G. E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128--142, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....common divisor (GCD) of two scalar polynomials is one of the central operations used in computer algebra systems. Supported by an Ontario Graduate Student Scholarship. The primary methods for computing scalar GCDs include the classic Euclidean algorithm [11] the subresultant GCD algorithm [7, 10], the modular GCD algorithm [7] and the heuristic algorithm [9] One sided GCDs also exist for noncommutative domains, including Ore domains such as the rings of differential or difference operators and domain of matrix polynomials. Algorithms corresponding to the subresultant and modular GCD ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128--142, January 1967.

....the Sturm sequence is an effective approach in some situations. But the length of the coefficients of the polynomials f i can grow exponentially with i. In fact, the coefficient growth is avoidable. The integer coefficients of f i share a large common factor. The subresultant algorithm of Collins [7] computes the coefficients of f i =c i , where c i is a positive integer dividing the coefficients of f i . The coefficient length of the modified sequence grows linearly. Further, the algorithm expresses the coefficients as matrix determinants. The matrices are submatrices of the resultant matrix ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. J. ACM, 14:128--142, 1967.

....demonstrate that the peak memory demands for larger problems of this class may exceed available resources. Therefore memory resource management becomes a key requirement for computer algebra systems. Algorithmic improvements are the best way to manage expression size. The subresultant algorithm [28, 18, 20] avoids intermediate expression swell in sequential computation of polynomial gcd s. Further algorithmic improvements have been developed by Corless et al. 32] where hierarchical representation tools are employed to reduce expression swell in perturbation problems. Such algorithmic improvements ....

George E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128--142, January 1967.

....in symbolic algebra systems. It is far better to determine a known common divisor and simply do the division without the need for forming fractions. This has been successively done in the case of fractionfree algorithms for solving linear systems [2] computation of scalar greatest common divisors [8, 13] and Pad e approximation [11] Our methods study the linear systems that are associated to the approximation problems. These matrices have the structure of block Sylvester matrices. By looking at these associated linear systems we are able to obtain recursive algorithms that control the size of ....

....coefficients of the polynomial reversed) Since our scalar algorithm computes Pad e approximants along an off diagonal path there is also a relation between our algorithm and fraction free computation of greatest common divisors. We expect that our algorithm gives the subresultant gcd algorithm [8, 13] as a special case. One can compare Example 6.3 to a wellknown gcd example first given by Knuth [15, Example7.6] as an example of a possible relationship between these two algorithms. A similar statement can be made regarding our algorithm and algorithms for fraction free solving of Hankel systems ....

G. Collins, Subresultant and Reduced Polynomial Remainder Sequences. J. ACM 14, (1967) 128-142

....is noteworthy because it uses only fractionfree arithmetic without coefficient GCD computations, while at the same time controls coefficient growth of intermediate computations. This is similar to the process used by the subresultant algorithm for computing the GCD of two scalar polynomials [9, 10, 11]. The algorithm is based on the FFFG fraction free method used in Beckermann and Labahn [8] which was developed for fraction free computation of matrix rational approximants, matrix GCDs and generalized Richardson extrapolation processes. In the scalar case the FFFG algorithm generalizes the ....

G. Collins. Subresultant and reduced polynomial remainder sequences. J. ACM, 14:128--142, 1967.

....algorithm. For matrix Pad e approximation the algorithm of Beckermann, Cabay and Labahn [10] uses a recursive procedure based on modified Schur complements of the associated linear equations to improve on Gaussian elimination. Finally the subresultant GCD algorithm of Brown and Collins [15, 24] gives a fast greatest common divisor algorithm in the case of scalar polynomials. In all cases our algorithm is also faster or at least as fast as those mentioned in special cases. In terms of linear algebra, we can view our problem as determining nullspaces of rectangular striped Krylov matrices ....

....simplifications. To illustrate this statement, take for instance the problem of computing a scalar GCD. Here several methods exist which avoid fractions (for a summary see, e.g. 29, Section 7.2] for instance the reduced PRS. However, only the subresultant GCD algorithm of Brown and Collins [15, 24] gives maximal Cramer solutions. We recall that, depending on the matrix C defined by our special rule (2) we may obtain a system of equations with a matrix of coefficients having a quite particular structure, for instance the following. Example 3.1 (Toeplitz and generalized Sylvester ....

