When are Two Numerical Polynomials Relatively Prime?

Transcription

1 J Symbolic Comptation (1998) 26, Article No sy When are Two Nmerical Polynomials Relatively Prime? BERNHARD BECKERMANN AND GEORGE LABAHN Laboratoire d Analyse Nmériqe et d Optimisation, Université des Sciences et Technologies de Lille, Villeneve d Ascq Cedex, France Department of Compting Science, University of Waterloo, Waterloo, Ontario, Canada Let a and b be two polynomials having nmerical coefficients We consider the qestion: When are a and b relatively prime? Since the coefficients of a and b are approximant, the qestion is the same as: When are two polynomials relatively prime, even after small pertrbations of the coefficients? In this paper we provide a nmeric parameter for determining whether two polynomials are prime, even nder small pertrbations of the coefficients Or methods rely on an inversion formla for Sylvester matrices to establish an effective criterion for relative primeness The inversion formla can also be sed to approximate the condition nmber of a Sylvester matrix c 1998 Academic Press 1 Introdction Let C[z] be the space of polynomials over the complex nmbers and let a, b C[z] be polynomials a(z) =a 0 + a 1 z + + a m z m, b(z) =b 0 + b 1 z + + b n z n, a m,b n 0 of degree m and n, respectively The greatest common divisor (GCD) of a and b is given by gcd(a, b)(z) = (z γ), where a(z) =a m (z α), b(z) =b n (z β) γ A B This is well defined by the fndamental theorem of algebra We are interested in the qestion: when are two polynomials, a, b relatively prime, that is, when do a and b have no common roots? In the case of exact arithmetic determining whether two polynomials are relatively prime is well-known This is not the case in the presence of finite precision arithmetic In this case a compter will not necessarily decide correctly whether two given polynomials with rational coefficients are coprime For instance, after transforming the coefficients of the polynomials α A a(z) =(z 1 3 )(z 5 3 )=z2 2z + 5 9, b(z) =z bbecker@anoniv-lille1fr glabahn@daisywaterlooca /98/ $3000/0 c 1998 Academic Press β B

2 678 B Beckermann and G Labahn into (decimal) floating-point nmbers, the reslting polynomials are coprime Also, the polynomials a(z) =50z 7, b(z) =z 1 7 are not coprime within a precision of two (decimal) digits A more reliable compter answer may be expected for the problem of deciding whether two polynomials remain coprime even after pertrbation of coefficients by qantities bonded in norm by some ɛ This is the type of problem that is of interest in applications sch as robotics and control theory (Kailath, 1980; Sederberg and Chang, 1993) where the inpt data are only known p to some fixed accracy or where noise is present in the inpt parameters In this paper we provide a parameter to determine coprimeness of two nmeric polynomials This parameter is based on qantities which are efficiently obtainable Indeed in Beckermann and Labahn (1998) we present an algorithm for compting this parameter that is both nmerically stable and at the same time is typically an order of magnitde faster than alternative methods Becase of this efficiency, compting this parameter as an initial test for coprimeness may always be done before starting the more expensive comptation of an ɛ-gcd (Corless et al, 1995; Emiris et al, 1997; Karmarkar and Lakshman, 1996; Noda and Sasaki, 1991; Schönhage, 1985) In fact, we are very mch interested in determining some non-trivial nmerical ɛ-gcd if the answer to the above qestion is no This problem has been treated by several athors each with a different notion of greatest common divisor These inclde methods that are based on optimization techniqes (Corless et al, 1995; Karmarkar and Lakshman, 1996) which are probably nmerically stable bt qite expensive and others which are more or less based on classical Eclidean concepts (Emiris et al, 1997; Noda and Sasaki, 1991) bt for which one is nable to garantee nmerical stability (Beckermann and Labahn, 1998) Finally we mention the qasi-gcd of Schönhage (1985) where the se of an oracle makes it difficlt to jdge the practical se It is well known that the Sylvester matrix of two polynomials plays a vital role in determining the GCD of two polynomials The magnitde of the inverse of the Sylvester matrix is important in determining the distance to the closest polynomials having a common root In or case, we se a new inversion formla for the Sylvester matrix to obtain an estimate of the magnitde of the inverse in terms of only the magnitde of the first and last colmns of the inverse We show that or estimate is better for determining the distance to the closest polynomials having a common root than that provided by the magnitde of the inverse of the Sylvester matrix The remainder of the paper is organized as follows In the next section we place or problem in a linear algebraic setting making se of Sylvester s matrix Section 3 gives a new inverse formla for Sylvester s matrix while or new coprime measre follows in Section 4 Section 5 gives a refinement of or primeness measre The final sections inclde some examples and give a conclsion Notation For the remainder of this paper we make se of the following notation: we denote the 1-Hölder vector norm on C n as well as the sbordinate matrix norm by For c C[z], c(z) =c c n z n we set c =(c 0,,c n ) T as the vector of coefficients Or norm on C[z] is given by c := c = j c j,

