Numerical Univariate Polynomial GCD Ioannis Z. Emiris, André Galligo, and Henri Lombardi Abstract. We formalize the notion of approximate GCD for univariate poly-
nomials given with limited accuracy and then address the problem of its computation. Algebraic concepts are applied in order to provide a solid foundation for a numerical approach. We exhibit the limitations of the euclidean algorithm through experiments, show that existing methods only solve part of the problem and assert its worst-case complexity. A rigorous geometrical point of view is given in the parameter space of all input polynomials and SVD computations on subresultants are applied in order to derive upper bounds on the degree of the approximate GCD. Then, we establish a certi cation theorem and state the conditions under which it determines the precise GCD degree.
1. Introduction
Recently, there has been a renewed interest in the study of univariate as well as multivariate polynomial systems whose coe cients are only imperfectly known. For instance, the input polynomials may be given by oating point coe cients to limited accuracy or only a certain number of signi cant digits may be obtainable e ciently. The usual algebraic problems, such as factoring, solving and root clustering, are not always well-de ned in this context. We are willing to accept an answer that is exact for a perturbed input set, which implies that the notion of a correct solution must be rede ned. On the one hand, we wish to use algebraic notions and exploit the mathematical veracity of symbolic computation, while on the other hand, we rely on techniques from numerical analysis and exploit the speed of numerical computation. In combining the two elds, we wish to provide a solid foundation for performing numerical calculations. In addition to the richness of mathematical issues involved, the answers to problems on imperfectly known polynomials have important practical rami cations. Whenever laboratory measurements are involved, data is presented together with a measure of tolerance. To mention only some elds within the realm of our experience, there is a multitude of robotics, vision as well as molecular kinematics problems where noise corrupts the input parameters [Fau93, Hav91, Mer92]. 1991 Mathematics Subject Classi cation. Primary 65Y20, 68Q40; Secondary 65D15. Work partially supported by PoSSo, ESPRIT BRA contract 6846.
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IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
Another domain of interest for approximative solutions is linear systems, control and network theory, see e.g. [Kai80]. This article addresses one of the most basic issues pertaining to polynomials with imprecise coe cients, namely the computation of an approximate Greatest Common Divisor (GCD) for two univariate polynomials. Our rst contribution is to pose a well-de ned problem in the presence of input uncertainty. Secondly, we propose algorithms that answer the question in certain cases and formally de ne these cases. This is the rst step in studying multivariate systems of polynomials and their approximate solutions, in the well-constrained as well as the overconstrained setting [Man94, MS96]. Our approach generalizes the usual notion of backward error, in the sense that we search for an exact solution to a slightly perturbed problem. The desired solutions should be compatible with dierent degrees of uncertainty; actually, a single solution has no longer any sense and a set of solutions as a function of the tolerance level is to be sought. Although backward error analysis has been primarily used for bounding the roundo error, here the latter is assumed to be insigni cant compared to the inaccuracy of the data. In this paper we are concerned with the following task. Problem Definition 1.1. Fix integers m; n and a metric jj on the spaces of univariate polynomials Pm ; Pn C [x] of degree bounded by m and n respectively. Given f 2 Pm , g 2 Pn and > 0 nd the maximum integer d such that there exist fb 2 Pm , bg 2 Pn , with jf ? fbj; jg ? bgj and deg gcd(f;b gb) = d.
Additionally, we may further consider bounding the error in the computed GCD, i.e., if we are given > 0, we would like to nd a polynomial g 2 Pd such that jg ? gcd(pb; qb)j . Trying to maximize the degree of the GCD is the natural approach in the presence of noise; of course, an exact computation on imperfect data would produce a unit GCD almost always, which excludes this approach. This task is quite dierent than the quasi-GCD studied in [Sch85], where the input polynomials have exact coe cients, given to arbitrary accuracy by some oracle. Problem 1.1 was considered in [CGTW95, HS96], but both papers have given only partial answers, as explained in subsequent sections. We wish to provide a more sound mathematical basis for the study of approximate GCD and specify clearly the results of the various algorithms. In addition, we distinguish between heuristics and theorems and propose our own methods. The paper is structured as follows. The following section lays out some definitions and notation, introduces the norms which we use and surveys existing work. Section 3 uses the euclidean algorithm as an e cient method for computing approximate common divisors, but which fails to guarantee anything but a lower bound on the -GCD degree, as shown in a concrete example. Therefore section 4 studies in further detail the geometry of polynomial spaces in order to give more rigorous arguments. This results in stable and eective methods for the -GCD in section 5, based on Singular Value Decompositions (SVD). Nonetheless, there exist cases where this method only provides upper bounds on the GCD degree, which leads us, in section 6, to consider further conditions on the validity of the algorithm's answers. A summary and directions for future work appear in section 7.
NUMERICAL UNIVARIATE POLYNOMIAL GCD
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2. Preliminaries
We are interested in oating point computations, but we shall usually ignore roundo error, assuming that the algorithms are executed in su ciently high precision to make the latter too small compared to the allowed tolerance. We could assume, therefore, that our computational model is that of the real-arithmetic Random Access Machine of [PS85], in which arithmetic between reals is exact.