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G. Collins, Subresultant and Reduced Polynomial Remainder Sequences. J. ACM 14, (1967) 128-142

....to the rules, provided that X is not smooth. Finding a singularity amounts to solving a system of algebraic equations in 3 unknowns over the algebraic closure of k. This can be done with Gr obner bases [Buc65, Buc85, BH91, BW93, Hon96a, Hon96b] as suggested in [Sch95] or with resultants [Col67, BT71, BCL82, Hon90, HON92b, HON92a, Mis93, HON95] An equation solver specialized to 3 unknowns can be found in [Kal95] The problem is not so easy, but it can be solved with a polynomial number (polynomial in the degree) of eld operations. If S is not de ned over the coecient eld k (but in ....

....turn out to be more complex with algebraic functions present. One of these is the problem of computing the singularities and multiplicities. Indeed, the algebraic functions must be treated as new unknowns computationally, which increases the costs for solving equation systems drastically (see [Col67] Another example is step 3, which becomes a nontrivial ideal membership problem, when algebraic functions are present. In short: if we perform blowing ups with Schicho s method, then we need to compute several Gr obner bases (or resultants) with an increasing number of variables during the ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. Journal ACM, 14:128-142, 1967.

....basis and they reach the optimal sequential complexity of O(n 2 ) arithmetical operations. 1. Introduction Developing algorithms for the Extended Euclidean scheme computation applied to two polynomials u(x) and v(x) over a field F is a highly important research area of Computer Algebra ( see Collins (1967), Brown (1971) Brown and Traub (1971) and Knuth (1981) and of its applications such as rational function arithmetic (Loos (1982) and squarefree and shiftfree factorizations (Loos (1983) and Yun (1976) Despite the wide literature dealing with the standard power form case, however a little ....

Collins G.E., (1967). Subresultants and reduced polynomial remainder sequences. J. ACM, 14, 128-142.

....connections between infinite Jacobi matrices, submatrices of finite order of such matrices and the associated continued fractions, see chapters IX; X;XI;XII in ( Wal48] ffl Collins proposed it as an attempt to keep small the magnitude of the coefficients in the polynomial G.C.D. computation [Col67]. He chose g = f 0 which is not a monic polynomial and k = 1. This algorithm is often used in computer algebra systems [Mig91] ffl Householder studied the same recurrence and provided relationships between the coefficients of the p.r.s. Hou74] ffl Hald seems to be the first to have ....

G. E. Collins. Subresultants and reduced polynomial remainder sequence and determinants. J. ACM, 14:128--142, 1967.

....: L; 1.3) where Gammar i 1 (x) is the remainder of the division of r i Gamma1 (x) and r i (x) and r L (x) is the greatest common divisor (gcd) of u(x) and v(x) The sequence fr i (x)g L i=0 is a polynomial remainder sequence (p.r.s. of u(x) and v(x) The sub resultant theory (see [7] [9] and [25] is the fundamental tool for computing any p.r.s. over an integral domain rather than on a field. The construction and the divisibility properties of sub resultants are generally stated in terms of determinantal relations. Differently, we feel that it is extremely illuminating to see the ....

....and so they should be avoided. An alternative approach which does not require gcd computations consists of performing pseudo divisions whereby factors known to exist are systematically removed from the coefficients of the pseudo remainders. The fundamental theorem of sub resultants (see [7] [9] and [25] relates by similarity a p.r.s. of u(x) and v(x) with the sub resultant chain of u(x) and v(x) defined as follows. The (n m) Theta (n m) resultant or Sylvester 16 matrix of u(x) and v(x) is given by S 0 = 2 6 6 6 6 6 6 6 6 6 6 6 4 un : u 0 . un : u 0 v m : ....

G.E. Collins, Subresultants and reduced polynomial remainder sequences, J. Assoc. Comput. Mach., 14 (1967), pp. 128-142.