3 When are Two Nmerical Polynomials Relatively Prime? 679 and on C[z] r s, the space of r s matrices with polynomial entries, by (c j,k ) := ( c j,k ) = max c j,k k Note that for all c, d C[z] wehave c d c d, and this ineqality also holds for polynomial matrices of appropriate size With this notation we can restate or problem as follows Definition 11 For a, b C[z] let ɛ(a, b) :=inf{ (a a,b b ) : (a,b ) have a common root, deg a m, deg b n, that is, any polynomials a, b satisfying (a a,b b ) ɛ<ɛ(a, b) and the above degree restrictions are coprime We will then refer to a, b as being ɛ-prime We are interested in compting approximately sharp simple lower bonds for ɛ(a, b) j 2 Inversion of Sylvester s Matrix It is well-known that the GCD problem can be placed in a linear algebra setting This has the advantage that it allows one to make se of concepts from nmerical linear algebra (sch as condition nmber) to give information on the nmerical GCD problem Let S(a, b) denote the Sylvester matrix for (a, b), that is, a b a 1 a 0 0 S(a, b) = a m a0 0 a m a 1 0 {{ 0 a m n b 1 b 0 0 b n b0 0 b n b {{ b n m C (m+n) (m+n) Sylvester s criterion from 1853 states that two polynomials are relatively prime iff S(a, b) is non-singlar (see, eg, Geddes et al (1992)) Nmerically the following lemma is known Lemma 21 For any two polynomials a and b we have 1 ɛ(a, b) S(a, b) 1 Proof According to or choice of matrix norms we have m m S(a, b) = max a j, b j = (a, b) j=0 Conseqently, sing a theorem from Gastinel (Higham, 1996, Theorem 65, p 123) we j=0

4 680 B Beckermann and G Labahn obtain ɛ(a, b) = inf{ S(a, b) S(a,b ) : S(a,b ) singlar 1 min{ S(a, b) B : B singlar = S(a, b) 1 Remark 22 In the case of the Eclidean norm, we have j =1,, m + n : σ j = min{ S(a, b) B 2 : defect(b) j, with σ 1 σ 2 σ m+n being the singlar vales of S(a, b) This allows one to define the ɛ-defect of the Sylvester matrix, which has been chosen by Corless et al (1995) as the degree of some ɛ-gcd Remark 23 From the proof of Lemma 21 we see that the qantity (a, b) /ɛ(a, b) may be considered as a strctred 1-condition nmber of S(a, b) in the class of Sylvester matrices (ie we consider only pertrbations of S(a, b) being themselves Sylvester matrices) More generally, the distance to the set of polynomials with GCD having a certain degree (see Corless et al (1995), Emiris et al (1997), Karmarkar and Lakshman (1996)) may be nderstood as a strctred singlar vale (with respect to the 1-Hölder norm) of a Sylvester matrix ɛ j (a, b) := min{ S(a, b) S(a,b ) : defect(s(a,b )) j = min{ (a, b) (a,b ) : degree of GCD of (a,b ) is at least j Lemma 21 states that if we pertrb the coefficients of or polynomials by any ɛ less than the reciprocal of the norm of the inverse of the Sylvester matrix then we still have relatively prime polynomials In fact, a test for coprimeness based on the size of the norm of the inverse of the Sylvester matrix is already inclded as a special case in the SVD GCD algorithm proposed by Corless et al (1995, p 198), and Emiris et al (1997) However, in or case we do not want to estimate the reciprocal of the norm of the inverse by singlar vale decomposition of the Sylvester matrix This decomposition is expensive and does not take advantage of the special strctre of a Sylvester matrix Or goal is to find an easily comptable bond that lies between ɛ(a, b) and the reciprocal of the norm of the inverse This gives a criterion for nmerical coprimeness that is both more precise and also less expensive to compte than previos methods Note that C[z] is a principal ideal domain, so that we have a, b = gcd(a, b) for any two polynomials a, b (where denotes the ideal generated by the specific elements) Ths, determining whether a and b are relatively prime is the same as solving Eqation (1) is the same as, v C[z], deg <m,deg v<n: a v + b =1 (1) [ v S(a, b) ] =(1, 0,,0) T (2) so that two polynomials are relatively prime iff one can determine the first colmn of the inverse of their corresponding Sylvester matrix That this is eqivalent to Sylvester s criterion is obvios from the next lemma which gives the inverse of a Sylvester matrix entirely in terms of the first colmn of its inverse

