Kaltofen, E., Krishnamoorthy, M., and Saunders, B. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. ALg. Disc. Math (1987), 683--690.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....p, so trying to compute the HNF of A modulo p does not give any useful information. Interestingly, the related problem of computing the HNF of a polynomial matrix (i.e. a matrix whose entries are polynomials with coe#cients in a field) can be solved in NC and therefore in polylogarithmic space [16, 23]. However, these techniques do not seem to apply to integer matrices and the problem of finding an NC algorithm for integer HNF is still open. The main contribution of this paper is a new practical algorithm to compute the Hermite Normal Form of integer matrices. The main advantage of the new ....

Kaltofen, E., Krishnamoorthy, M., and Saunders, B. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. ALg. Disc. Math (1987), 683--690.

....matrices [15] this bounds the degrees of the polynomials involved during the calculations, but seems to be inadequate to bound the coefficients of those polynomials in particular over the rational polynomials. The first polynomialtime algorithm to compute the Smith form over Q[x] appeared in [13], it is based on the Chinese remainder algorithm. Then it has been established in [26] that the form and associated unimodular transformations can be computed over any ring K[x] with coefficient lengths remaining polynomially bounded over Q[x] A drawback of this latter method is that a field ....

....of the polynomials in input, and consequently the unimodular transformations are not obtained over K[x] but involve elements of the extension. As for the computation of the Frobenius form, randomization can remove the sequential iterations (successive triangularizations) This has been used in [13,14,24]. The key idea is somehow equivalent to the one used for the Frobenius form: after a random transformation of the input matrix only one triangularization is sufficient with high probability. This gives a probabilistic parallel solution for the problem and speed the sequential methods themselves. ....

[Article contains additional citation context not shown here]

E. Kaltofen, M.S. Krishnamoorthy, and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomials matrices. SIAM J. Alg. Disc. Meth., 8 4, pp 683-690, 1987.

....a unimodular triangularization. ffl Parallel point of view. Very few algorithms exist that compute the Hermite form fast using polynomially many processors. The solutions in [18,16] are elimination processes and are thus highly sequential. The problem has been shown to belong to the class NC in [22]. Nevertheless, this latter approach involves O(n d) structured linear systems of dimension n d. Since quite prohibitive, the cost has not been precisely computed by the authors. For a number of parallel steps in O(log ) one can evaluate the number of needed processors to be in O( ....

E. Kaltofen, M.S. Krishnamoorthy, and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomials matrices. SIAM J. Alg. Disc. Meth., 8 4, pp 683-690, 1987.

....p, so trying to compute the HNF of A modulo p does not give any useful information. Interestingly, the related problem of computing the HNF of a polynomial matrix (i.e. a matrix whose entries are polynomials with coecients in a eld) can be solved in NC and therefore in polylogarithmic space [16, 23]. However, these techniques do not seem to apply to integer matrices and the problem of nding an NC algorithm for integer HNF is still open. The main contribution of this paper is a new practical algorithm to compute the Hermite Normal Form of integer matrices. The main advantage of the new ....

Kaltofen, E., Krishnamoorthy, M., and Saunders, B. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. ALg. Disc. Math (1987), 683-690.

....log 2 kAk) 4 Computing rational forms of integer matrices bit operations using standard arithmetic. However, Ozello does not address the question of choosing good primes p modulo which the Frobenius form of A mod p equals F mod p, and the algorithm stated there may produce incorrect results. Kaltofen et al. 1987, 1990) demonstrate fast parallel algorithms for computing the Frobenius and rational Jordan forms of matrices over abstract elds, nite elds and the rationals. While these algorithms are not particularly fast sequentially, we employ some of the techniques developed here in our fast sequential ....

