Bernhard Beckermann Villeneuve d'ascq Cedex, France. and

Transcription

1 When are two nmerical polynomials relatiely prime? Bernhard Beckermann Laboratoire d'analyse Nmeriqe et d'optimisation, Uniersite des Sciences et Technologies de Lille, 5955 Villenee d'ascq Cedex, France e{mail and George Labahn Department of Compting Science Uniersity of Waterloo, Waterloo, Ontario, Canada e{mail Jan, 998 Abstract Let a and b be two polynomials haing nmerical coecients We consider the qestion when are a and b relatiely prime? Since the coecients of a and b are approximant, the qestion is the same as when are two polynomials relatiely prime, een after small pertrbations of the coecients? In this paper we proide a nmeric parameter for determining that two polynomials are prime, een nder small pertrbations of the coecients Or methods rely on an inersion formla for Sylester matrices to establish an eectie criterion for relatie primeness The inersion formla can also be sed to approximate the condition nmber of a Sylester matrix Introdction Let C[z] be the space of polynomials oer the complex nmbers and let a; b C[z] be polynomials a(z) = a 0 + a z + + a m z m ; b(z) = b 0 + b z + + b n z n ; a m ; b n = 0 of degree m and n, respectiely The GCD of a and b is gien by gcd (a; b)(z) = Y A\B(z? ); where a(z) = a m Y A(z? ); b(z) = b n Y B (z? ) This is well dened by the Fndamental Theorem of Algebra We are interested in the qestion when are two polynomials, a; b relatiely prime, that is, when do a and b hae no common roots? In the case of exact arithmetic determining if two polynomials are relatiely prime is well known This is not the case in the presence of nite precision arithmetic In this case a compter will not necessarily decide correctly whether two gien polynomials with rational coecients are coprime For instance, after transforming the coecients of the polynomials a(z) = (z? )(z? 5 ) = z? z ; b(z) = z? into (decimal) oating point nmbers, the reslting polynomials are coprime Also, the polynomials a(z) = 50z? ; b(z) = z?

2 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 een after pertrbation of coecients by qantities bonded in norm by some This is the type of problem that is of interest in applications sch as robotics and control theory [, ] where the inpt data is only known p to some xed accracy or where noise is present in the inpt parameters In this paper we proide a parameter to determine coprimeness of two nmeric polynomials This parameter is based on qantities which are eciently obtainable Indeed in [] 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 alternate methods Becase of this eciency, compting this parameter as an initial test for coprimeness may always be done before starting the more expensie comptation of an {GCD [5,, 0,, 5] In fact, we are ery mch interested in determining some non{triial nmerical {GCD if the answer to the aboe qestion is no This problem has been treated by seeral athors each with a dierent notion of greatest common diisor These inclde methods that are based on optimization techniqes [5, 0] which are probably nmerically stable bt qite expensie and others which are more or less based on classical Eclidean concepts [, ] bt for which one is nable to garantee nmerical stability [] Finally we mention the qasi-gcd of Schonhage [5] where the se of an oracle makes it diclt to jdge the practical se It is well known that the Sylester matrix of two polynomials plays a ital role in determining the greatest common diisor of two polynomials The magnitde of the inerse of the Sylester matrix is important in determining the distance to the closest polynomials haing a common root In or case, we se a new inersion formla for the Sylester matrix to obtain an estimate of the magnitde of the inerse in terms of only the magnitde of the rst and last colmns of the inerse We show that or estimate is better for determining the distance to the closest polynomials haing a common root than that proided by the magnitde of the inerse of the Sylester matrix The remainder of the paper organized is as follows In the next section we place or problem in a linear algebraic setting making se of Sylester's matrix Section gies a new inerse formla for Sylester's matrix while or new \coprime measre follows in Section Section 5 gies a renement of or primeness measre The nal sections inclde some examples and gie a conclsion Notation For the remainder of this paper we make se of the following notation we denote the {Holder ector norm on C n as well as the sbordinate matrix norm by jj jj For c C[z], c(z) = c c n z n we set ~c = (c 0 ; ; c n ) T as the ector of coecients Or norm on C[z] is gien by jjcjj = jj~cjj = X j jc j j; and on C[z] rs, the space of r s matrices with polynomial entries, by jj(c j;k )jj = jj(jjc j;k jj)jj = max k X j jjc j;k jj Note that for all c; d C[z] we hae jjc djj jjcjj jjdjj, and this ineqality also holds for polynomial matrices of appropriate size With this notation we can restate or problem as follows Denition For a; b C[z] let (a; b) = inffjj(a? a ; b? b )jj (a ; b ) hae a common root, deg a m; deg b ng;