2.1. Previous Work. Among the classical algorithms that compute exact GCDs, it is known that the subresultant version is the most e cient, since it strikes a balance between coe cient growth and computational eort. Comparative presentations, in order of increasing detail, can be found in [DST88, ch. 2], [Zip93, ch. 8, 9] and [Loo82]. The introduction of subresultant chains in the eld is usually attributed to [Col67, Bro71], although the same objects had been studied in [Hab48]; see also [LRGVR95]. The main advantage of the subresultant algorithm is the linear growth in the coe cients, whereas the plain euclidean algorithm may cause an exponential swell. A signi cant portion of the literature is devoted to methods derived from Euclid's algorithm and its extensions. Schönhage [Sch85] proposes ways to compute the quasi-GCD, under the assumption that the coe cients of the given polynomials can be given to arbitrary accuracy by some oracle. This problem de nition is in sharp contrast to our own, where it is precisely these coe cients that cannot be known exactly. To handle long integer multiplication a dierent model is used, namely that of a pointer machine. Schönhage shows that, given p; q and > 0, one can nd polynomials p1; q1 ; u; v and quasi-GCD g such that jp ? p1gj ; jq ? q1 gj ; jup + vq ? gj : The algorithm is not simple as it uses a special change of variable called pivoting in order to control coe cient size. A lot of eort has been spent on optimizing its complexity in terms of pointer time. An -GCD is sought by Hribernig and Stetter [HS96], who use the classical euclidean algorithm in order to identify the clusters of polynomial roots. All computation is in oating-point, but the roundo error is assumed to be negligible. Since their method is very similar to our technique for deriving lower bounds on the GCD degree, we examine it in detail in section 3. It must be noted, though, that this approach oers no guarantee that an -GCD has been found; it only returns a common divisor within the prescribed tolerance, as exempli ed by a speci c polynomial pair later. Noda and Sasaki [NS91] introduce a scaled euclidean algorithm, which is e cient, but again does not solve the same problem. Approximate GCDs have been studied in relation to various applications, including the computation of proper parametrizations and rational curve degree reduction [Sed86, SC93]. Numerical GCDs have been studied in the control theory literature, where numerical computation, even with rational input, is bound to produce some error in the result [PB73, KM94]. The latter paper by Karkanias and Mitrouli uses standard backward error analysis techniques to show that the numerical GCD obtained by their method will be su ciently close to the exact one. Their setting is somewhat dierent, as they assume that the error source is roundo error, not imprecision of input.
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IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
Their main tool is SVD on Sylvester's matrix, which has long been known within the numerical computation community to be rather stable; see [MD95] for an application. The SVD method was also proposed by Corless, Gianni, Trager and Watt recently [CGTW95] in the setting of seminumerical computation. This is a promising approach and can be very powerful; we examine it in detail in section 5. For now we note that their problem is slightly dierent, since the a priori bound is not guaranteed to bound the L2 -norm of the input perturbation; see remark 5.6. Corless et al. have also extended this work to polynomial systems, taking advantage of resultant formulations for approximating the common roots. A signi cant contribution of [CGTW95] is the eort towards posing well-de ned problems in the presence of error. In this task, they use backward error analysis to show that the outcome is relevant for a perturbed input, while assuming that roundo error is insigni cant. We take up several of their ideas, including backward error analysis. We concentrate on monomial bases for convenience. Dierent bases, such as the Bernstein polynomials [FR87], may improve stability but are beyond the scope of this paper.
2.2. Matrix and Polynomial Norms. This subsection contains a series of properties useful in our presentation. More material can be found in [Zip93, ch. 11] and [BBEM90]. We equip the space of all linear transformations from C n+m?2r to C n+m?r with the following euclidean norm. k ; for any mapping A and vector x 2 C n+m?2r ; kAk = sup kkAx x xk where kxk represents the L2 -norm of x. If we denote by ai;j the entries of the matrix of A it can be shown that X kAk2 jai;j j2 ; i;j
where jcj is the module of the complex number c. We de ne polynomial norms ? jP jl = jp0 jl + jpd jl 1=l ;
jP j1 = maxfjp0 j; : : : ; jpd jg;
where P =
d X i=0
l 1; and pi xd?i 2 C [x]:
We shall denote jP j2 simply as jP j. The following relations are known. jP j1 jP j jP j1 (d + 1)jP j1; p pjP j1 jP j d + 1 jP j1 : (2.1) d+1 For any P1; P2 2 C [x], let d stand for the sum of their degrees. [BBEM90, thm. 1.1] shows that (2.2) jP1 P2jl 2d(1?1=l) jP1 jl jP2 jl ; 1 l 1: When P1; P2 are monic, [Gel60] showed p (2.3) jP1j1 jP2j1 d + 1 2d jP1 P2j1 :
NUMERICAL UNIVARIATE POLYNOMIAL GCD
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Corollary 2.1. For any P1; P2 2 C [x] with respective degrees d1 ; d2 , p
jP1 jjP2 j
(d1 + 1)(d2 + 2)(d1 + d2 + 1) 2d1 +d2
jP1 P2 j 2(d1+d2 )=2 jP1jjP2 j:
Proof. The rst part follows from above and the fact that jcP j = jcjjP j, for any polynomial P and complex c. The second part is a special case of inequality (2.2). We let the norm of a polynomial pair be the maximum of the two norms, for any norm considered, and denote it j; jl . In particular, for two polynomials P1 and P2, jP1 ; P2 j = maxfjP1j; jP2 jg: Weighted norms, sometimes called Bombieri norms [BBEM90], assign dierent weights to dierent coe cients and, hence, take into account the importance of middle coe cients in polynomial multiplication. The weighted L2-norm is de ned as follows. v u d 2 uX hP i = t j?pi j : d i=0 i
The rst advantage of this norm is that it remains invariant, for bivariate homogeneous polynomials, under unitary changes of variables. The second is that the norm of the product is very tightly bound above and below by small scalar multiples of the product of the norms.