....Frobenius form F . If p is a bad prime such that the Frobenius form F p of A p = A mod p 2 Z n n p does not satisfy F p = F mod p) 2 Z n n p , and k is the maximal index for which f (p) k 6= f k mod p, then deg f (p) k deg f k . Giesbrecht Storjohann 5 Proof. The proof is similar to Kaltofen et al. 1987), Lemma 4.1. Since i is monic, we clearly have i mod p dividing (p) i , so deg (p) i deg i and (p) i = i mod p in case the degrees are equal. Since f i = i = i 1 we have deg f i = deg i deg i 1 . The maximal k 2 N for which f (p) k 6= f k mod p is precisely the ....

E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebraic and Discrete Methods 8, pp. 683-690, 1987.

....prime moduli to obtain correct Smith normal form of a given matrix. The answer is known only for special classes of matrices. INRIA Parallelism in Hermite and Smith normal forms 9 Recently new methods has been developed to compute Smith normal form via randomization technique as is described in [28, 35]. The question if the number of bits for the largest matrix entry is polynomialy bounded in the Bradley algorithms for Hermite normal form and Smith normal form is still open even if some experiments [18] seem to show it is not. RR n2077 10 T. Hr uz et D. Fortin 2 Upper triangular unimodular ....

Erich Kaltofen, M. S. Krishnamoorthy, B. David Saunders, Fast Parallel Computation of Hermite and Smith Forms of Polynomial Matrices, SIAM J. Alg. Disc. Meth., Vol. 8, No. 4, October 1987, pp. 683-690.

....a unimodular triangularization. ffl Parallel point of view. Very few algorithms exist that compute the Hermite form fast using polynomially many processors. The solutions in [18,16] are elimination processes and are thus highly sequential. The problem has been shown to belong to the class NC in [22]. Nevertheless, this latter approach involves O(n 2 d) structured linear systems of dimension n 3 d. Since quite prohibitive, the cost has not been precisely computed by the authors. For a number of parallel steps in O(log 3 ) one can evaluate the number of needed processors to be in O( ....

E. Kaltofen, M.S. Krishnamoorthy, and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomials matrices. SIAM J. Alg. Disc. Meth., 8 4, pp 683-690, 1987.

....we write kvk = kvk1 , for B 2 Z m Thetan we write kBk = kBk Delta = max ij jB ij j, and for g = P 0it b i x i 2 Z[x] we write kgk = max i jb i j. We give two algorithms here. The first is for sparse matrices A, and is based on a combination of techniques developed in Wiedemann (1986) Kaltofen et al. 1987, 1990) and Kaltofen Saunders (1991) An extended abstract of a similar algorithm to this first appeared in Giesbrecht (1996) The second algorithm is for dense matrices and is based on some of the same techniques with asymptotically fast matrix arithmetic replacing Wiedemann s sparse matrix ....

E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders, Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebraic and Discrete Methods 8 (1987), 683--690.

....command uses a variation which also computes transforming matrices of the modular determinant approach. UMVR4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 [0] 0] 0] 1] 0] 0] 0] 0] 1] 0] 0] 0] 1] 0] 0] 0] 0] 1] 0] 0] 0] 2] 0] 1] 1] 0] 1] 0] 4] 0] 7] 0] 5] 5] 3] 6] [12] [16] 12] 19] 11] 17] 17] 15] 18] 12] 17] 13] 20] 11] 18] 18] 16] 19] 19] 27] 22] 30] 23] 28] 28] 26] 29] 29] 37] 32] 39] 32] 38] 38] 36] 39] 30] 38] 32] 40] 33] 39] 39] 37] 40] 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and VMVR4 2 6 6 6 6 6 6 6 6 6 6 6 6 ....

....uses a variation which also computes transforming matrices of the modular determinant approach. UMVR4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 [0] 0] 0] 1] 0] 0] 0] 0] 1] 0] 0] 0] 1] 0] 0] 0] 0] 1] 0] 0] 0] 2] 0] 1] 1] 0] 1] 0] 4] 0] 7] 0] 5] 5] 3] 6] 12] 16] [12] [19] 11] 17] 17] 15] 18] 12] 17] 13] 20] 11] 18] 18] 16] 19] 19] 27] 22] 30] 23] 28] 28] 26] 29] 29] 37] 32] 39] 32] 38] 38] 36] 39] 30] 38] 32] 40] 33] 39] 39] 37] 40] 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and VMVR4 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ....