5 When are Two Nmerical Polynomials Relatively Prime? 681 Lemma 24 Let f(z) =f 1 z f 1 m n z 1 m n = (z) a(z) + O(z m n ) z Then S(a, b) is invertible with inverse given by v b v n 1 v b n 1 b a m a m 1 a (3) 0 f 1 f 1 m n 0 0 f 1 Proof Note that eqation (1) gives (z) a(z) + v(z) b(z) = 1 a(z) b(z) = O(z m n ) z and so f(z) = v(z) b(z) + O(z m n ) z Thswehave a b a 1 b a m a0 b n b0 0 0 a1 0 b S(a, b)= 0 0 a m 0 0 b n 0 f 1 f 1 m n v f 1 m 1 0 vn v0 0 0 m v n 1 The inverse formla follows directly by mltiplying the right-hand side of the previos eqation with the matrix on the left of eqation (3) Remark 25 We note that for or Sylvester inversion formlae it is not important that b has precise degree n In fact in the case m = deg a n deg b all formlas remain valid Remark 26 Similar inversion formla can also be derived for matrices that express

6 682 B Beckermann and G Labahn information abot the existence of common roots, in particlar for the Bézot matrix of two polynomials If we assme, withot loss of generality, that m = deg a deg b, and choose m = n, then the Sylvester matrix S(a, b) has size 2m 2m, and we may partition it into for sqare blocks as follows [ ] L(a) L(b) S(a, b) = U(a) U(b) In this case, the matrix B(a, b) :=U(a) L(b) U(b) L(a) coincides p to some reordering of colmns and rows with the Bézot of a and b as considered by Fiedler (1986, Chapter 7, p 164ff) By making some block maniplations and sing a similar argment as in Lemma 24 we obtain B(a, b) 1 = f m f m 1 f 1 2m f 1 m f m f 2 2m f 1 f 2 f m In other words, the inverse of B(a, b) is a (Toeplitz) block fond in the factorization of the inverse of S(a, b) 3 Coprime Parameters For or prposes we se or inversion formla to obtain information on the magnitde of the inverse of a Sylvester matrix In this section we give an pper bond for the norm of the inverse of a Sylvester matrix This gives s an (initial) nmerical parameter that can be sed to determine whether two polynomials are coprime Theorem 31 Let, v be polynomials of degrees at most m 1 and n 1 respectively, solving eqation (1) Then ] ] v [ S(a, b) 1 v [ +2 f (a, b) (4) Proof Since (v, ) defines the first colmn of the inverse of S(a, b) the ineqality on the left of eqation (4) follows directly from the definition of or polynomial and matrix norms The bond on the right follows from or inverse formla Theorem 31 gives a bond for the norm of the inverse of the Sylvester matrix in terms of the cofactors, v, and the easily comptable first coefficients f j of the power series /a However, it still remains to determine how good (or bad) sch a bond will be In particlar, we need to determine the size of the coefficients f j As a first step we note that Sylvester s matrix has a certain interesting dality property Namely, let a(z) =z m a(1/z), b(z) =z n b(1/z), (5) that is, a(z) =a m + a m 1 z + + a 0 z m, b(z) =b n + b n 1 z + + b 0 z n The Sylvester matrices S(a, b) and S(a,b) are the same p to reordering of rows and colmns In particlar, their inverses have the same matrix norms As sch it is of interest