3. A Lower Bound on the Degree by the Euclidean Algorithm
This section takes an applied approach in studying the behavior of the degree of the approximate GCD. Modular algorithms are naturally excluded, since we are interested in oating-point calculations, so we concentrate on algorithms derived from Euclid's procedure for integer GCD. We show that they only produce lower bounds on the degree and substantiate our claims with a counterexample. The variants of Euclid's algorithm oer an e cient method for bounding the degree as well as computing an approximate common divisor, which is, in several cases, a good approximate GCD. An -divisor of f is a polynomial p such that jf ? pqj , for some polynomial q. We are interested in generalized Polynomial Remainder Sequences (PRS) in order to set the discussion in the most general possible context. The input polynomials are f = F1 ; g = F2 2 C [x] and a generalized division step is written, for j > 1, (3.1) aj Fj?1 = Qj Fj + bj Fj+1 ; Qj ; Fj 2 C [x]; aj ; bj 2 C : Dierent choices for aj ; bj produce dierent sequences, but the nal result is always the GCD polynomial within some scalar factor. The sequence terminates with Fr = 0; for some r > 2; which implies Fr?1 = gcd(F1; F2 ): The PRS corresponding to the plain euclidean algorithm is obtained by aj = bj = 1; j > 1: If l(F ) denotes the leading coe cient of polynomial F and j?1 = deg Fj?1 ? deg Fj > 0, then the pseudo-division PRS is obtained by aj = l(Fj )j?1+1 ; bj = 1; j > 1:
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IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
Lastly, for the subresultant PRS we set aj = l(Fj )j?1+1 ; bj = (?1)j?1 l(Fj?1) ( j ) ; j > 1; (3.2) where the j are de ned recursively by 1 = 1 and j = l(Fj )j?1 = j?j?11?1 . We de ne another two sequences of polynomials R(ji) 2 C [x], for i = 1; 2 and j i ? 1: (3.3) R(i?i)1 = 0; R(ii) = 1; i = 1; 2; and (3.4) aj+1 R(ji) = Qj R(ji?) 1 + bj?1 R(ji?) 2 ; i = 1; 2; j > 1:
The polynomials R(ji) are therefore functions of the quotients Qk and can be used to express the two input polynomials. Lemma 3.1. For i = 1; 2 ai+1Fi = aj+1 R(ji) Fj + bj R(ji?) 1 Fj+1 ; 8j > i:
(2) Moreover, deg R(1) j = deg Fj ?1 ? deg F2 , and deg Rj = deg Fj ?1 ? deg F3 , for j i + 1. Proof. By recursion on j . The basis of the recursion for i = 1; j = 2 is (1) a2 F1 = a3 R(1) 2 F2 + b2 R1 F3 (1) (1) = (Q2R1 + b1 R0 )F2 + b2 R(1) by (3:4) 1 F3 = Q 2 F2 + b 2 F3 by (3:3); which is relation (3.1) for j = 2. Similarly, for i = 2; j = 3 we get a3 F2 = Q3F3 + b3 F4, which is relation (3.1) for j = 3. For the recursion step, for i = 1; 2 and j > i, we obtain ai+1 Fi = aj+1 R(ji) Fj + bj R(ji?) 1 Fj+1 = (Qj R(ji?) 1 + bj?1 R(ji?) 2 )Fj + bj R(ji?) 1 Fj+1 by (3:4) = R(ji?) 1 (Qj Fj + bj Fj+1 ) + bj?1 R(ji?) 2 Fj = R(ji?) 1 aj Fj?1 + bj?1R(ji?) 2 Fj by (3:1); which is the recursion hypothesis. To establish the degrees, again a recursion on j will do, applying equation (3.4). (2) This gives deg R(1) j = deg Qj + + deg Q2 and deg Rj = deg Qj + + deg Q3 . To rewrite the degree recall that deg Qj = deg Fj?1 ? deg Fj .
Algorithm 3.2. Input: Polynomials f; g and precision . Output: An divisor of f; g. 1. Initialize F1 = f , F2 = g and j = 2. 2. Compute scalars aj and bj according to the chosen algorithm. 3. Division step: Compute Qj and Fj+1 such that aj Fj?1 = Qj Fj + bj Fj+1 . Also compute R(ji) , i = 1; 2 as de ned by equation (3.4). (2) 4. Termination test: If the maximum of jbj R(1) j ?1 Fj +1 j and jbj Rj ?1 Fj +1 j is bounded by , then return Fj . Otherwise increment j and go to step 2.
NUMERICAL UNIVARIATE POLYNOMIAL GCD
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Proposition 3.3. Suppose that after executing any variant of Euclid's algorithm on polynomials F1 ; F2 , we obtain polynomials Fr and R(ri?) 2 for i = 1; 2, such that jbr?1R(ri?) 2 Fr j ; i = 1; 2: Then
deg ( ? gcd(F1; F2 )) deg Fr?1: If di = deg Fr + deg R(ri?) 2; i = 1; 2, then (
)
2di (di + 1)3=2 ; jFr j imin =1;2 jbr?1 j jR(i) j r?2 where for r > 2, d1 = deg Fr +deg Fr?2?deg F2 and d2 = deg Fr +deg Fr?2?deg F3. Proof. The termination test can be also written jai+1 Fi ? ar R(ri?) 1 Fr?1j = jbr?1R(ri?) 2 Fr j . Fr?1 is a divisor of both F1 and F2 within and therefore, by de nition, the degree of the approximate GCD cannot be lower than deg Fr?1. For the second statement, by the termination condition, jbr?1R(ri?) 2 Fr j for
i = 1; 2. Then apply corollary 2.1. To rewrite the sum of the two degrees, use lemma 3.1.
Despite the exponential numerator, the bound on jFr j is usually close to , especially when the exponent of 2 is not positive. We substantiate this claim below experimentally. It is possible to compute an a posteriori estimate of the total amount of roundo error by bounding the norm of ai+1 Fi ? br?1 R(ri?) 2Fr ? ar R(ri?) 1 Fr?1, maximized over i = 1; 2. Remark 3.4. It is important to realize that any variant of Euclid's algorithm will only produce, in general, a lower bound on the degree of the -GCD, in contrast to various claims in the literature, e.g. [HS96, sect. 3] where this algorithm is applied in order to compute the -GCD. The intuitive reason is that euclidean algorithms consider only special kinds of perturbations of the input polynomials.
To show that a loose bound on the degree may be also obtained in practice, we have implemented the extended euclidean algorithm in Maple V; the code is available upon request. We use this code to give a counterexample, where the lower bound is inferior to the degree of the -GCD. For this, we use the polynomial pair examined in [CGTW95, sect. 2.7]. Example 3.5.