[Article contains additional citation context not shown here]

Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods 8 (1987), 683--690.

....matrices [15] this bounds the degrees of the polynomials involved during the calculations, but seems to be inadequate to bound the coefficients of those polynomials in particular over the rational polynomials. The first polynomialtime algorithm to compute the Smith form over Q[x] appeared in [13], it is based on the Chinese remainder algorithm. Then it has been established in [26] that the form and associated unimodular transformations can be computed over any ring K[x] with coefficient lengths remaining polynomially bounded over Q[x] A drawback of this latter method is that a field ....

....of the polynomials in input, and consequently the unimodular transformations are not obtained over K[x] but involve elements of the extension. As for the computation of the Frobenius form, randomization can remove the sequential iterations (successive triangularizations) This has been used in [13,14,24]. The key idea is somehow equivalent to the one used for the Frobenius form: after a random transformation of the input matrix only one triangularization is sufficient with high probability. This gives a probabilistic parallel solution for the problem and speed the sequential methods themselves. ....

[Article contains additional citation context not shown here]

E. Kaltofen, M.S. Krishnamoorthy, and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomials matrices. SIAM J. Alg. Disc. Meth., 8 4, pp 683-690, 1987.

....U and V such that UAV = S is in Smith normal form within the same asymptotic complexity as they require to produce S alone. One reason for producing multipliers is to verify correctness. In particular, Kaltofen, Krishnamoorthy and Saunders have given a Monte Carlo probabilistic algorithm in [11] that computes the Smith normal form but does not produce pre and post multipliers. The drawback of the KKS Monte Carlo algorithm is that it may return an incorrect result which cannot be detected easily. The only other algorithm that we are aware of that solves for the Smith normal form without ....

....The drawback of the KKS Monte Carlo algorithm is that it may return an incorrect result which cannot be detected easily. The only other algorithm that we are aware of that solves for the Smith normal form without multipliers is also given by Kaltofen, Krishnamoorthy and Saunders in [11]. Here, the authors give a proof that computing the Smith normal form over Q[x] is in P , the class of polynomial time algorithms. Their algorithm uses the fact, a consequence of Kannan [13] that computing the Smith normal form over GF(p) x] is in the computational class P . Given a nonsingular A ....

[Article contains additional citation context not shown here]

Erich Kaltofen, M. S. Krishnamoorthy, and B. David Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods, 8:683--690, 1987.

....2 c 1 Delta Delta Delta x c l 2 c l . xn ) x c1 n c 1 Delta Delta Delta x c l n c l Theorem10. S c1 ; S c2 ; S c l have solutions iff S is ACUNh unifiable and (s i ) ACUNh (t i ) 1 i k. Since solvability of linear equations over Z 2 [h] is in P [14], we get Corollary11. Elementary ACUNh unifiability problem can be solved in polynomial time. General ACUNh unifiability is again NP complete. The above method of solving the ACUNh unification problem can be generalized to allow more than one homomorphism. However, this involves solving ....

E. Kaltofen, M.S. Krishnamoorthy and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods 8 (4) October 1987, 683--690.

....algorithm replacing division. It is well known that this procedure results impractical for its bad numerical behavior. The idea of converting triangularization over the ring of polynomials to that of solving linear systems over the field of reals appears to have first been used by Kaltofen et al. [10]. The goal of the authors was to establish a parallel complexity result and Storjohann [19, Chapter 4] reexamined their approach to obtain sequential results. The key point was to note that one is led to a linear system that does not involve unknown coefficients of the triangular form, but only ....

E. Kaltofen, M. S. Krishnamoorthy and B. D. Saunders "Fast Parallel Computations of Hermite and Smith Forms of Polynomial Matrices", SIAM Journal of Algebraic and Discrete Methods, Vol. 8, pp. 683--690, 1987.