7 to look at soltions to the diophantine eqations When are Two Nmerical Polynomials Relatively Prime? 683 a(z) ṽ(z)+b(z) ũ(z) =1 (6) with ũ, ṽ being polynomials of degrees m 1 and n 1, respectively Letting (z) = z m 1 ũ(1/z) and v(z) =z n 1 ṽ(1/z), eqation (6) is the same as [ ] a(z) v(z)+b(z) (z) =z m+n 1 v, that is, S(a, b) =(0,,0, 1) T (7) The polynomials v, define a Padé approximant (Cabay and Meleshko, 1993) of type (m 1,n 1) for the power series b(z)/a(z) Let [ κ := v v ] { [ v = max ], v [ ] We may combine the reslts of Theorem 31 and eqation (7), and obtain at the same time an pper bond for f Corollary 32 With, v and,v soltions of eqations (1) and (7) we have κ S(a, b) 1 κ +2 f (a, b), (8) where f = v v Frthermore, f κ 2 Proof The two ineqalities in eqation (8) are clear from Theorem 31 To determine f we have that f(z) z 1 m n and so [v(z) (z) (z) v(z)] = (z) a(z) z1 m n [v(z) (z) (z) v(z)] + O(z m n ) z a(z) = z1 m n [b(z) (z) (z)+a(z) (z) v(z)] + O(z m n ) z a(z) = z 1 m n (z) a(z) + O(z m n ) z = O(z m n ) z, f(z) =z 1 m n [v(z) (z) (z) v(z)] Nmerical experiences seem to indicate (Cabay and Meleshko, 1993; Cabay et al, 1997) that, for correctly scaled a and b, the qantity S(a, b) 1 is proportional to κ and not of size κ 2 A slight generalization of Corollary 32 gives s more information abot a class of polynomials (a, b) where this property is tre First notice that for any Larent polynomial g(z) =g 1 z 1 + g 2 z 2 + we have (v(z)+g(z) b(z)) (z) ((z) g(z) a(z)) v(z) =z m+n 1 f(z)+g(z) = z m+n 1 f(z)+o(z 1 ) In other words, the polynomial part of the left-hand side eqals f, and f (, v) T (v + g b, v + g a), where we may choose a g to improve the pper bond for f given

8 684 B Beckermann and G Labahn in Corollary 32 More generally, let g a, g b be polynomials verifying deg g a <m+ n, deg g b <m+ n, z n a(z)g a (z)+z m b(z)g b (z) = z 2(m+n) 1 + O(z m+n 1 ) z (9) Then we have that (g a, g b ) (, v) T = z 2(m+n) 1 f + O(z m+n 1 ) z, again allowing for an estimate of f Using Theorem 31 we obtain Corollary 33 Denote by ρ m+n (a, b) the minimm of the set of all prodcts (a, b) (g a,g b ) T where the pair (g a,g b ) verifies eqation (9) Then with, v soltions of eqation (1) we have [ ] v ) [ ] S(a, b) ( ρ m+n (a, b) v Note that ρ m+n (a, b)/ (a, b) may be estimated above for instance by (,v) T in terms of the cofactors of the diophantine eqation (7), or by ρ m+n (a, 0)/ a (resp ρ m+n (0,b)/ b ), the norm of the polynomial obtained by the first m + n coefficients of the power series at zero of a(z) 1 (and of b(z) 1, respectively) Therefore, the qantity ρ m+n (a, b) may be close to one even if the Sylvester matrix S(a, b) is ill-conditioned (see for instance the nmerical reslts of Beckermann and Labahn (1998)) 4 Closest Common Roots In the previos section we obtained an pper bond (cf Corollary 32) for the norm of the inverse of the Sylvester matrix Assming, for the time being, that the comptation of both (v, ) and (v,) can be done in an efficient way (cf Beckermann and Labahn (1998)), we will have an effective method of determining when two polynomials are relatively prime The only drawback to the above method is that or parameter (in this case 1/(κ +2 f (a, b) ) which is a lower bond for 1/ S(a, b) 1 and hence for ɛ(a, b)), may be too small since it cold potentially be of the order of 1/κ 2 In order to obtain a more precise bond we reqire a more detailed stdy for determining ɛ(a, b) The following statement is probably well-known, however for the sake of completeness we provide a proof Theorem 41 We have ( ) ɛ(a, b) = inf a(z) (1,z m ), b(z) (1,z n (10) ) z C where C := C { The infinm on the right-hand side is attained for a z =: ccr(a, b) C (called the closest common root ) Proof Let h(a, b, z) = ( a(z) (1,z m ), b(z) (1,z ) ) To see that ɛ(a, b) h(a, b, z) for some n z C, let a and b have the common root z From Hölder s ineqality we get a(z) = a(z) a (z) a a 1 (1, z,, z m ) T = a a max{1, z m with a similar ineqality for b Therefore (a a,b b ) = max{ a a, b b h(a, b, z)