3 that is, any polynomials a, b satisfying jj(a?a ; b?b )jj < (a; b) and the aboe degree restrictions are coprime We will then refer to a; b as being {prime We are interested in compting approximately sharp \simple lower bonds for (a; b) Inersion of Sylester's Matrix It is well-known that the greatest common diisor problem can be placed in a linear algebra setting This has the adantage that it allows one to make se of concepts from nmerical linear algebra (sch as condition nmber) to gie information on the nmerical gcd problem Let S(a; b) denote the Sylester Matrix for (a; b), that is, S(a; b) = a a a 0 0 a m a0 0 a m a 0 0 a m {z } n b b b 0 0 b n b0 0 b n b 0 0 b n 5 {z } m C (m+n)(m+n) Sylester's criterion from 85 states that two polynomials are relatiely prime if and only if S(a; b) is non{singlar (see, eg, [8]) Nmerically it is known that Lemma For any two polynomials a and b we hae Proof (a; b) jjs(a; b)? jj According or choice of matrix norms we hae jjs(a; b)jj = maxf mx j=0 ja j j; mx j=0 jb j jg = jj(a; b)jj Conseqently, sing a Theorem of Gastinel [9, Theorem 5, p] we obtain (a; b) = inffjjs(a; b)? S(a ; b )jj S(a ; b ) singlarg minfjjs(a; b)? Bjj B singlarg = jjs(a; b)? jj Remark In the case of the Eclidean norm, we hae j = ; ; m + n j = minfjjs(a; b)? Bjj defect(b) jg; with m+n being the singlar ales of S(a; b) This allows one to dene the {defect of the Sylester matrix, which has been chosen by Corless, Gianni, Trager and Watt [5] as the degree of some {GCD

4 Remark From the proof of Lemma we see that the qantity jj(a; b)jj=(a; b) may be considered as a strctred {condition nmber of S(a; b) in the class of Sylester matrices (ie, we consider only pertrbations of S(a; b) being themseles Sylester matrices) More generally, the distance to the set of polynomials with GCD haing a certain degree (see [5,, 0]) may be nderstood as a strctred singlar ale (with respect to the {Holder norm) of a Sylester matrix j (a; b) = minfjjs(a; b)? S(a ; b )jj defect(s(a ; b )) jg = minfjj(a; b)? (a ; b )jj degree of GCD of (a ; b ) is at least j g Lemma states that if we pertrb the coecients of or polynomials by any less than the reciprocal of the norm of the inerse of the Sylester matrix then we still hae relatiely prime polynomials In fact, a test for coprimeness based on the size of the norm of the inerse of the Sylester matrix is already inclded as a special case in the SVD GCD algorithm proposed by Corless, Gianni, Trager and Watt [5, p98], and in [, Algorithm ] of Emiris, Galligo and Lombardi Howeer, in or case we do not want to estimate the reciprocal of the norm of the inerse by the singlar ale decomposition of the Sylester matrix This decomposition is expensie and does not take adantage of the special strctre of a Sylester matrix Or goal is to nd an easily comptable bond that lies between (a; b) and the reciprocal of the norm of the inerse This gies a criterion for nmerical coprimeness that is both more precise and also less expensie to compte than preios methods Note that C[z] is a principal ideal domain, so that we hae < a; b >=< gcd (a; b) > for any two polynomials a; b (where < > denotes the ideal generated by the specic elements) Ths, determining if a and b are relatiely prime is the same as soling Eqation () is the same as 9 ; C[z]; deg < m; deg < n a + b = () S(a; b) ~ ~ = ; 0; ; 0) T () so that two polynomials are relatiely prime if and only if one can determine the rst colmn of the inerse of their corresponding Sylester matrix That this is eqialent to Sylester's criterion is obios from the next lemma which gies the inerse of a Sylester matrix entirely in terms of the rst colmn of its inerse Lemma Let f(z) = f? z? + + f?m?n z?m?n inertible with inerse gien by b0 0 0 n? bn? b ?a0 0 0 m? 0 0 0?am??a0 0 0 = (z) a(z) + O(z?m?n ) z! f? f?m?n 0 0 f? Then S(a; b) is 5 () Proof Note that eqation () gies (z) a(z) + (z) b(z) = a(z)b(z) = O(z?m?n ) z! and so