....deg f = 0, that is, with f 2 F . When #F deg N there may not exist a solution f 2 F . A standard approach taken when the ground field F is too small is to work over an algebraic extension field K of F which has a sufficient number of elements. For example, the randomized algorithms proposed in [6, 7, 10] for computing normal forms of matrices over F [x] as well as the deterministic normal form algorithm in [12] all take this approach. A drawback of working over K is that the final result may This work has been supported by grants from the Swiss Federal Office for Education and Science in ....

Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods 8 (1987), 683--690.

....O (r log kAk) bits. These integers converge on the determinantal divisors of A as the algorithm proceeds. Previous algorithms, which work with dense m Theta n matrices in Z d , require O(nm log d) or O (nm 2 log kAk) bits of storage. Our Monte Carlo algorithm is more akin to the methods of Kaltofen et al. 1987) for computing Smith normal forms of matrices of polynomials than to the modulo determinant algorithms discussed. Hafner McCurley (1991) conjectured that these methods might be adapted to work for integer matrices. Our algorithm demonstrates this, showing that we can find the Smith normal form ....

....S = diag(s1 ; sr ; 0; 0) 2 Z m Thetan . 2 2 A faster, space efficient Smith form algorithm In this section we present our new, faster and more space efficient Monte Carlo algorithm for the Smith normal form of an A 2 Z m Thetan of rank r. The algorithm is related to that of Kaltofen et al. 1987) for computing Smith normal forms of matrices of polynomials over a field K. They show that for A 2 K[x] n Thetan , to compute the kth determinantal divisors dk you need only consider the GCD of the leading minors of two random perturbations of A, and not of the Gamma n k Delta 2 minors of ....

E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebraic and Discrete Methods 8, pp. 683--690, 1987.

....matrices functions, solving matrix equations or differential equations and systems. The Jordan form has been widely studied from a theoretical point of view [10] and sequential polynomial time algorithms are known [20, 14, 13] From a parallel point of view, the first fast parallel algorithms [17, 13] are randomized. Those results have been improved in[21, 22] where algorithms are given to compute the Jordan normal form in parallel arithmetic time O Gamma log 3 n Delta using n O(1) processors for any field F . The later algorithm [22] is based on algebraic number computation in a ....

....the Jordan normal form in parallel arithmetic time O Gamma log 3 n Delta using n O(1) processors for any field F . The later algorithm [22] is based on algebraic number computation in a parallel D5 [7, 8, 9] arithmetic manner and computes the symbolic Jordan normal form as defined in [17]. The main tool involved in parallel D5 arithmetic is the computation of gcd free bases [16] of polynomials. Using algorithms proposed in [17, 1] this computation requires O Gamma log 3 n Delta arithmetic steps which dominates the cost of the computation of the Jordan form in [21] This ....

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Kaltofen, E., Krishnamoorthy, M., and Saunders, B. D. Fast Parallel Computation of Hermite and Smith Forms of Polynomial Matrices. SIAM J. Alg. Disc. Meth. 8, 4 (Oct. 1987), 683--690.

.... computing Hermite normal forms over Q[x] is in P (the class of polynomial time algorithms) was Kannan in [13] A fast parallel algorithm for computing the Hermite normal form and pre multiplier matrix for a square nonsingular polynomial matrix is given by Kaltofen, Krishnamoorthy and Saunders in [11] and a generalization that works for rectangular input matrices in [12] We remark that the modulo arithmetic algorithms for matrices over the integers presented in [3, 7, 9] can be modified to work for input matrices over F[x] but suffer from excessive coefficient growth when F = Q. A recent ....

....method for computing the Hermite normal form of a polynomial matrix is given by Labhalla, Lombardi and Marlin in [14] Their approach is to convert the problem to one of triangularizing a large matrix over the coefficient field. It is important to note that the Hermite normal form algorithms in [3, 9, 11, 13] are initially presented for the special case of square nonsingular input matrices. Hafner and McCurley present in [7] a generalization of the modulo arithmetic approach that works for rectangular matrices but they are not able to directly compute a candidate for a premultiplier matrix. To handle ....