....Example (iii) uses only polynomial operations and has a minimal coefficient growth. However it accomplishes this by using greatest common divisor calculations making it equivalent to the classic Euclidean algorithm for our purposes. The remaining two methods the Reduced PRS algorithm of Collins [21] and the Subresultant PRS algorithm of Brown and Collins [16, 21] both use only polynomial operations and has moderate coefficient growth. While the coefficient growth is not minimal it does have the advantage that the cost to reduce coefficient growth is minimal, namely a simple division by a ....

....coefficient growth. However it accomplishes this by using greatest common divisor calculations making it equivalent to the classic Euclidean algorithm for our purposes. The remaining two methods the Reduced PRS algorithm of Collins [21] and the Subresultant PRS algorithm of Brown and Collins [16, 21] both use only polynomial operations and has moderate coefficient growth. While the coefficient growth is not minimal it does have the advantage that the cost to reduce coefficient growth is minimal, namely a simple division by a known divisor, exactly the process followed in fraction free ....

G. Collins, Subresultant and Reduced Polynomial Remainder Sequences. J. ACM 14, (1967) 128-142

....package GlobSol. In the final part of the paper, we suggest an optimization formulation of the multivariate problem. 2 Previous Work Algorithms for computing the exact GCD of two polynomials are well understood. Many of these algorithms are based on the Euclidean algorithm and subresultants ([2, 4]) The field of approximate GCD computation, however, is not so well developed, though there have been significant advances in the past decade. In 1985, Schonhage ( 19] proposed the notion of quasiGCD, but assumes that it is possible to get more figures of accuracy for the polynomial ....

Collins, G. E. Subresultants and reduced polynomial remainder sequences. J. ACM 14, 1 (1967), 128--142.

....formulations has led to a subresultant theory, developed simultaneously by G.E. Collins and W.S. Brown and J. Traub. The subresultant theory produced an efficient algorithm for computing polynomial GCDs and their resultants, while controlling intermediate expression swell [Brown and Traub 1971, Collins 1967, Knuth 1981a] It should be noted that by adopting an implicit representation for symbolic objects, the intermediate expression swell introduced in many symbolic computations can be palliated. Recently, polynomial GCD algorithms have been developed that use implicit representations and thus avoid ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. J. ACM, 14:128-- 142, 1967.

....of the main theorem. 2 Main Result The main goal of this section is to give a precise statement of the relations mentioned in the introduction. For this, we briefly recall some basic notions such as resultants, principal subresultant coefficients, subresultants, and composition. For details, see [2, 1, 4, 5]. For those who are familiar with them can skip the following definitions. Throughout this paper, let F stand for a field. Let A = P m i=0 a i x i and B = P n i=0 b i x i be two non zero polynomials over F of degree m and n. Definition 1 (Sylvester Matrices) The Sylvester matrix of A and B, ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. Journal ACM, 14:128-- 142, 1967.

.... basis method (Buchberger, 1965; Buchberger, 1985; Gao and Chou, 1992; Kalkbrener, 1990; Hoffmann, 1989; Winkler, 1988) Collins cylindrical algebraic decomposition method (Collins, 1975; Hong, 1990; Collins and Hong, 1991) Ritt Wu s characteristic set method (Wu, 1986) and the resultants (Collins, 1967; Brown and Traub, 1971) etc. However one can often devise a more efficient simpler method for a particular problem class by taking advantage of its special structure. One such method was given by one of the authors (Hong, 1996; Hong, 1995) for a certain sub class of trigonometric curves. The ....

....p Gamma1 : C R. This is a general way to invert rational functions, e.g. in (Schicho, 1995) Note that the gcd is the last nonvanishing polynomial in the polynomial remainder sequence of P (s) Gamma x; Q(s) Gamma y (over K) Then, by a theorem of Collins and Habicht (see (Habicht, 1948; Collins, 1967)) the first subresultant of P (s) Gamma x; Q(s) Gamma y (still over K) is the gcd. Subresultants commute with homomorphisms. Therefore, R 0 R 1 s is linear modulo R, and GammaR 0 =R 1 represents the rational function p Gamma1 . 2 ....

Collins, G. E. (1967). Subresultants and reduced polynomial remainder sequences. Journal ACM, 14:128--142.