5 f(z) =? (z) + b(z) O(z?m?n ) z! Ths we hae f? f?m?n 0 0 f? 5 S(a; b) = a0 0 0 b0 0 0 a b 0 0 am a0 bn b0 0 a 0 b 0 0 am 0 0 bn 0 0 0? ?n? ?0 0 0 m? 0 0?n? m? The inerse formla follows directly by mltiplying the right side of the preios eqation with the matrix on the left of eqation () 5 Remark 5 We note that for or Sylester inersion formla it is not important that b has precise degree n In fact in the case m = deg a n deg b all formlas remain alid Remark Similar inersion formla can also be deried for matrices that express information abot the existence of common roots, in particlar for the Bezot matrix of two polynomials If we assme, withot loss of generality, that m = deg a degb, and choose m = n, then the Sylester matrix S(a; b) has size m m, and we may partition it into for sqare blocks as follows S(a; b) = L(a) U(a) L(b) 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 Bezot of a and b as considered by Fiedler [, Chapter, p] By making some block maniplations and sing a similar argment as in Lemma we obtain B(a; b)? = f?m f?m? f?m f?m f?m f?m f? f? f?m In other words, the inerse of B(a; b) is a (Toeplitz) block fond in the factorization of the inerse of S(a; b) 5 Coprime Parameters For or prposes we se or inersion formla to obtain information on the magnitde of the inerse of a Sylester matrix In this section we gie an pper bond for the norm of the inerse of a Sylester 5

6 matrix This gies s a (initial) nmerical parameter that can be sed to determine if two polynomials are coprime Theorem Let ; be polynomials of degrees at most m? and n? soling eqation () Then jjs(a; b)? jj + jjfjj jj(a; b)jj () Proof Since (; ) denes the rst colmn of the inerse of S(a; b) the ineqality on the left of () follows directly from the denition of or polynomial and matrix norms The bond on the right follows from or inerse formla Theorem gies a bond for the norm of the inerse of the Sylester matrix in terms of the cofactors ;, and the easily comptable rst coecients f j of the power series =a Howeer it still remains to determine how good (or bad) sch a bond will be In particlar, we need to determine the size of the coecients f j let that is, As a rst step we note that Sylester's matrix has a certain interesting dality property Namely, a(z) = z m a(=z); b(z) = z n b(=z); (5) a(z) = a m + a m? z + + a 0 z m ; b(z) = b n + b n? z + + b 0 z n The Sylester matrices S(a; b) and S(a; b) are the same p to reordering of rows and colmns In particlar, their inerses hae the same matrix norms As sch it is of interest to look at soltions to the diophantine eqations a(z) ~(z) + b(z) ~(z) = () with ~; ~ being polynomials of degrees m? and n?, respectiely Letting (z) = z m? ~(=z) and (z) = z n? ~(=z), eqation () is the same as a(z) (z) + b(z) (z) = z m+n? ; that is, S(a; b) ~ ~ = 0; ; 0; ) T () The polynomials ; dene a Pade approximant [] of type (m? ; n? ) for the power series?b(z)=a(z) Let = = maxf ; We may combine the reslts of Theorem and (), and obtain at the same time an pper bond for jjfjj g Corollary With ; and ; soltions of () and () we hae where jjfjj = jj? jj Frthermore, jjfjj jjs(a; b)? jj + jjfjj jj(a; b)jj; (8)