F1 = x5 + 5:503 x4 + 9:765 x3 + 7:647 x2 + 2:762 x + 0:37725; F2 = x4 ? 2:993 x3 ? 0:7745 x2 + 2:007 x + 0:7605: Performing any variant of the euclidean algorithm in su cient accuracy (at least 15 decimal digits), Maple produces the following polynomial remainder sequence for any 0:00056. The maximum norm, which serves as the termination condition appears in the second-to-last column while the last column shows the polynomial
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IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
norm jFj j. F3 = 35:968 x3 + 12:220 x2 ? 15:050 x ? 6:0840; 41:310; 41:310; F4 = 0:77623 x2 + 0:78164 x + 0:19677; 10:226; 1:1190; F5 = ?0:0018829 x ? 0:00051873; 0:00056718; 0:0019530; F6 = 0:040344; 0:36110; 0:040343; F7 = 0; 0; 0: ? 4 For any 5:6 10 , algorithm 3.2 outputs 0 as the degree. However, if is bounded below by 1:6 10?4, the approximate GCD is quadratic, according to [CGTW95, sect. 2.7], which computes x2 +1:007 x +0:2534; such that it divides exactly Fi + Fi; i = 1; 2, with jF1 j; jF2 j 1:6 10?4. n o (2) Remark 3.6. The test value max jR(1) j ?2 Fj j; jRj ?2 Fj j ; for j 3; is not necessarily a monotonic function of j , as illustrated in the example. This implies that a certain lower bound may require a smaller perturbation than an even lower one, which is a positive feature. In the example above, if = 5:6 10?4, we have a strong indication that the -GCD degree is at least 2, while no produces a linear GCD. jFj j is also not monotonic but it is small when the test value is small. Our implementation oers the option of choosing among certain variants of the euclidean procedure, namely the plain, pseudodivision and subresultant algorithms. We experimented with the latter and compared it to the plain algorithm, because the subresultant PRS is known to achieve the best tradeo between coe cient growth and the complexity for rational inputs [Loo82, sect. 4]. This implies that we should obtain clearer indications of where the approximation may be minimized when the accuracy of the calculation is limited. On the other hand, the additional computation of scalars aj ; bj de ned in equation (3.2) adds to the roundo error, so the choice is not always clear. When the precision is large enough, the plain version is preferred since it gives the same results at a lower cost. For instance, in example 3.5 above we obtain the same PRS running either algorithm, when accuracy is set at 15 digits or more. We did observe dierences at lower precision, e.g. of magnitude 4% at 12 digits of accuracy. We may conclude that, although Euclid's method has roughly quadratic complexity and hence is pretty e cient, it is not powerful enough for our task, as it oers no guarantee even on the GCD degree. We are thus led to a deeper study of the problem and the search of numerically more stable routines. The rst step is to analyze the structure of the polynomial spaces that come into play.
4. The Geometry of Polynomial Con guration Spaces
We are now interested in a tighter determination of the degree of the approximate GCD. We start with describing the abstract setting of polynomial spaces parametrized by the coe cients or roots. Hence we consider con guration spaces of higher dimension. 4.1. Monic Univariate Polynomials. We consider the set Tn of monic univariate polynomials of degree n, with coe cients in the complex eld C . Let us recall the correspondence between roots and coe cients. (4.1)
f = xn +
n X i=1
ai xn?i =
n Y
i=1
(x ? i ):
NUMERICAL UNIVARIATE POLYNOMIAL GCD
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The ai are symmetric functions of the roots j , and they are homogeneous polynomials of degree i. The roots are de ned up to reordering by a permutation. We have the following diagram, which de nes the con guration space Tn . We denote by Sn the permutation group of n elements and by f 1 ; : : : ; n g the multiset of roots. n n T = C n =S ?! Cn Cn ?! n n ( 1; : : : ; n ) 7! f 1 ; : : : ; n g 7! (a1; : : : ; an ) The map n is a surjection, whereas n is a homeomorphism. Tn is thus naturally endowed with the structure of a smooth analytical, and even algebraic, variety. The analytic functions on Tn are the symmetric functions on C n and they admit an analytic expression as functions of the ai . The surjection n is a branched covering with n! folds. Its critical locus Cn consists of hyperplanes. Its discriminant locus Dn , the image of Cn by n , is an hypersurface in Tn whose equation in terms of ai is a polynomial called the discriminant. We can equip Tn with dierent metrics. By n we get the usual distances on the vectors of coe cients. By n, the Minkowski distance between sets of roots may be de ned as a minimization over distances between sequences: ? dist (f j gj ; f j gj ) = min dist ( ) ; ( ) : j j j ( j ) 2S n
The behavior of these two kinds of metrics is made numerically precise through Ostrowski's inequalities which we recall hereafter. Note that they require some complementary information. Theorem 4.1. [Ost66, app. A] Let f and g be two elements of Tn . Denote by ai and bi their coe cients as in equation (4.1), i and i their roots and the maximum of the modules of these roots. It is known that (4.2)
2 maxfjai j1=i ; jbi j1=i : 1 i ng: There exists a permutation 2 Sn , such that
j i ? (i) j < 2n
v u n uX n t
i=1
jbi ? ai j n?i;
i = 1; : : : ; n:
Without further information this bound is quite tight, as seen when we take f = xn and g = xn ? n , for some very small , 0 < 1. Then i = 0 and i = e2i=n , so = . The inequality of the theorem then becomes < 2n. This example illustrates the di culty of distinguishing the roots, even if they are clearly apart, when the two polynomials get very close together as becomes small. Now, we prove a transformed version of this theorem that we will use below and is particularly useful for the case of two polynomials lying very close together. Definition 4.2. We will say that a polynomial g is almost monic if its leading coe cient b0 satis es the inequality jb0 ? 1j 1=2. Corollary 4.3. With the previous notations, suppose polynomial f ispmonic and polynomial g is almost monic, with 1. Suppose also that jf ? gj n + 1 and n 2. Then, there exists a permutation 2 Sn such that, for any i = 1; : : : ; n
we have:
qp j i ? (i) j < 4n ( + 1) n n + 1jf ? gj:
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IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
Proof. We apply the theorem to monic polynomials f and bg = g=b0 . Note that, by de nition, jb0 j 1=2. Let t = 1=b0 ? 1, hence jtj 2jb0 ? 1j 2jf ? gj, where the last inequality follows from the fact that 1 ? b0 is the leading coe cient of f ? g. If bbi are the coe cients of bg we have:
A=
n X b i i=1
b ?a
i
n?i n
n X i=1
jbi ? ai j + jtj
n X i=1
!