....the most eOEcient, since it strikes a balance between coeOEcient growth and computational eoeort. 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 coeOEcients, whereas the plain euclidean algorithm may cause an exponential swell. A signi cant portion of the literature is devoted to ....

G.E. Collins, Subresultants and reduced polynomial remainder sequences, J. ACM 14 (1967), 128142.

....of n greatest common divisors of coefficients, each of which will requirek (if that is their degree) greatest common divisors of their coefficients, and so on. Since k seems to double each step, this problem seems insuperable. Fortunately, there are two solutions available. The first, due to Collins [1967] and Brown [1971] is to predict the cancellations, by the sub resultant method, and thus to eliminate the recursive greatest common divisor computations. The second, also due to Brown [1971] and Collins [1971] with a Hensel s Lemma variation due to Moses Yun [1973] abandons direct computation ....

Collins, G.E., Subresultants and Reduced Polynomial Remainder Sequences. J. ACM14 (1967) pp. 128-142.

....Sylvester formulations has led to a subresultant theory, developed simultaneously by G.E. Collins and W.S. Brown and J. Traub. The subresultant theory produced an efficient algorithm for computing polynomial GCDs and their resultants, while controlling intermediate expression swell [Bro71, BT71, Col67, Col71, Knu81] It should be noted that by adopting an implicit representation for symbolic objects, the intermediate expression swell introduced in many symbolic computations can be palliated. Recently, polynomial GCD algorithms have been developed that use implicit representations and thus ....

G. E. Collins. Subresultants and reduced polynomial remainder sequences. J. ACM, 14:128--142, 1967.

....approximate GCDs. In the final part of the paper, we suggest an optimization formulation of the multivariate problem. 2 Previous Work Algorithms for computing the exact GCD of two polynomials are well understood. Many of these algorithms are based on the Euclidean algorithm and subresultants ([4, 6]) The field of approximate GCD computation, however, is not so well developed, though there have been significant advances in the past decade. In 1985, Schonhage ( 22] proposed the notion of quasiGCD, but assumes that it is possible to get more figures of accuracy for the polynomial ....

Collins, G. E. Subresultants and reduced polynomial remainder sequences. J. ACM 14, 1 (1967), 128--142.

....polynomial systems are generated (by replacing the dividing polynomial with its initial and reductum whenever pseudo division is performed) and to each of them the same procedure is applied recursively. As another algorithm, we propose using subresultant polynomial remainder sequences (SPRS, cf. [1, 3]) For any given polynomial system [P; Q] we perform an elimination top down from x n to x 1 with two kinds of splitting. For each x k , a single polynomial P with lv(P ) x k is produced by means of forming SPRS iteratively; this polynomial is then used to pseudo divide the polynomials in Q. ....

....according to the following fact: Let P 1 ; P 2 ; P s be an SPRS of P 1 and P 2 wrt x k and let I i be the leading coefficient of P i wrt x k for each i. Then P 1 = 0; P 2 = 0 and I 1 I 2 6= 0 if and only if P i = 0; I i 1 = 0; I s = 0 and I 1 I 2 I i 6= 0 for some 2 i s (see [1, 3, 8]) In each case of splitting, one produced system is taken to update the current system [P; Q] and the others are collected and to be treated similarly. In this way, we can compute a zero decomposition of the same form (2) with the formation of SPRS as basic operation. By renaming the variables, a ....

Collins, G. E. Subresultants and reduced polynomial remainder sequences. J. ACM 14 (1967), 128--142.

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G. E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128--142, 1967.

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G. Collins. Subresultant and reduced polynomial remainder sequences. J. ACM, 14:128-142, 1967.

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G. Collins, Subresultant and Reduced Polynomial Remainder Sequences. J. ACM 14, (1967) 128-142

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George E. Collins, Subresultants and Reduced Polynomial Remainder Sequences. Journal of the ACM 14(1) (1967), 128-142.

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Collins, G. (1967), "Subresultants and Reduced Polynomial Remainder Sequences," JACM, Vol. 14, No. 1, Jan. 1967, pp. 128-142.

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