9 When are Two Nmerical Polynomials Relatively Prime? 685 Taking the infinm on both sides leads to the first half of or assertion Note that the fnction z h(a, b, z) is continos over C, and therefore attains its minimm on the nit disk Also, we have h(a, b, z) =h(a,b, 1/z), leading to { inf h(a, b, z) = min z C inf h(a, b, z), inf h(a,b,z) z 1 z 1 showing that the infinm in eqation (10) is attained To show eqality in eqation (10), sppose that z is the closest common root of a, b, and consider { a (z) =a(z) a(z 1, if z ) < 1, (z/z ) m, otherwise, along with a similar b Remark 42 We see from the proof of Theorem 41 that ccr(a, b) is in fact the common root of the polynomials a,b which nder all pairs of non-coprime polynomials have minimal distance to a, b Here ccr(a, b) = is eqivalent to saying that deg a m 1 and deg b n 1 Also, = min (a(z),b(z)) min (a(z),b(z)) if ccr(a, b) 1, z 1 z 1 ɛ(a, b) = min (a(z),b(z)) min (a(z),b(z)) if ccr(a, b) 1, z 1 z 1 with a,b as in eqation (5) Remark 43 A statement similar to Theorem 41 can also be made for other Hölder vector norms, and one may in addition consider weighted norms (sefl, for example, in cases where only some of the coefficients may have inaccracies) For instance, let α, β C[z] be of degree m, and n, respectively, with positive coefficients Then (compare also Corless et al (1995, Remark 4)) { m a j a n j inf 2 b j b j + 2 : (a,b ) have a common root, α j j=0 j=0 β j deg a m, deg b n a(z) 2 = inf z C α( z 2 ) + b(z) 2 β( z 2 ) Corless et al (1995, Section 26) and Karmarkar and Lakshman (1996) proposed to apply standard optimization algorithms for calclating a nmerical GCD, and in particlar for determining sch a 2-conterpart of ɛ(a, b) Of corse, for the problem of coprimeness it is preferable to take the above expression on the right since the nmber of free parameters is redced from m + n + 1 to 1 One easily shows that ɛ(a, b) =ɛ(b, a) =ɛ(a,b), and that ɛ(a, b) min{ a, b Also, it seems to be clear that a, b may not be ɛ-prime if they have zeros that are too close In fact, denoting by z a a zero of a, and by z b a zero of b, respectively, we may show the estimate z a z b ɛ(a, b) max{m a,n b max{1, z a max{1, z b, where the distance of zeros is measred in some chordal metric,