jb i j :
Now use the triangle inequality jb0 j 1 + jf ? gj and express the coe cients of bbi as symmetric functions of the roots i . n X i=1
jbi j = jb0 j
n X b i i=1
b (1 + jf ? gj)
n X X i=1
j j1 : : : jl j;
Combining the above and using relation (2.1), we have ?p A n n + 1jf ? gj + 2jf ? gj(1 + jf ? gj)( + 1)n : Thus we obtain the claimed inequality, because for n 2 and 1 the following holds p p p n + 1 + 2(1 + n + 1)( + 1)n 4 n + 1( + 1)n: This concludes the proof. 4.2. The Space Gr . Let m and n be two integers with m n. In Tn Tm , we consider for r = 1; : : : ; m the sets Gr formed by pairs of polynomials (f; g) such that deg(gcd(f; g)) r. We have Tn T m G 1 G 2 G m ' Tm : We now view Gr in three dierent ways. We consider the Sylvester matrix of f and g, and we express the fact that its rank is less than or equal to n + m ? r by setting to zero the corresponding minors. This proves that Gr is an algebraic subvariety of Tn Tm identi ed with C m+n , and that Gr+1 is a closed subvariety of Gr . Gr is the image by n m of the subvariety Sr of C n C m de ned as a union of (n + m ? r)-planes: [ f (1) = (1) ; : : : ; (r) = (r) g: Sr = 2Sn ; 2Sm The singular locus of Sr is
formed by the two-by-two intersections of these (n + m ? r)-planes. Alternatively, it equals the union of Sr+1 with the trace on Sr of the critical locus of n m. Consider the following mapping. Tr Tn?r Tm?r ! Tn Tm (p; fr ; gr ) 7! (p fr ; p gr ): It is clear that Gr is its image. Denote by r the corresponding mapping r : Tr Tn?r Tm?r ! Gr : r is an isomorphism outside the preimage of Gr+1. We can say that r is a desingularization of Gr . We note that the singular locus of Gr is not the full image of the singular locus of Sr by n m .
NUMERICAL UNIVARIATE POLYNOMIAL GCD
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Figure 1. The dashed curve is subvariety Gr+1 , while the entire umbrella zx2 = y2 depicts Gr .
4.3. Singularities and Neighborhoods. This subsection sketches the singularities of a pair (Gr ; Gr+1 ) and will be useful later in nding examples and counterexamples of the behavior of approximate GCD. A generic point of Gr+1 is a pair (f; g) whose GCD p is of degree r + 1 and has simple roots. Near a generic point, the variety Gr consists of the transverse intersection of r + 1 folds along Gr+1 , each of them corresponding to one of the monic divisors of p of degree r. We can say that the situation looks like that of the pair Sr ; Sr+1 near a preimage of (f; g). Near a nonsingular point (f; g) of Gr+1 such that p = gcd(f; g) has multiple roots the geometry is more intricate. More precisely, assume p has q distinct roots of multiplicity mi; i = 1; : : : ; q. Then, locally, the variety Gr is the transverse intersection of q folds, which are smooth if mi = 1. They look like the generalized Whitney (or Cartan) umbrella singularities, i.e., those of equation zi xmi = ymi , where the zi are some parameters on Gr+1. Figure 1 depicts the situation for mi = 2. Near a singular point of Gr+1 , the geometry of the pair (Gr ; Gr+1 ) is a degenerate situation of the previous ones. Tn Tm can be identi ed with R2(n+m). Then, let Fr be a continuous function from Tn Tm to R, which vanishes on Gr , thus the set Fr is a neighborhood of Gr . If Fr admits an analytical expression, then this neighborhood tends to enlarge near the singularities of Gr . In that case, the distance from a point of this neighborhood to Gr (near a singularity) is bounded by the ojacewicz inequality. The ojacewicz exponent increases with the multiplicity of the speci c singular point. An illustration is given at gure 2, where the bold lines depict this neighborhood which indeed gets larger near the singularity of Gr . 4.4. De nition of Approximate GCD. Once a metric has been xed on Tn Tm , the interpretation of the degree of an approximate GCD of a pair (f; g) is straightforward. Consider the set [ Gr; = B ((f; g); ); r = 1; : : : ; m; 1 > > 0; (f;g)2Gr
where B (; ) denotes a ball of radius . This provides a double ltration by r and of Tn Tm . Definition 4.4. For (f; g) 2 Tm Tn and the set of polynomial pairs Gr; de ned above for r = 1; : : : ; m and 1 > > 0, we set deg (-gcd(f; g)) r , (f; g) 2 Gr;:
12
IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
G
r, ε
Figure 2. The thick curves are the smooth approximations Gr; of Gr around a singularity that contains Gr+1 . The dotted line separates the two parts of Gr; that are closer to each of the two branches of Gr . The discontinuity may arise at points near the
dotted line whose projections lie on dierent branches and hence at a large distance.
The -gcd will be an element of Gr n Gr+1 , computable within some approximation . We interpret this as a retraction from Gr; n Gr+1; to Gr n Gr+1 , which is not necessarily continuous but has only discontinuities of norm less than . Figure 2 illustrates the di culty. We see that there are points of Gr; n Gr+1; equidistant from two elements of Gr . In their neighborhood there will be a discontinuity of the -GCD. This must be allowed by a su ciently large tolerance , for instance such that the distance between the corresponding points in Gr is at least 2. This de nition is equivalent to de nition 1.1. Remark 4.5. An analogous di culty may arise near nonsingular points of the variety which have large curvature with respect to . In fact, one can try to construct an eective continuous retraction which is not the orthogonal projection. This is a di cult task considered for a singularity with normal crossing in [Cam75]. A not-so-elegant way of getting rid of the rst problem is to increase the tolerance with the degree r, so that outside Gr+1;(r+1) the orthogonal projection from Gr;(r) on Gr is uniquely de ned, for small enough. We will not consider this possibility in the rest of this paper. We are led to a family of unconstrained optimization problems. Indeed, x some pair (f; g), and for each r, evaluate (or simply compare) the distance in Tn Tm between (f; g) and Gr nGr+1 which is given by a parametrization r . These optimization problems are ill-conditioned and will be simpli ed if we can simply bound the corresponding degree r either by almost exact algebraic computations or by standard and more stable numerical computations. The next sections are devoted to this task.
5. An Upper Bound on the Degree by SVD Computations
This section discusses methods for deriving upper bounds on the degree of the approximate GCD and compares our approach to this of [CGTW95]. A classical result says that any linear map F between C p and C q equipped with their usual hermitian norms can be written, after a suitable orthonormal change of coordinates, as a matrix whose only nonzero entries are real, non-negative and on the diagonal.
NUMERICAL UNIVARIATE POLYNOMIAL GCD
13
The ordered diagonal elements 1 2 : : : min(p;q) are called the singular values of the map F . SVD stands for Singular Value Decomposition, a method which provides the coordinate changes and the singular values and whose numerical stability has been extensively studied. In particular, it is known that the singular values are numerically stable. It is implemented in various numerical libraries, e.g. [SBD+ 76, ABB+ 92]. Here we sketch the basic properties of the singular values; a more extensive treatments can be found in [GL89]. The singular values of F describe how the map F deforms the objects from the source space to the target space. 1 is the norm of the map F . Denoting by S1 (F ) the unit sphere of any subspace F of C p , we have
r+1 = r =
(
inf
sup (kF (x)k) ;
F C p :codimF =r x2S1 (F )
sup
)
inf (kF (x)k) :
F C p :dim F =r x2S1 (F )
If r > 0 then the rank of is larger than r. In the real case, if p q, the rst p elements of the new base of Rq are given by the principal axes of the ellipsoid image of the unit ball. The knowledge of the singular values allows a discussion on the numerical rank of F . It is easily shown that if we perturb F by a linear map F of norm kFk, such that r > kFk > r+1, then the rank of F + F may get down to r but cannot reach r ? 1. On the other hand, it may go up to min(p; q) but this does not depend on the norm of F . 5.1. Subresultant Mappings. To every pair (f; g) in Tn Tm and to every r m n we associate the subresultant mapping Syr (f; g) given by Syr (f; g) : Tm?r?1 Tn?r?1 ! Tm+n?r?1 (u; v) 7! uf + vg: Its matrix in the usual monomial bases has m ? r columns corresponding to f , n ? r columns corresponding to g and a total of m + n ? r rows: 2 3 1 0 0 1 0 0 6 a1 1 0 b1 1 0 77 6 6 .. 7 . . . . . .. . . .. .. . . 6 . 7 6 7 6 7 b m 6 7: Syr (f; g) = 6 an 7 1 0 1 6 7 6 0 an 7 6 7 6 . 7 . . . ... ... ... 4 .. 5 0 an 0 bm If r = 0, the Sylvester mapping and the respective matrix is denoted Sylv(f; g). The following properties are classical. Sylv(f; g) is of rank m + n ? r if and only if deg(gcd(f; g)) = r Syr (f; g) has full rank i.e. m + n ? 2r if and only if deg(gcd(f; g)) r. Let us denote by r or r (f; g) the last singular value of Syr (f; g) and by
m?1 : : : 0 0 the m last singular values of Sylv(f; g).
14
IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
Lemma 5.1. We have
m?1 : : : 1 0 = 0 Proof. Since r , resp. r+1 , is the smallest norm of the image of an element of the unit ball, we have, for some polynomial pair (u; v): j jxuf + xvgj r+1 = jufju;+vvg j = jxu; xvj r :
5.2. An Upper Bound. For any pair (f; g) of Tn Tm , we denote by jf; gj its norm, and we consider its matrix Syr (f; g). The de nition of matrix norm k k in section 2.2 and the special form of this matrix implies that kSyr (f; g)k2 (m ? r)jf j2 + (n ? r)jgj2 (m + n ? 2r)jf; gj2 : The set Syr+1 (Gr) is contained in the set of mappings of rank smaller than m+n?2r, and with the properties presented in the previous subsection, we get p m + n ? 2r dist((f; g); Gr+1 ) dist(Syr (f; g); Syr (Gr+1)) r (f; g): We have the same kind of properties with the singular values r of Sylv(f; g). This establishes the following lemma. Lemma 5.2.
r (f; g) : dist((f; g); Gr+1) p r (f; g) ; dist((f; g); Gr+1 ) p m+n m + n ? 2r Remark 5.3. This kind of reasoning cannot provide reversed inequalities, since Sylvester matrices do not ll the entire space of linear transformations from C n+m?2r to C n+m?r . Proposition 5.4.
p r (f; g) implies deg -gcd(f; g) r: m + n ? 2r pm r+ n implies deg -gcd(f; g) r ? 1: This readily leads to an eective method for bounding the degree from above, namely by computing the SVD of the Sylvester matrix and looking at the singular values. This is essentially the approach of [KM94, CGTW95], and can only return an upper bound on the degree of the -GCD. We now combine the method of section 3 with the current discussion to obtain a more rigorous algorithm. Corollary 5.5. For some (f; g) 2 Tn Tm , it is possible to compute polynomials f;b gb such that jf ? fbj , jg ? bgj and deg gcd(f;b gb) pr, say by a variant of p Euclid's algorithm. We can, moreover, test whether r = m + n ? r b gb) = r . or r = m + n and, if so, conclude that deg gcd(f; This oers a good heuristic for solving the original problem, although it is not guaranteed to succeed in every case. The next section examines the additional information required to circumvent this problem. For now we examine a nasty case.
NUMERICAL UNIVARIATE POLYNOMIAL GCD (f,g)
15 γ r< ε
ε
(f+∆ f, g+∆ g)
γr+1< ε
m+n
Figure 3. The fat point represents Gr+2 within variety Gr+1 , represented by the straight line. The top curve is the boundary of neighborhood
r and the closed curve is neighborhood p
r+1 m + n: Here, the degree of b-GCD may be superior to r + 1, suggested by SVD with the given . Remark 5.6. It seems to us that claim 2 on the output of algorithm 2.4 in [CGTW95] is not demonstrated. Namely, it states that the algorithm computes an b-GCD, where b is an a posteriori bound, but the given proof is incomplete. Indeed, with our notation, nothing forbids that we have p
r (f; g) < r+1 (f; g) = m + n b; as well as < dist((f; g); Gr+2 ) b; where b is an a posteriori bound computed as the maximum of the two perturbations maxfjf j; jgjg. Thus computing a common divisor of (f + f; g + g) with jf; gj b does not imply that it is an b-GCD of (f; g). We illustrate this phenomenon in gure 3. The algorithm would then return r +1 but switching from the input tolerance to the a posteriori bound b, may cause the degree of b-GCD to grow to r + 2. To be rigorous, one would have to restart the whole process with b . On the other hand, if b < the above corollary applies; then the problem is successfully solved and it is identical to our problem.