7 Proof The two ineqalities in (8) are clear from Theorem To determine jjfjj we hae that f(z)? z?m?n [(z) (z)? (z) (z)] = (z)? a(z) z?m?n [(z) (z)? (z) (z)] + O(z?m?n ) z! a(z) = z?m?n [b(z) (z) (z) + a(z) (z) (z)] + O(z?m?n ) z! a(z) = z?m?n (z) a(z) + O(z?m?n ) z! = O(z?m?n ) z! ; and so f(z) = z?m?n [(z) (z)? (z) (z)] Nmerical experiences seem to indicate [, ] that, for correctly scaled a and b, the qantity jjs(a; b)? jj is proportional to and not of size A slight generalization of Corollary gies 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? z? + g? z? + we hae ((z) + g(z) b(z)) (z)? ((z)? g(z) a(z)) (z) = z m+n? f(z) + g(z) = z m+n? f(z) + O(z? ) In other words, the polynomial part of the left hand side eqals f, and jjfjj jj(; ) T jj jj( + g b; + g a)jj, where we may choose a g to improe the pper bond for jjfjj gien in Corollary More generally, let g a, g b be polynomials erifying 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 (m+n)? + O(z m+n? ) z! (9) Then we hae that (g a ;?g b ) (; ) T = z (m+n)? f + O(z m+n? ) z!, again allowing for an estimate of jjfjj Using Theorem we obtain Corollary Denote by m+n (a; b) the minimm of the set of all prodcts jj(a; b)jjjj(g a ; g b ) T jj where the pair (g a ; g b ) eries (9) Then with ; soltions of () we hae jjs(a; b)? jj + m+n (a; b) Note that m+n (a; b)=jj(a; b)jj may be estimated aboe for instance by jj(; ) T jj in terms of the cofactors of the diophantine eqation (), or by m+n (a; 0)=jjajj (resp m+n (0; b)=jjbjj), the norm of the polynomial obtained by the rst m + n coecients of the power series at zero of a(z)? (and of b(z)?, respectiely) Therefore, the qantity m+n (a; b) may be close to one een if the Sylester matrix S(a; b) is ill{conditioned (see for instance the nmerical reslts of []) Closest Common Roots In the preios section we obtained an pper bond (cf Corollary ) for the norm of the inerse of the Sylester matrix Assming, for the time being, that the comptation of both (; ) and (; ) can be done in an ecient way (cf []), we will hae an eectie method of determining when two polynomials are relatiely prime The only drawback to the aboe method is that or parameter (in this case =( + jjfjj jj(a; b)jj) which is a lower bond for =jjs(a; b)? jj and hence for (a; b)), may

8 be too small since it cold potentially be on the order of = In order to obtain a more precise bond we reqire a more detailed stdy for determining (a; b) The following statement is probably well known, howeer for the sake of completeness we proide a proof Theorem We hae (a; b) = inf zc a(z) jj(; z m )jj ; b(z) (0) jj(; z n )jj where C = C [ fg The inmm on the right side is attained for a z = ccr (a; b) C (called the \closest common root) a(z) Proof Let h(a; b; z) = jj(;z m )jj ; a and b hae the common root z From Holder's ineqality we get b(z) jj(;z n )jj To see that (a; b) h(a; b; z) for some z C, let ja(z)j = ja(z)? a (z)j jj~a? ~a jj jj(; z; ; z m ) T jj = jja? a jj maxf; jzj m g with a similar ineqality for b Therefore jj(a? a ; b? b )jj = maxfjja? a jj; jjb? b jjg h(a; b; z) Taking the inmm on both sides leads to the rst half of or assertion Note that the fnction z! h(a; b; z) is continos oer C, and therefore attains its minimm on the nit disk Also, we hae h(a; b; z) = h(a; b; =z), leading to inf h(a; b; z) = minf inf h(a; b; z); inf h(a; b; z)g; zc showing that the inmm in (0) is attained To show eqality in (0), sppose that z is the closest common root of a; b, and consider ( a (z) = a(z)? a(z if jz ) j <, (z=z ) m otherwise, along with a similar b Remark We see from the proof of Theorem that ccr (a; b) is in fact the common root of the polynomials a ; b which nder all pairs of not coprime polynomials hae minimal distance to a; b Here ccr (a; b) = is eqialent to saying that deg a m? and deg b n? Also, (a; b) with a; b as in (5) 8 >< > = min = min jj(a(z); b(z))jj min jj(a(z); b(z))jj if jccr (a; b)j, jj(a(z); b(z))jj if jccr (a; b)j, jj(a(z); b(z))jj min Remark A statement similar to Theorem can also be made for other Holder ector norms, and one may in addition consider weighted norms (sefl, for example, in cases where only some of the 8

