W. Brown and J. Traub. On Euclid's algorithm and the theory of subresultants. Journal of the Association for Computing Machinery , vol. 18:pp. 505-514, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

W.S. Brown and J.F. Traub. On euclid's algorithm and the theory of subresultants. Journal of the ACM, 18(4):505--514, October 1971.

....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 ....

W. Brown & J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18 (1971) 505-514.

....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 ....

W. Brown and J. Traub. On Euclid's algorithm and the theory of subresultants. J. ACM, 18:505--514, 1971.

....their own right. Perhaps the most prevalent class of structured matrices are Toeplitz matrices and Toeplitz block matrices, which arise in many computations on polynomials. Examples of Toeplitz block matrices are Sylvester and their submatrices known as subresultant matrices (see Brown and Traub [19]) which arise in polynomial greatest common divisors (GCD) LCD and univariate resultant computations. It is well known (see [19] and [15] that Toeplitz matrices and matrices of bounded displacement rank also arise naturally in many other algebraic computation and signal processing applications, ....

....arise in many computations on polynomials. Examples of Toeplitz block matrices are Sylvester and their submatrices known as subresultant matrices (see Brown and Traub [19] which arise in polynomial greatest common divisors (GCD) LCD and univariate resultant computations. It is well known (see [19] and [15] that Toeplitz matrices and matrices of bounded displacement rank also arise naturally in many other algebraic computation and signal processing applications, such as: linear prediction (see Makhoul [55] decoding error correcting codes and linear feedback shift register synthesis (see ....

[Article contains additional citation context not shown here]

W.S.Brown and J.F.Traub, On Euclid's Algorithm and the Theory of Subresultants, J. ACM 18 (1971), 505--514.

....j = deg (D ) We shall prove that t x he condition j I = deg (D ) is impossible. Assume that this condition is satisfied, i.e. 0 j I. By S (y ,x ) we j e denote the j th subresultant of f and g and writ S (y ,x ) s (y )x . s (y ) j j 0 j j p with s (y ) e Z[y] 0 p j (cf. Brown and Traub 71, Sec. 5] Since D divides S it follows that s (y ) 0. From j j 0 d lemma 4.1 and the fact that g is the polynomial of smalles egree solving (3.1) we conclude that D (0,b) 0. This t implies that y divides s (y ) However, we can show tha j 0 ven y divides s (y ) Let f(y,x) x a (y )x ....

Brown, W.S., Traub, J.F.: On Euclid's Algorithm - 5 and the Theory of Subresultants. J. ACM 18, 505 14 (1971).

....rational numbers or functions and GCD operations on the arising numerators and denominators. The relationship between the solution of Toeplitz systems, Pade approximations, and the Euclidean algorithm is classical. Fraction free versions [3] can be obtained from the subresultant PRS algorithm [2]. Dornstetter [6] gives an interpretation of the Berlekamp Massey algorithm as a partial extended Euclidean algorithm. We map the subresultant PRS algorithm onto Dornstetter s formulation. We note that the Berlekamp Massey algorithm is more e#cient than the classical extended Euclidean algorithm. ....

....we present a fraction free version of the Berlekamp Massey algorithm, which never introduces an element in the field of fractions of D, and in which all divisions are exact. Recall the pseudo division of polynomials [15, pp. 425 426, also pp. 428 429] in the fundamental theorem of subresultants [2]. Based on the equivalence between the Berlekamp Massey algorithm and the extended Euclidean algorithm on polynomials x and a0x a1x , Dornstetter [6] interpreted the discrepancies # i as the coe#cients in their polynomial remainder sequence (PRS) By computing the corresponding ....

Brown, W. S., and Traub, J. F. On Euclid's algorithm and the theory of subresultants. J. ACM 18 (1971), 505--514.