10 686 B Beckermann and G Labahn From Lemma 21 and Theorem 41 we have ɛ(a, b) 1 S(a, b) 1 = min y 0 y S(a, b), ɛ(a, b) = min y z C y(z) S(a, b), y(z) where y(z) =(1, z,, z m+n 1 ) At present or coprimeness parameter reqires the potentially large overestimate for S(a, b) 1 given by Theorem 31 and Corollary 32 Can we improve this, for example by the following? {{ ɛ(a, b) 1 { κ = min 1 S(a, b) 1 e 1, 1 S(a, b) 1? e m+n In other words, can the norm of the inverse be replaced by only the norm of the first and/or last colmn of the inverse? Corollary 44 There holds ɛ(a, b) 1 κ, and, more precisely, min (a(z),b(z)) 1 z 1 (v, ) T, min (a(z),b(z)) 1 z 1 (v,) T Proof In view of Remark 42, the estimate for ɛ(a, b) is a conseqence of the other two estimates In order to prove the second one, notice that [ ] v(z) (a(z),b(z)) (z) 1 min (a(z),b(z)) min [ ] z 1 z 1 v(z) ] v (z) [ Here we have sed the fact that, for every polynomial matrix U, there holds max U(z) U z 1 Finally, the third estimate follows by symmetry Remark 45 From Remark 42 and the proof of Corollary 44 we see that, provided ccr(a, b) 1, we have the estimate ɛ(a, b) 1/ (,v ) T for any polynomials,v satisfying a v + b = 1, even if the degree constraints of (1) are not valid Ths, the bonds of Corollary 44 may be improved by considering (,v )=(, v)+α (a, b), where α C[z] is chosen in order to minimize the norm of (,v ) T In this context it is interesting to mention that by the Corona Theorem (Nikol skii, 1986, Appendix 3) we may find fnctions #,v # analytic and bonded in the nit disk (ie elements of the Hardy space H ) sch that a v # + b # =1, max # (z) 2 + v # (z) 2 1 z 1 ɛ + 7 log 1/ɛ +20 log 1/ɛ ɛ 2, (11) provided that ɛ a(z) 2 + b(z) 2 1 for all z 1 Conseqently, in the case (a, b) T 1 it seems to be possible to find polynomials (,v ) by a sitable limiting procedre with 1/ (,v ) T lying between ɛ(a, b) and roghly its sqare Remark 42 and Corollary 44 tell s that it is sfficient to solve only a single diophantine eqation in order to determine an effective bond for ɛ(a, b), provided that we know

11 When are Two Nmerical Polynomials Relatively Prime? 687 in advance that the closest common root lies in or otside the nit disk In some cases, sch a localization of the closest common root may be given Lemma 46 Sppose the roots of a and b all lie in the nit disk Then ccr(a, b) 1 Proof From (5) and Theorem 41 we know that ɛ(a, b) = min{min (a(z),b(z)), min (a(z),b(z)) z 1 z 1 Ths for the assertion of Lemma 46 it is sfficient to show that a(z) a(z) (and analogosly that b(z) b(z) ) for all z < 1, where as sal z denotes the complex conjgate of z) If x 1,,x m are the roots of a, then this follows from a(z) a(z) = a(z) m z m a(1/z) = j=1 z x j 1 z x j being less than or eqal to one for any z < 1, since z x / 1 x z 1 for all x, z lying in the nit disk For example, sppose that the roots of the polynomial c lie in the nit disk By the Gaß Lcas Theorem (Marden, 1966, p 22), the zeros of the derivative of c lie in the convex hll of the set of its zeros, and hence also in the nit disk Ths if a, b are any (higher order) derivatives of c, then ccr(a, b) 1 5 Examples In order to illstrate and to compare the findings of the preceding sections, we consider the following three simple examples Example 51 Let a(z) =z n 1, and b(z) =b b n z n with b 1 Then [ ] [ ] In L S(a, b) = and S(a, b) 1 (U + L) 1 U (U + L) = 1 L I n U (U + L) 1 (U + L) 1 Therefore S(a, b) 1 [ 1 1 S(a, b) 1 =2, (L + U) 1 n, 2 ] n 2 The matrix L + U is circlant, with eigenvales b(ω j ), j =0, 1,,n 1, where ω is a primitive n-th root of nity Moreover, L + U is normal, and therefore (L + U) 1 2 = max{1/ λ : λ is an eigenvale of L + U From Lemma 21 and Theorem 41 we then have B := min b(ω j 1 ) ɛ(a, b) j S(a, b) 1 B 1 2 n Note also that Corollary 33 applies in this context with ρ 2n (a, b) 4, giving [ ] [ ] v S(a, b) v