9 coecients may hae inaccracies) For instance, let ; C[z] be of degree m, and n, respectiely, with positie coecients Then (compare also [5, Remark ]) inf n m X j=0 ja j? a j j j + nx j=0 ja(z)j = inf zc (jzj ) + jb(z)j (jzj ) jb j? b j j j (a ; b ) hae a common root, deg a m; deg b n Corless et al [5, Section ] and Karmarkar et al [0] proposed to apply standard optimization algorithms for calclating a nmerical GCD, and in particlar for determining sch a {conterpart of (a; b) Of corse, for the problem of coprimeness it is preferable to take the aboe expression on the right since the nmber of free parameters is redced from m + n + to o One easily shows that (a; b) = (b; a) = (a; b), and that (a; b) minfjjajj; jjbjjg Also, it seems to be clear that a; b may not be {prime if they hae zeros that are too close In fact, denoting by z a a zero of a, and by z b a zero of b, respectiely, we may show the estimate (a; b) maxfm jjajj; n jjbjjg where the distance of zeros is measred in some \chordal metric From Lemma and Theorem we hae (a; b) jjs(a; b)? jj = min y=0 jz a? z b j maxf; jz a jg maxf; jz b jg ; jjy S(a; b)jj ; (a; b) = min jjyjj zc jjy(z) S(a; b)jj ; jjy(z)jj where y(z) = (; z; ; z m+n? ) At present or \coprimeness parameter reqires the potentially large oerestimate for jjs(a; b)? jj gien by Theorem and Corollary Can we improe this, for example by the following (a; b)? z} { = min n jjs(a; b)? e jj ; jjs(a; b)? e m+n jj In other words, can the norm of the inerse be replaced by only the norm of the rst and/or last colmn of the inerse? o? Corollary There holds (a; b), and, more precisely, min jj(a(z); b(z))jj jj(; ) T jj ; min jj(a(z); b(z))jj jj(; ) T jj Proof In iew of Remark, the estimate for (a; b) is a conseqence of the other two estimates In order to proe the second one, notice that min jj(a(z); b(z))jj min (a(z); b(z)) (z) (z) (z) (z) Here we hae sed the fact that, for eery polynomial matrix U, there holds max jju(z)jj jjujj z Finally, the third estimate follows by symmetry 9

10 Remark 5 From Remark and the proof of Corollary we see that, proided jccr (a; b)j, we hae the estimate (a; b) =jj( ; ) T jj for any polynomials ; satisfying a + b =, een if the degree constraints of () are not alid Ths, the bonds of Corollary may be improed by considering ( ; ) = (; ) + (a;?b), where C[z] is chosen in order to minimize the norm of ( ; ) T In this context it is interesting to mention that by the Corona Theorem [, Appendix ] we may nd fnctions ; analytic and bonded in the nit disk (ie, elements of the Hardy space H ) sch that a + b = ; max qj p (z)j + j (z)j + log = + 0 log = ; () proided that p ja(z)j + jb(z)j for all jzj Conseqently, in the case jj(a; b) T jj it seems to be possible to nd polynomials ( ; ) by a sitable limiting procedre with =jj( ; ) T jj lying between (a; b) and roghly its sqare Remark and Corollary tell s that it is scient to sole only a single diophantine eqation in order to determine an eectie bond for (a; b), proided that we know in adance 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 gien Lemma Sppose the roots of a and b all lie in the nit disk Then jccr (a; b)j Proof From (5) and Theorem we know that (a; b) = min n min jj(a(z); b(z))jj; min jj(a(z); b(z))jj o Ths for the assertion of Lemma it is scient to show that ja(z)j ja(z)j (and analogosly that jb(z)j jb(z)j) for all jzj <, where as sal z denotes the complex conjgate of z If x ; ; x m are the roots of a, then this follows from ja(z)j ja(z)j = ja(z)j jzj m ja(=z)j = m Y j= jz? x j j j? z x j j being less or eqal to one for any jzj <, since jz? xj=j? x zj 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 Ga{Lcas Theorem [, p], the zeros of the deriatie of c lie in the conex hll of the set of its zeros, and hence also in the nit disk Ths if a; b are any (higher order) deriaties of c, then jccr (a; b)j 5 Examples In order to illstrate and to compare the ndings of the preceding sections, we consider the following three simple examples 0