[Article contains additional citation context not shown here]

Erich Kaltofen, M. S. Krishnamoorthy, and B. David Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM Journal of Algebraic and Discrete Methods, 8:683--690, 1987.

....procedure results impractical for its bad behavior with respect to intermediate expression swell. The idea of converting triangularization over the ring of polynomials to that of solving linear systems over the field of reals appears to have first been used by Kaltofen, Krishnamoorthy and Saunders [8]. The goal of the authors was to establish a parallel complexity result and Storjohann [15] reexamined their results to obtain a more efficient sequential solution. The key point was to note that the linear system constructed does not involve unknown coefficients of the triangular form, but only ....

....control literature. It should thus seem surprising that, up to the authors knowledge, no results are available so far concerning an interpolation approach to triangularization of polynomial matrices. The objective of this paper is to fill this gap by using the linear system approach described in [8, 15, 11]. It aims to extend the domain of application of interpolation techniques by showing that they can also be applied to solve the general problem of the reduction into polynomial matrix triangular form. Since the interpolation approach generally leads to efficient and easy to implement numerical ....

E. Kaltofen, M. S. Krishnamoorthy and B. D. Saunders "Fast Parallel Computations of Hermite and Smith Forms of Polynomial Matrices", SIAM Journal of Algebraic and Discrete Methods, Vol. 8, pp. 683--690, 1987.

.... applications a la fois dans le cas o u l anneau des coefficients est Z [12] et dans le cas o u cet anneau est un anneau de polynomes a une variable [3, 6] D un point de vue algorithmique, on sait calculer la forme en temps s equentiel polynomial sur les entiers [10] et sur les polynomes [9, 7, 13]. En parall ele, trouver un algorithme rapide est toujours une question ouverte dans le cas entier, question qui repose sur la parall elisation du pgcd d entiers [4] Dans le cas polynomial, on ne connaissait jusqu alors que des algorithmes rapides mais probabilistes [7, 8, 5] le probl eme ....

.... point of view [5] the form is well known and has many applications, both in the integer case [17] and in the polynomial case [5,8] From an algorithmic point of view, polynomial time algorithms are known for a sequential computation of the SNF with integer entries [12] or polynomial ones [11,9,18]. In parallel, finding a fast algorithm is still a difficult question in the integer case, since the problem relies on integer gcd computations [6] In the polynomial case, fast algorithms may be found in [9,10,7] but they use random choices, so that the problem is only known to be in RNC. The ....

[Article contains additional citation context not shown here]

E. Kaltofen, M.S. Krishnamoorthy, and B.D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomials matrices. SIAM J. Alg. Disc. Meth., 8 4, pp 683-690, 1987.

....p modulo which the Frobenius form of A mod p equals F mod p. Part of our contribution here is to show how to identify these good primes quickly. Ozello (1987b) also proposes a probabilistic algorithm, but the estimate of the probability of success is insufficient to prove a lower expected cost. Kaltofen et al. 1987, 1990 demonstrate fast parallel algorithms for computing the Frobenius and rational Jordan forms of matrices over abstract fields, finite fields and the rationals. While these algorithms are not particularly fast sequentially, we employ some of the techniques developed here in our fast sequential ....

.... p does not satisfy Fp j F mod p, and the degree of the ith invariant factor of Ap is d (p) i for 1 i l, then (d1 ; dk ) is lexicographically larger than (d (p) 1 ; d (p) l ) A similar lemma to the above, for Smith normal forms of matrices of polynomials, is shown by Kaltofen et al. 1987), Lemma 4.1. Now we have a criteria for rejecting bad primes assuming we have one good prime: the degree sequence (d (p) 1 ; d (p) k ) of the invariant factors of A modulo a good prime p will be lexicographically greater than the degree sequence of A modulo a bad prime p. How many good ....

E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Algebraic and Discrete Methods 8, pp. 683--690, 1987.