....generator of #1,# 2, by the Berlekamp Massey algorithm. The argument makes use of the interpretation of a Berlekamp Massey algorithm as the extended Euclidean algorithm on the polynomials F 1 = X N and F0 = #1X N 1 #2X N 2 [5] combined with the fundamental theorem on subresultants [2]. Here N is the number of elements that are considered for determining the linear generator. Dornstetter shows that #r in Step 3 of the Berlekamp Massey algorithm of subsection 2.1 is the leading coe#cient in a remainder, F i , in a polynomial remainder sequence (PRS) of F 1 and F0 . The ....

....where i # 0, then #r #=0whenever 2L r and L t, where t is the degree of the linear generator. Appealing now to the fundamental theorem of subresultants, the PRS is normal if and only if the leading coe#cient of the N i 1 st subresultant of F 1 and F0 does not vanish. By definition [2], this is the determinant of a (2i 1) 2i 1) matrix shown in figure 1. det # # # # # # # # # # # # # # # # # # # # # # # # 10 00 . 10 . 0 10 0 # 1 # i # i 1 #2i 1 . #1 . 00# 1 # i 1 # # # # # # # # # # # # # # # # # # # # # # # # ....

Brown, W. S., and Traub, J. F. On Euclid's algorithm and the theory of subresultants. J. ACM 18 (1971), 505--514.

....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 ....

[Article contains additional citation context not shown here]

W. Brown & J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18 (1971) 505-514.

....can achieve that our assumption is ful lled. 2.2 The main mathematical tools In this section we brie y describe the main mathematical ingredients we use in our algorithm. Resultants: One important concept is that of resultants and subresultants. For further information see for example [7] and [2]. Let f(x1 ; xn) l x l n l 1 x l 1 n 0 ; l 6= 0 g(x1 ; xn) mx m n m 1 x m 1 n 0 ; m 6= 0; be polynomials with coecients i ; i 2 Q[x1 ; xn 1 ] The resultant res(f; g; xn) of f and g with respect to xn is a polynomial in ....

W. Brown and J. F. Traub. On Euclid's algorithm and the theory of subresultants. Journal of the ACM, 18:505-514, 1971.

....faster and faster. However, using pseudo division in every step of the Euclidean Algorithm causes exponential coecient growth. This was suspected in the late 1960 s. Collins (1967) p. 139 writes: Thus, for the Euclidean algorithm, the lengths of the coecients increases exponentially. In Brown Traub (1971) we nd: Although the Euclidean PRS algorithm is easy to state, it is thoroughly impractical since the coecients grow exponentially. An exponential upper bound is in Knuth (1981) p. 414: Thus the upper bound [ would be approximately N 0:5(2:414) n , and experiments show that the ....

....uni es many results in the literature on various types of PRS which can be derived as corollaries from this theorem. In Section 6 we apply it to the various de nitions of polynomial remainder sequences already introduced. This yields a collection of results from Collins (1966, 1967, 1971, 1973) Brown (1971, 1978) Brown Traub (1971) Lickteig Roy (1997) and von zur Gathen Gerhard (1999) Lickteig Roy (1997) found a recursion formula for polynomial subresultants not covered by the Fundamental Theorem. We translate it into a formula for scalar subresultants and use it to nally solve an open ....

[Article contains additional citation context not shown here]

W. S. Brown and J. F. Traub, On Euclid's Algorithm and the Theory of Subresultants. Journal of the ACM 18(4) (1971), 505-514.

....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 some ....

W. S. Brown and J. F. Traub. On Euclid's algorithm and the theory of subresultants. Journal ACM, 18(4):505-514, October 1971.

....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 ....

W. Brown & J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18 (1971) 505-514.

....Toeplitz or Toeplitz like linear systems of equations. The list of such problems Parallel Complexity of Polynomial GCD and Toeplitz like. 3 includes the computation of the resultant, the Sturm and subresultant sequences, and the least common multiple (lcm) for a pair of univariate polynomials ([BT71], BGY80] BP94] sections 2.8 2.10) as well as the shift register synthesis and linear recurrence computation [Be68] Ma75] inverse scattering [BK87] adaptive filtering [K74] H91] modelling of stationary and nonstationary processes [KAGKA89] KVM78] K87] L AK84] L AKC84] numerical ....