6. Certi cation of the Degree
In this section we describe a condition on the gap between two singular values r (f; g) and r+1 (f; g), which allows certi cation of the degree of -GCD being equal to r + 1.
6.1. Approximate Common Multiples and Common Divisors.
Definition 6.1. Given a pair (f; g) 2 Tn Tm ; r m n; > 0; > 0, we say that the 4-tuple (u; f; v; g) is an (r; ; )-common multiple of (f; g) if and only if u is a monic polynomial of degree m ? r, v is an almost monic polynomial of degree n ? r, (see section 4.1), the modules of the roots of u; v; f; g are bounded by , and we have juf ? vgj . Definition 6.2. Given a pair (f; g) 2 Tn Tm ; r m n; > 0, we say that (f; g) admits an -common divisor of degree r if and only if there exist three monic polynomials f;b gb; pb such that jf ? fbj ; jg ? gbj ; deg(pb) = r and pb divides fb and g. b
16
IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI m
m-r
r
n-r
n
Figure 4. The correspondence between the roots of uf and vg in m+n-r
proposition 6.3. Top-bottom pairs represent roots close to each other.
Proposition 6.3. Let (u; f; v; g) be an (r; ; )-common multiple of (f; g), with
1, then, for b satisfying the equality below, (f; g) admits an b-divisor of degree r. Let deg f = m, deg g = n and d = n + m ? r, 2 m n and set b = d2 ( + 1)2d 1=d : Proof. Let us denote by 1 ; : : : ; n and 1 ; : : : ; m the roots of f and g respectively. Their modules, as well as those of the roots of u; v, are all bounded by
. For any permutation s of 1; : : : ; m + n ? r, s(f1; : : : ; ng) \ f1; : : : ; mg contains at least m ? (m ? r) = r elements; this is shown in gure 4. Apply corollary 4.3 of Ostrowski's theorem to products uf and vg. Then there exists a permutation s as above such that there exist r-tuple (i1; : : : ; ir ) and r-tuple (j1; : : : ; jr ) satisfying, for l = 1; : : : ; r , il n; jl m; jl = s(il ), j il ? jl j < 4d ( + 1)(d + 1)1=d 1=d : Let l = ( il + jl )=2 and consider polynomials
P= and
fb = P
Y
i6=il ;1lr
r Y l=1
(x ? l)
(x ? i ); gb = P
Y
j 6=jl ;1lr
(x ? j ):
We show that P is an b-common divisor of f; g. By de nition, deg P = r and P divides f;b gb. P The coe cients of fPsatisfy (?1)lal = i1 il . The coe cients of polynomial fb are bai , i.e. fb = ni=1 bai xn?i + xn , therefore we can write them in terms of symmetric functions: X (?1)lbal = j1 jk m1 ml?k : This shows X X jal ? bal j j it ? t j l?1 = 12 j it ? jt j l?1 ; t
t
where the last equation follows by the de nition of t. Now take the sum of all squares jal ? bal j2 , which expresses the square of the norm of the polynomial dierence: ? jf ? fbj < 2prd ( + 1)(d + 1)1=d1=d 1 + 2 + : : : + 2d?2 < d2 ( + 1)2d 1=d :
NUMERICAL UNIVARIATE POLYNOMIAL GCD
17
This establishes the claim for jf ? fbj. An analogous argument can be used for jg ? bg j, thus concluding the proof. 6.2. A Lower Bound on the Degree. Let us denote by r and r+1 the last singular values of the mappings Syr (f; g) and Syr+1 (f; g) respectively. We are primarily interested in the case that r is very small but r+1 is well detached from zero. p Lemma 6.4. If r+1 2 r (4z1 + 1), there exists a (r + 1; ; z)-common multiple of (f; g) where f; g are as above and p ? p z1 = 2 maxfjf j2 ; jgj2 g ? 1; z2 = zp1 = r+1 ? 2 r (2z1 + 1) ; z = maxfz1; z2 ; 1g; = 2z2 r : Proof. For any pair (u0; v0 ) of polynomials, not necessarily monic, of degree m ? r ? 2 and n ? r ? 2 respectively, we have ju0 f ? v0gj r+1 ju0 ; v0 j. There exists a 4-tuple u0; v0 ; u0 ; v0 de ned up to a multiplicative constant such that u = u0xm?r?1 + u0 ; v = v0xn?r?1 + v0 and juf ? vgj = r ju; vj: We choose ju0 j = jv0j = 1 and we wish to nd a lower bound for ju0 j and jv0 j in order to bound the modules of the roots of u and v. If ju0 j > 1, the modules of the roots of u are less than 1 and the proofpis complete by p relation (4.2). So we suppose that ju0 j 1; jv0 j 1. Hence juj 2 and jvj 2. This gives p juf ? vgj 2 r ; ju0 f ? v0gj r+1 : p We rst suppose r+1 > 2 r . Since (uf ? vg) ? (u0f ? v0g) = u0xm?r?1f ? v0xn?r?1g; we get, by applying triangular inequalities: p r+1 ? 2 r ju0 f ? v0gj ? juf ? vgj j(u0f ? v0 g) ? (uf ? vg)j ju0 xpm?r?1f j + jv0 xnp?r?1gj ju0 j jfnj2 ? 1 + jv0 j jgj2 ? 1 o p p (6.1) 2 max ju0 j jf j2 ? 1; jv0 j jgj2 ? 1 : Suppose that the rst argument in the maximum is larger, then p ? r +1 ju0 j p 2 2 r : (6.2) 2 jf j ? 1 In addition, p juf ? vgj = j(u0 ? v0)xm+n?r?1 + j 2 r ; which implies p p ju0 ? v0 j 2 r ) jv0 j ju0j ? 2 r : (6.3) Equations (6.2) and (6.3) imply p p 2 r (2 jf j ? 1 + 1) ; r +1 ? 2p jv0 j 2 jf j2 ? 1
18
IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
where the numerator is positive by the hypothesis on r+1 . So a lower bound on jv0 j is obtained by substituting jf j2 by maxfjf j2 ; jgj2 g. So far we have relied on the rst argument of relation (6.1) being minimum. But we arrive at a general conclusion: minfju0j; jv0 jg 1=z2: Now, we consider monic polynomial U = u=u0 and polynomial V = v=u0, whose roots p have modules less than or equal to z2. V is almost monic because r+1 2 r (4z1 + 1). Moreover, p jUf ? V gj 2z2 r : p By (4.2) the modules of the roots of f are less than or equal to maxf jf j2 ? 1; 1g, thus bounded by z = maxf1; z1 ; z2 g. This concludes the proof. Putting together the two previous results of this section, the following proposition follows. Proposition 6.5. We use the previous notation and de ne : p p z1 = 2 maxfjf j2 ; jgj2 g ? 1; z2 = zp1=( r+1 ? 