg(u) = deg g 1 deg(g) k, deg(v) = deg(f 1 ) deg(f) k, and uf + vg = Hence, the dimension of the solution space of uf + vg = is at least 1 Theefoe, the ank deficiency of S k is at least 1 a,,,m b,,,n S k = a,,,1,m 1 a,,,m a m,, a m 1,1,, a m,,, a,,,m 1 a,,,1,m 2 a,,,m 1 a,,, } {{ } β(n k,) Fo k = 1, S k = S is the Sylveste matix of f and g b,,,1,n 1 b,,,n b n,,cdots, b n 1,1,,cdots, b n,,, b,,,n 1 b,,,1,n 2 b,,,n 1 b,, } {{ } β(m k,) Theoem 34 Given f, g C[x 1, x 2,, x ], deg(f) = m, deg(g) = n 1 k min(m, n) S(f, g) is the Sylveste matix of f and g, S k is the k th Sylveste matix of f and g Then deg(gcd(f, g)) k if only if the ank deficiency of S k is at least one Poof of Theoem 34: Suppose deg(gcd(f, g)) k Denote p = GCD(f, g), f = p u and g = p v, whee GCD(u, v) = 1, then d = deg(p) k, deg(u) m k and deg(v) n k and [v f + u g = Accoding to the constuction of S k, it has a non-zeo solution which is geneated by the coefficients of u, v Theefoe, the ank deficiency of S k is at least one

9 396 LZhi and ZYang On the othe hand, suppose the ank deficiency of S k is at least one, then thee exists a vecto w C (m+n+2 2k) 1 such that S k w = We can fom polynomials v and u fom w, whee deg(u) m k and deg(v) n k It is easy to check that v f + u g = Suppose p = GCD(f, g), then since f/p is a facto of u, we have deg(f/p) deg(u) m k = deg(p) k Now we can efomulate the PROBLEM 21 as : min f f 2 deg(gcd( f, g)) k 2 + g g 2 2 min f f g g 2 2 dim Nullspace( S k ) 1 whee S k is the matix geneated by f and g Theoem 35 Given f, g C[x 1, x ] with deg(f) = m, deg(g) = n Suppose S k is the matix geneated by f and g Then the following statements ae equivalent: (a) deg(gcd(f, g)) k (b) We can choose a column b k and patition S k as S k =[A k1 b k A k2 ] such that A k x = b k has a solution, whee A k = [A k1 A k2 ] Poof of Theoem 35: Suppose A k x = b k have a solution, then b k Range(A k ) Since b k is the column of S k Theefoe, it is obvious that the dimension of null space of S k = [A k1 b k A k2 ] is not less than one Accoding to Theoem 34, deg(gcd(f, g)) k On the othe hand, suppose deg(gcd(f, g)) k Simila to the poof of Theoem 31, we can find the column b k such that A k x = b k has a solution Fo simplicity of discussion, we choose b k as the fist column of S k Solving the ovedetemined system A k x b k (15) whee S k = [b k A k ], we obtain a minimum Sylveste stuctued petubation f k and E k such that b k + f k Range(A k + E k ) Theefoe, S k = [b + f A k + E k ] is a solution having Sylveste stuctue and ank deficiency of S k is at least one The Sylveste stuctue petubation [f k E k ] of S k can still be epesented by the vecto (η 1, η 2,, η β(m,)+β(n,) 1, η β(m,)+β(n,) ) T As in the subsection 31, we can find a matix P k C β(m+n k,) (β(m,)+β(n,)) such that f k = P k η P k = Q m+1 Q m Q1 (16)

10 ( Ii ) Appoximate GCD of Multivaiate Polynomials 397 whee Q i =, i = 1, 2, m + 1, and I i is an i i identity matix, is a (n k) i zeo matix Using the simila way as constucting Y 1, we can find a Sylveste stuctue matix Y k C (m+n k+1) (m+n+2) satisfies Y k η = E k x Now we can apply STLN method to solve the following minimization poblem ( ) ( ) ( ) min w(yk P k ) w(a k + E k ) η w x η + I β(m,)+β(n,) x η (17) 2 EXAMPLE 33 Suppose m = n = 3, k = 2, the Sylveste matix S is geneated by the bivaiate polynomials f and g, whee f = a + a 1 y + a 1 x + a 2 y 2 + a 11 xy + a 2 x 2 + a 3 y 3 + a 12 xy 2 + a 21 x 2 y + a 3 x 3, g = b + b 1 x + b 1 x + b 2 y 2 + b 11 xy + b 2 x 2 + b 3 y 3 + b 12 xy 2 + b 21 x 2 y + b 3 x 3 Then the matix S 2 = [b 2 A 2 ] is A 2 = Suppose x = [x 1, x 2, x 3, x 4, x 5 ] T, then Y 2 = b 3 a 3 b 12 b 3 a 12 b 21 b 12 a 21 b 3 b 21 a 3 b 3 a 3 b 2 b 3 a 2 a 12 b 11 b 2 b 12 a 11 a 21 b 2 b 11 b 21 a 2 a 3 b 2 b 3 a 2 b 1 b 2 a 1 a 11 b 1 b 1 b 11 a 1 a 2 b 1 b 2 a 1 b b 1 a a 1 b b 1 a b, b = a 3 a 12 a 21 a 3 a 2 a 11 a 2 a 1 a 1 a x 3 x 1 x 4 x 3 x 1 x 4 x 3 x 1 x 4 x 3 x 1 x 4 x 2 x 5 x 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 x 5 x 4 x 2 x 5 x 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 x 5 x 4 x 2 x 5 x 3 x 2 x 1 x 5 x 4 x 2 x 5