....author) This paper appears in Linear Algebra and Its Applications 136, pp. 189 208 (1990) The results in Section 2 and 4 were first announced in the Proc. EUROCAL 87, Springer Lect. Notes Comput. Sci 378, pp. 317 322 (1989) 2 This paper is third in a series on canonical forms of matrices [13], 12] Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix in F [x ] The algorithm has two important advantages over our previous one. The multipliers relating the Smith form to the input matrix are computed and the algorithm is of Las Vegas ....

....that by multiplying the input matrix with a certain randomly chosen matrix, the new randomized matrix will require with high probability only two Hermite steps before the Smith normal form appears. The proof of this fact uses ideas similar to those for our Monte Carlo Smith normal form algorithm [13], but is more complicated. An unlucky premultiplication is discovered immediately if after two Hermite steps we do not obtain a Smith normal form. The point is now that if we do, we must have the unique Smith normal form of the input matrix together with the unimodular pre and post multipliers. ....

[Article contains additional citation context not shown here]

Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D., "Fast parallel computation of Hermite and Smith forms of polynomial matrices," SIAM J. Alg. Discrete Meth., vol. 8, pp. 683-690, 1987.

....This paper appears in Linear Algebra and Its Applications 136, pp. 189 208 (1990) The results in Section 2 and 4 were first announced in the Proc. EUROCAL 87, Springer Lect. Notes Comput. Sci 378, pp. 317 322 (1989) 2 This paper is third in a series on canonical forms of matrices [13], 12] Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix in F [x ] m n . The algorithm has two important advantages over our previous one. The multipliers relating the Smith form to the input matrix are computed and the algorithm is of Las ....

....that by multiplying the input matrix with a certain randomly chosen matrix, the new randomized matrix will require with high probability only two Hermite steps before the Smith normal form appears. The proof of this fact uses ideas similar to those for our Monte Carlo Smith normal form algorithm [13], but is more complicated. An unlucky premultiplication is discovered immediately if after two Hermite steps we do not obtain a Smith normal form. The point is now that if we do, we must have the unique Smith normal form of the input matrix together with the unimodular pre and post multipliers. ....

[Article contains additional citation context not shown here]

Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D., "Fast parallel computation of Hermite and Smith forms of polynomial matrices," SIAM J. Alg. Discrete Meth., vol. 8, pp. 683-690, 1987.

....[82, 98] asymptotically fastest polynomial factorization algorithm over high algebraic extensions of finite fields [93] 2. 2 Linear Algebra ffl Processor efficient parallel algorithms for solving general linear systems over a field [62, 66] ffl Parallel algorithms for matrix canonical forms [32, 54]. ffl Probabilistic analysis [80] and implementation [76, 89, 101] of the block Wiedemann parallel sparse linear system solver. ffl Probabilistic analysis of the Lanczos sparse linear system solver over finite fields [94] 2.3 Divisions in Algebraic Complexity Theory ffl Polynomial length ....

E. Kaltofen, M. S. Krishnamoorthy, and B. D. Saunders. Fast parallel computation of Hermite and Smith forms of polynomial matrices. SIAM J. Alg. Discrete Math., 8:683--690, 1987. Ka86:stoc

.... in the tutorial is Chistov s (1985) The randomized matrix rank algorithm is due to Borodin et al. 1982) while a deterministic solution was given by Mulmuley (1987) Such randomization techniques apply to matrix canonical form computations, such as the rational and Jordan normal forms; see (Kaltofen et al. 1987) and (Kaltofen et al. 1989) Implementations I am aware of three Ph.D. theses written on implementation issues of algebraic algorithms on parallel machines, Watt s (1985) Johnson s (1988) and Ponder s (1988) Efforts to date are still scattered, among them the Reduce implementation on the CRAY ....

Kaltofen, E., Krishnamoorthy, M. S., and Saunders, B. D., "Fast parallel computation of Hermite and Smith forms of polynomial matrices," SIAM J. Alg. Discrete Math. 8, pp. 683--690 (1987).

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