2 r (2z1 + 1)); z = maxfz1; z2 ; 1g; = 2 z2 r ; ( r ; r+1 ) = d2 (z + 1)2d 1=d : p If r+1 > z1 r = 2 and b = ( r ; r+1 ), then deg b- gcd(f; g) r + 1: Remark 6.6. The function is strictly increasing with respect to r and decreasing with respect to r+1. Thus the condition r+1 > ( r ; r+1 ) will be satis ed as soon as r is small enough. Remark 6.7. The lower bound on r+1 in the previous proposition can be replaced by the following condition which is shorter but slightly stronger. p r+1 > 4 2jf; gj r : 6.3. A Certi cation Theorem. Now, we are ready to state a theorem, which is essentially the rst mathematically rigorous result in our topic. Although it is still too rough and pessimistic to be useful in practice, it allows some understanding of the encountered di culties and provides insights for future research. We aim at satisfying simultaneously the conditions of proposition 6.5 together with the hypothesis in the rst statement of proposition 5.4. To get a clearer result, we get rid of some factors by giving rougher conditions. Theorem 6.8. The same notation is used as throughout the section. If r+1 > j d3( + 1)3d r1=d and r+1 ? 4jf; gj r < 1; r < 1 with = r+12?jf;g 4jf;gj r then, for any such that p r+1 > d ? r > d3( + 1)3d r1=d we have deg -gcd(f; g) = r + 1: Proof. It is a straightforward consequence of the previous results where we have inserted the following upper bounds. We have replaced z1 by 2jf; gj and z2 j by r+12?jf;g 4jf;gj r . Then the conditions of proposition 6.5 and of proposition 5.4 for r + 1 follow from the one stated.
NUMERICAL UNIVARIATE POLYNOMIAL GCD
19
Remark 6.9. As we said above, the given estimates are not sharp. There are clearly two directions for getting weaker conditions. First, in search of more adapted norms in order to lower the factor of r1=d , one may experiment with weighted (or Bombieri) norms which are submultiplicative, as discussed in section 2.2. Second, the exponent of r re ects the use of Ostrowski's inequality in the worst case where all zeros of the considered polynomial are closely clustered. This is too pessimistic and will typically not happen in practice. Moreover, the exponent should be controlled by a measure of the closeness of f and g to the discriminant locus.
7. Conclusion and Directions of Research
In this article, we addressed the task of computing an approximate GCD for two univariate polynomials with complex coe cients known with xed and limited precision. We gave precise de nitions and provided some analysis useful for designing algorithms capable to deal with this task. Our approach basically consists in providing, for a xed tolerance , lower and upper bounds for the degree of an -GCD, and then proceeding to minimizations in (possibly large) parameter spaces. We have shown that previous presentations and solutions were incomplete. In general, this is a complicated task, and we have presented dierent points of view in order to reach a better understanding. Indeed, the di culty of the problem is far from being uniform among all possible input data. Therefore some heuristics may give good results for some examples and not for others. We gave a brief description of some varieties naturally attached to the GCD in the considered parameter space. This description revealed that even if is small, the minimization problem could have several distant solutions and hence be badly conditioned near some singular or discriminant locus. In a future work, this geometrical analysis deserves to be developed further, both qualitatively and quantitatively. The minimization problem depends upon the choice of a norm for univariate polynomials. In this paper we chose the L2 norm of the vector of coe cients in order to be coherent with previous works against which we wished to compare our own. However, it is possible that other norms may be more adequate for our purpose, such as weighted norms. It will be interesting to distinguish between dierent possible behaviors in the computation of bounds and in the minimization problems by calculating some new invariant attached to the pair of input data. This invariant may measure the distance of the polynomial pair to some discriminant variety, possibly in the sense of [SS93]. It should provide more precise certi cation theorems adapted to dierent subcases; such theorems should not be too pessimistic in order to be useful in practice. This is a target not yet reached. In this paper we mostly assumed roundo error is negligible. A more detailed approach should attempt to incorporate computational accuracy, namely in the case where this is signi cant with respect to the prescribed tolerance. This would assert the cost attached to increasing the precision so that the error is negligible for the speci c computation. The literature on bounding roundo error is quite rich, see e.g. [Wil63]. Typically, the GCD is used in order to derive some information on the common roots of the given polynomials. It is conceivable to invert this process and compute
20
IOANNIS Z. EMIRIS, ANDRÉ GALLIGO, AND HENRI LOMBARDI
an approximate GCD once the roots have themselves been approximated. In this approach, we can apply the rich literature on root isolation as well as polynomialtime matching in order to avoid the a priori exponential complexity of the combinatorial problem.
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E-mail address:
[email protected]
Laboratoire de Mathématiques, Université de NiceSophia Antipolis, Nice 06108, and Projet S.A.F.I.R., I.N.R.I.A., B.P. 93, Sophia-Antipolis 06902, France
E-mail address:
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Laboratoire de Mathématiques, Université de Franche-Comté, Besançon 25030, France
E-mail address:
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