11 398 LZhi and ZYang 4 Appoximate GCD Algoithm and Expeiments The following algoithm is designed fo solving the poblem (21) Algoithm AppSylv-k Input - A Sylveste matix S geneated by two polynomials f and g of degees m n espectively An intege 1 k n and a toleance tol Output- Two polynomials f and g with dim Nullspace( S( f, g) ) β(k 1, ) and the Euclidean distance f f g g 2 2 is educed to a minimum 1 Compute the k-th Sylveste matix S k as the above section, set b k the fist column of S k and A k the emainde columns of S k Let E k =, f k = 2 Compute x fom min A k x b k 2 and = b k A k x Fom P k and Y k as showed as the above section 3 Repeat (a) min x η ( w(yk P k ) w(a k + E k ) I β(m,)+β(n,) (b) Set x = x + x, η = η + η ) ( η x ) + ( w η (c) Constuct the matix E k and f k fom η, and Y k fom x Set A k = A k + E k, b k = b k + f k, = b k A k x until ( x 2 tol and η 2 tol) 4 Output the polynomials f and g fomed fom [b k A k ] EXAMPLE 41 Given polynomials p = 3 + y + 5x + 5y 2 + xy + 2x 2, ) f = p (3 + 4y + 3x) + 4y + 3x + 1xy + 4y 3, g = p (2 + y + 5x) xy + 6y 2 + 4x 3, and ɛ = 1 2 We ae going to compute the ɛ-gcd of f and g Set tol = 1 3 and k = 2, β(1, 2) = 3, S is the Sylveste matix of f and g The last seveal singula values of S ae: , , 8342, 4921,

12 Appoximate GCD of Multivaiate Polynomials 399 Constuct the 2 th Sylveste matix S 2 S 2 = Applying the algoithm AppSylv-k to the matix S 2, afte thee iteations of the step 2, we obtain the minimum petubed S 2 of ank deficiency 1 S 2 = The last seveal singula values of S2 ae: We can fom the polynomials f, g fom S 2 : , 47762, , f = y + 242x y xy + 218x y xy x 2 y + 613x 3, g = y x y xy x 2 The minimum petubation is +513y xy x 2 y + 115x 3 f f g g 2 2 = 5831

13 4 LZhi and ZYang It is easy to un the algoithm again fo k = 3, we find the least petubation fo f and g have a degee thee GCD is 3583 So fo ɛ = 1 2, we can say that the highest degee of ɛ-gcd of f and g is 2 Moeove, this GCD can be computed fom S 2 as shown in [6] p = y + 4x + 434y xy x 2 The fowad eo of this ɛ-gcd is: p 1249 p 2 / p 2 = 779 Ex deg(f), coeff iteative backwad backwad deg(g) k eo num eo(zeng) eo σ β(k 1,) σ β(k 1,) 1 3, e e 8 221e 8 173e , e e 8 35e 8 814e , e e 2 529e 3 389e , , e e 3 259e 4 119e , e e 3 626e 4 224e , e e 2 352e 2 155e , e e 1 139e 1 378e Table 1 Algoithm pefomance on benchmaks(bivaiate case) In Table 1, we show the pefomance of ou new algoithms fo computing ɛ-gcd of bivaiate polynomials andomly geneated in Maple deg(f) and deg(g) denote the degee of the polynomials f and g k is a given intege coeff eo is the ange of petubations we add to f and g iteative num denotes the numbe of iteations by AppSylv-k algoithm σ k and σ k ae the last β(k 1, )-th singula value of S(f, g) and S( f, g) espectively We show that the backwad eo computation Suppose the polynomial p is GCD(f,g), u and v ae the pimay pat of f and g espectively Then the backwad eo in Table 1 is f p u g p v 2 2, whee backwad eo(zeng) denotes the backwad eo by Zeng s algoithm[16] and backwad eo is the backwad eo computed by ou algoithm Refeences [1] Anda, AA and Pak, H Fast plane with dynamic scaling SIAM J Matix Anal Appl vol15, pp , 1994 [2] Anda, AA and Pak, H Self-scaling fast otations fo stiff and equality-constained linea least squaes poblems Linea Algeba and its Applications vol234, pp , 1996 [3] PChin, RMColess, CFColess Optimization stategies fo the appoximate GCD poblems In Poc 1998 Intenat Symp Symbolic Algebaic Comput (ISSAC 98) pp [4] Moody TChu, ERobet and JRobetPlemmons Stuctued Low Rank Appoximation, manuscipt, 1998 [5] RMColess, PMGianni, BMTage, and SMWatt The singula value decomposition fo polynomial systems In Poc 1995 Intenat Symp Symbolic Algebaic Comput (IS- SAC 95) pp [6] SGao, EKaltofen, JMay, ZYang and LZhi Appoximate factoization of multivaiate polynomials via diffeential equations In Poc 24 Intenat Symp Symbolic Algebaic Comput (ISSAC 4) pp

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