12 688 B Beckermann and G Labahn Example 52 Let a(z) =z m, and b(z) =b b n z n with b 1 In this case the cofactors can be obtained explicitly: from b(z) (z) =1+O(z m ) z 0 via the Taylor expansion of 1/b at zero and v as a corresponding remainder Then [ ] 0 L S(a, b)=, S(a, b) I n U 1 =1, [ ] S(a, b) 1 U L 1 I = n S(a, b) 1 1 L 1, [1, 2], 0 the latter observation being in accordance with Corollary 33 since ρ m+n (a, b) =1For example, if b(z) =(1 2z)/3 then m 1 1 S(a, b) 1 ɛ(a, b) a(1/2)=2 m, 1 a conseqence of (z) =3 (1+2z +4z 2 + +(2z) m 1 ) Example 53 b = 1, and with This gives With the same setting as in Example 52, let b(z) = ( 1 z 2 )m Then m 1 ( m 1+j ) (z) =2 m (1 z) m + O(z m )=2 m z j j =2 m j=0 ( 2m 1 ) 23m 1 m 1 π m πm 8 m S(a, b) 1 1 ɛ(a, b) =b(1/3)=3 m Ths the criterion of Corollary 44 does not always yield sharp bonds, since for large m we have 1/κ 1/ S(a, b) 1 1 ɛ(a, b) From Example 53 we also see that a small ɛ(a, b) in general does not imply that a has a root which is close to one of the roots of b 6 Conclsion We have considered the problem of determining when two polynomials are nmerically relatively prime A parameter has been given that improves a previos existing measre for nmerical primeness A sharper measre can be given in the case where it is known that the two polynomials have all their roots in the nit disk This parameter is based on qantities which are efficiently obtainable in a nmerically stable way (Beckermann and Labahn, 1998) The efficiency and nmerical correctness of sch a comptation makes a good initial test for coprimeness before starting the more expensive comptation of an ɛ-gcd References Beckermann, B, Labahn, G (1998) A fast, nmerically stable Eclidean-like algorithm for detecting relatively prime nmerical polynomials J Symb Compt 26, Cabay, S, Jones, A R, Labahn, G (1997) Experiments with a weakly stable algorithm for compting Padé Hermite and simltaneos Padé approximants ACM Trans Math Softw, 23,

13 When are Two Nmerical Polynomials Relatively Prime? 689 Cabay, S, Meleshko, R (1993) A weakly stable algorithm for Padé approximants and the inversion of Hankel matrices SIAM J Matrix Anal and Applications, 14, Corless, R M, Gianni, P M, Trager, B M, Watt, S M (1995) The singlar vale decomposition for polynomial systems Proceedings ISSAC 95, pp New York, ACM Press Emiris, I, Galligo, A, Lombardi, H (1997) Certified approximate nivariate GCDs J Pre Appl Algebra, 117, Fiedler, M (1986) Special Matrices and their Application in Nmerical Mathematics Dordrecht, Martins Nijhoff Geddes, K O, Czapor, SR, Labahn, G (1992) Algorithms for Compter Algebra Boston, MA, Klwer Higham, N J (1996) Accracy and Stability of Nmerical Algorithms Philadelphia, SIAM Kailath, T (1980) Linear Systems, Englewood Cliffs, NJ, Prentice-Hall Karmarkar, N, Lakshman, Y N (1996) Approximate polynomial greatest common divisors and nearest singlar polynomials Proceedings ISSAC 96 pp New York, ACM Press Marden, M (1966) Geometry of polynomials, Math Srveys 3, Providence, RI, American Mathematical Society Nikol skii, N K (1986) Treatise of the Shift Operator Berlin, Heidelberg, Springer Noda, M-T, Sasaki, T (1991) Approximate GCD and its applications to ill-conditioned algebraic eqations JCAM, 38, Schönhage, A (1985) Qasi-GCD comptations, J Complexity, 1, Sederberg, T W, Chang, G Z (1993) Best linear common divisors for approximate degree redction, Compt Aided Des, 25, Originally Received 15 March 1997 Accepted 8 Janary 1998