11 Example 5 Let a(z) = z n?, and b(z) = b b n z n with jjbjj Then?In L S(a; b) = and S(a; b)? = U Therefore I n jjs(a; b)jj = ;?(U + L)? U (U + L)? L (U + L)? (U + L)? jjs(a; b)? jj [ p ; p n] jj(l + U)? jj n The matrix L + U is circlant, with eigenales b(! j ), j = 0; ; ; n?, where! is a primitie n-th root of nity Moreoer, L + U is normal, and therefore jj(l + U)? jj = maxf=jj is an eigenale of L + Ug From Lemma and Theorem we then hae B = min jb(! j )j (a; b) j jjs(a; b)? jj B p n Note also that Corollary applies in this context with n (a; b), giing jjs(a; b)? jj 9 Example 5 Let a(z) = z m, and b(z) = b b n z n with jjbjj In this case the cofactors can be obtained explicitly from b(z) (z) = + O(z m ) z!0 ia the Taylor expansion of =b at zero and as a corresponding remainder Then 0 L S(a; b) = U I n ; jjs(a; b)jj = ; S(a; b)? =?U L? I n L? 0 ; jjs(a; b)? jj jjjj [; ]; the latter obseration being in accordance with Corollary since m+n (a; b) = For example, if b(z) = (? z)= then m? jjs(a; b)? jj (a; b) a(=) =?m ; a conseqence of (z) = ( + z + z + + (z) m? ) Example 5 With the same setting as in Example 5, let b(z) = (?z )m Then jjbjj =, and with This gies (z) = m (? z)?m + O(z m ) = m jjjj = m X m? j=0 m? p m? m? m m? + j j z j p m 8?m jjjj jjs(a; b)? jj (a; b) = b(=) =?m Ths the criterion of Corollary does not always yield sharp bonds, since for large m we hae = =jjs(a; b)? jj (a; b) From Example 5 we also see that a \small (a; b) in general does not imply that a has a root which is \close to one of the roots of b

12 Conclsion We hae considered the problem of determining when two polynomials are nmerically relatiely prime A parameter has been gien that improes a preios existing measre for nmerical primeness A sharper measre can be gien in the case where it is known that the two polynomials hae all their roots in the nit disk This parameter is based on qantities which are eciently obtainable in a nmerically stable way [] The eciency and nmerical correctness of sch a comptation makes a good initial test for coprimeness before starting the more expensie comptation of an {GCD References [] B Beckermann, The stable comptation of formal orthogonal polynomials, Nmerical Algorithms (99) - [] B Beckermann and G Labahn, A fast, nmerically stable Eclidean-like algorithm for detecting relatiely prime nmerical polynomials Jornal of Symbolic Comptation (this isse) [] S Cabay and R Meleshko, A weakly stable Algorithm for Pade Approximants and the Inersion of Hankel matrices, SIAM J Matrix Analysis and Applications (99) 5-5 [] S Cabay, A R Jones and G Labahn, Experiments with a Weakly Stable Algorithm for Compting Pade-Hermite and Simltaneos Pade Approximants, ACM Trans of Mathematical Software (TOMS) () (99) 9-0 [5] RM Corless, PM Gianni, BM Trager & SM Watt, The Singlar Vale Decomposition for Polynomial Systems, Proceedings ISSAC '95, ACM Press (995) 95-0 [] I Emiris, A Galligo and H Lombardi, Certied approximate niariate GCDs, J Pre and Applied Algebra, (99) 9-5 [] M Fiedler, Special matrices and their application in nmerical mathematics, Martins Nijho Pblishers, Dordrecht (98) [8] KO Geddes, SR Czapor and G Labahn, Algorithms for Compter Algebra (Klwer, Boston, MA, 99) [9] NJ Higham, Accracy and Stability of Nmerical Algorithms (SIAM, Philadelphia, 99) [0] N Karmarkar and YN Laksmann, Approximate polynomial greatest common diisors and nearest singlar polynomials, Proceedings ISSAC '9, ACM Press (99) 5- [] T Kailath, Linear Systems, Prentice{Hall (980) [] M Marden, Geometry of Polynomials, Math Sreys (Amer Math Soc Proidence, RI, 9) [] NK Nikol'skii, Treatise of the shift operator, (Springer, Berlin, Heidelberg, 98) [] M-T Noda & T Sasaki, Approximate GCD and its applications to ill{conditioned algebraic eqations, JCAM 8 (99) 5-5 [5] A Schonhage, Qasi{GCD Comptations, J Complexity (985) 8- [] TW Sederberg and GZ Chang, Best linear common diisors for approximate degree redction, Compter-Aided Design 5 (99) -8