W. S. Brown, J. F. Traub, On Euclid's Algorithm and the Theory of Subresultants, J. of ACM, 18, 4 , 505--514, 1971.

....quite inefficient. This is due to the fact that the size of the coefficients needed in each successive polynomial grows exponentially [Knu81] This is a fairly well studied problem in polynomial gcds, and has led to the development of the subresultant polynomial remainder sequence algorithm [BT71] With this approach, the coefficients of the terms of the polynomial sequence can be formulated as the determinant of a submatrix of a Sylvester matrix. Notice that since we evaluate the polynomials at u = 0, we are only interested in the constant terms of each polynomial in the sequence. ....

J.S. Brown and J.F. Traub. On euclid's algorithm and the theory of subresultant. Journal of ACM, 18(4):505--514, 1971.

....step. The 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 [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 ....

W. S. Brown and J. F. Traub. On Euclid's algorithm and the theory of subresultants. J. ACM, 18:505--514, 1971.

.... 1) and t 2 1;2 Gammaj = Gamma 0 B B a 2 1 a 2 . a j 1 1 C C A : 2 Gammaj 1) Theta1 We now extend the method of (Brent et al. 1980) to find the rank of T ; if it is singular. The algorithm uses a polynomial remainder sequence and the fundamental theorem of subresultants (Brown and Traub 1971). Let F 0 = x 2 1 and F 1 = a 2 x 2 : a 1 x a 0 . Let S j (F 0 ; F 1 ) denote the j th subresultant of F 0 and F 1 , i.e. S j (F 0 ; F 1 ) det( 0 B B B B B B B B B B B 1 0 : 0 a 2 0 : 0 0 1 : 0 a 2 Gamma1 a 2 : 0 . ....

Brown, W. S. and Traub, J. F., "On Euclid's algorithm and the theory of subresultants," J. ACM 18, pp. 505--514 (1971).

....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, ....

W. S. Brown and J. F. Traub. On Euclid's algorithm and the theory of subresultants. Journal ACM, 18(4):505--514, October 1971.

....matrix. We shall refer to this representation as the SigmaLU representation of a matrix, and to ff as the displacement rank. For pure Toeplitz matrices, for example, we have ff 2. Toeplitz like matrices are ubiquitous in symbolic computation as resultants and subresultants have this form (Brown and Traub 1971, Sasaki and 226 PASCO 94: First International Symposium on Parallel Symbolic Computation Furukawa 1984, Hong 1993) Furthermore, block matrices are used, for instance, as a parallelization technique by reducing the dimensions while increasing the running time of the arithmetic operations on the ....

Brown, W. S. and Traub, J. F., "On Euclid's algorithm and the theory of subresultants," J. ACM 18, pp. 505--514 (1971).

No context found.

W. Brown and J. Traub. On Euclid's algorithm and the theory of subresultants. Journal of the Association for Computing Machinery , vol. 18:pp. 505-514, 1971.

No context found.

W. Brown and J. Traub. On Euclid's algorithm and the theory of subresultants. Journal of the ACM, 18(4):505--514, 1971.

No context found.

W. Brown and J. Traub. On Euclid's algorithm and the theory of subresultants. J. ACM, 18:505-514, 1971.

No context found.

W. Brown & J.F. Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18 (1971) 505-514.

No context found.

W.S.Brown and J.F.Traub, On Euclid's Algorithm and the Theory of Subresultants, J. ACM 18 (1971), 505--514.

No context found.

W.S.Brown and J.F.Traub, On Euclid's algorithm and the theory of subresultants, J. ACM 18 (1971), 505--514.

No context found.

J. ACM, 18, 478-504. Brown W.S., Traub J.F., (1971). On Euclid's algorithm and the theory of subresultants. J. ACM, 18, 505-514.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC