P.D. Domich, R. Kannan, L.E. Trotter, Hermite normal form computation using modulo determinant arithmetic, Math. Op. Res. 12 (1987), 50-59.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....column operations over the integers. Their procedure is successively improved by Chou and Collins [2] who prove better bounds for both the time and space used, and Iliopoulos [15] who extends this procedure with modular techniques. For the case of square matrices, Domich, Kannan and Trotter [5], limit the size of the entries by using computation modulo suitably chosen numbers. Essentially, they describe an algorithm which in a first stage performs Gaussian elimination modulo the determinant, and then recovers the Hermite normal form of the original matrix. A technique useful to improve ....

....form of the original matrix. A technique useful to improve the time and space e#ciency of all these algorithms is introduced [4] However, the technique only applies for the case when a factorization of the determinant of the input matrix is known. Hafner and McCurley [12] extend the results of [5] to nonsquare matrices and also show how to use fast matrix multiplication in computing a triangular form of the input matrix. However, the result produced by this algorithm is not necessarily in Hermite normal form since o# diagonal elements may be bigger than the corresponding diagonal element. ....

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Domich, P. D., R.Kannan, and L.E.Trotter. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research 12, 1 (Feb. 1987), 50--59.

....geometry that can be reduced to the determinant question; the reader may refer to [11, 12, 9, 10, 46, 43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41, 28, 23, 51] or in computational number theory [16] In this paper we survey the known major results for computing the determinant and its sign and give the corresponding references. Our discussion focuses on theoretical computational complexity This material is based on work supported in part by the ....

P.D. Domich, R. Kannan, and L.E. Trotter Jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):5059, 1987.

....column operations over the integers. Their procedure is successively improved by Chou and Collins [2] who prove better bounds for both the time and space used, and Iliopoulos [15] who extends this procedure with modular techniques. For the case of square matrices, Domich, Kannan and Trotter [5], limit the size of the entries by using computation modulo suitably chosen numbers. Essentially, they describe an algorithm which in a rst stage performs Gaussian elimination modulo the determinant, and then recovers the Hermite normal form of the original matrix. A technique useful to improve ....

....form of the original matrix. A technique useful to improve the time and space eciency of all these algorithms is introduced [4] However, the technique only applies for the case when a factorization of the determinant of the input matrix is known. Hafner and McCurley [12] extend the results of [5] to nonsquare matrices and also show how to use fast matrix multiplication in computing a triangular form of the input matrix. However, the result produced by this algorithm is not necessarily in Hermite normal form since o diagonal elements may be bigger than the corresponding diagonal element. ....

[Article contains additional citation context not shown here]

Domich, P. D., R.Kannan, and L.E.Trotter. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research 12, 1 (Feb. 1987), 50-59.

....Special issue 3 December 2001 to the determinant question; the reader may refer to [11,12,9,10,46,43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41,28,23,51] or in computational number theory [16] In this paper we survey the known major results for computing the determinant and its sign and give the corresponding references. Our discussion focuses on theoretical computational complexity aspects. For an input matrix A # Z nn with infinity matrix ....

P.D. Domich, R. Kannan, and L.E. Trotter Jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):50--59, 1987.

....n unimodular matrix R such that H = AR is a lower triangular with positive diagonal elements. Further, each off diagonal element of H is non positive and strictly less in absolute value than the diagonal element in its row. H is called the Hermite normal form of A. It is known (see for example [7]) that the Hermite normal form of a matrix A is unique and the right unimodular matrix is unique too . Elementary operations from Definition 1 can be realized through multiplication of the matrix A by special matrices called elementary matrices. Elementary matrices are unimodular. 2 Theorem 3 ....

.... log 2 jj A jj) Chou and Collins in 1982 introduced a new algorithm for Hermite normal form where by reordering the computation in Kannan Bachem process they achieved an upper bound of n(log 2 (n) log 2 jj A jj) In 1987 Domich, Kannan and Trotter introduced a modulo determinant arithmetic [7]. They proved an upper bound n(log 2 (n) log 2 jj A jj) on the number of bits for their modular method. A modulo determinant computation has one very important feature, it can be combined with any other method of computing a Hermite normal form of matrix. Because Hermite normal form of an integral ....

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P. D. Domich, R. Kannan and L. E. Trotter, Jr. Hermite normal form computation using modulo determinant arithmetic, Math. of Operating Research, Vol. 12, No. 1, February 1987, pp. 50-59. 5

....See also Havas and Majewski (1994) Mulders and Storjohann: On Lattice Reduction for Polynomial Matrices 16 2. Modulo determinant methods, performing arithmetic modulo a multiple of a determinant. The modular arithmetic avoids exponential growth of intermediate expressions (Domich (1989) Domich et al. 1987); Hafner and McCurley (1991) Iliopoulos (1989) Storjohann and Labahn (1996) 3. Coecient methods for polynomial matrices, translating the Hermite problem to a problem over the coecient ring (Bitmead et al. 1978) Labhalla et al. 1996) Villard (1996) Some of these algorithms use fast matrix ....

P. D. Domich, R. Kannan, and L. E. Trotter, Jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):50-59, 1987.

....column operations over the integers. Their procedure is successively improved by Chou and Collins [2] who prove better bounds for both the time and space used, and Iliopoulos [14] who extends this procedure with modular 2 techniques. For the case of square matrices, Domich, Kannan and Trotter [4], limit the size of the entries by using computation modulo suitably chosen numbers. Essentially, they describe an algorithm which in a first stage performs Gaussian elimination modulo the determinant, and later from the result that is obtained it recovers the Hermite normal form of the original ....

....is obtained it recovers the Hermite normal form of the original matrix. A technique useful to improve the space efficiency of all these algorithms is introduced [3] for the case when a factorization of the determinant of the input matrix is known. Hafner and McCurley [11] extend the results of [4] to nonsquare matrix, but the main result of their paper is an algorithm for matrix triangularization based on fast matrix multiplication. It is not shown how to efficiently compute the Hermite normal form of such a matrix. Storjohann [19] gives a fast procedure which does this, and obtains the ....

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P. D. Domick, R. Kannan, and L. T. Jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):50--59, Feb. 1987.

....of a 20 Theta 20 matrix with all its entries in the range [0,10] which yields entries of more than 5000 digits when the standard process of first going to a triangular matrix and then reducing modulo the diagonal entries is used in calculating the Hermite normal form. Domich, Kannan and Trotter [12] obtained a more computationally feasible algorithm for HNF and SNF. Hafner and McCurley have modified this to give the following algorithm: Choose a number h which is divisible by det A. Do row and column operations mod h. For the SNF we proceed as follows: Using row and column operations mod h ....

P. D. Domich, R. Kannan and L. E. Trotter, Hermite normal form computation using modulo determinant arithmetic. Math. Oper. Res. 12, (1987) 50--59.

....running time the potential for exponential growth of intermediate entries (see [6] has to be avoided ( 3, 9, 12] See also [8] ffl Modulo determinant methods, performing arithmetic modulo a multiple of a determinant. The modular arithmetic avoids exponential growth of intermediate expressions ([4, 5, 7, 10, 17]) ffl Coefficient methods for polynomialmatrices, translating the Hermite problem to a problem over the coeficient ring ( 2, 13, 18] Some of these algorithms use fast matrix and or fast polynomial arithmetic to improve their complexity. The best known complexity result is O(n (nd) 1 ffl ....

P. D. Domich, R. Kannan, and L. E. Trotter, Jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):50--59, 1987.

.... Gamma5000x 3 10x 4 = Gamma5 After the gaussian elimination of x 1 , it becomes: 14x 2 Gamma2000x 3 41x 4 Gamma29 Gamma8x 2 19999x 3 Gamma40x 4 29 6x 2 Gamma14999x 3 30x 4 Gamma35 Gamma410x 2 1000003x 3 Gamma2x 4 = 1012 13 This is the well known problem of expression swell (see [DKT87]) which must not be underestimated. However, in most systems coming from data dependence analysis as well as others, the size of coefficients a ij is very small (less than 10) and should not make a data overflow appear. As for the right hand sides b i which may have greater values than those of ....

P. D. Domich, R. Kannan, and L. E. Trotter.JR. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1), February 1987. 49

....1 Delta 1 k Delta Delta 1 Delta Delta 1 Delta Delta 1 i j 1 C C C C C C C A Figure 1: Elementary operations in matrix form coresponding to S c (i; j) M c (i) A c ( Gamma i; j; k) resp. A c ( Gamma i; j; k) type operations. It is known (see for example [25]) that the Hermite normal form of a matrix A is unique and also the right unimodular matrix is unique. Elementary operations from Definition 1 can be realized through multiplication of the matrix A by special matrices called elementary matrices. Elementary matrices are unimodular as can be seen in ....

....Hermite normal form is in simplified form illustrated in Figure 3. From that time on all proposed algorithms use Euclidean operations as a basic tool and the main attention is given to the analysis of expression swell even if the algorithm of Rosser is experimentally investigated (see for example [25, 31]) But the theoretical analysis remains open and there are no bounds proved on algorithms using only elementary operations. Some notes can be found in [21] and [9] Kannan and Bachem proposed the first algorithms with a known polynomial expression swell in 1979 [20] The order of computation of ....

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P. D. Domich, R. Kannan and L. E. Trotter, Jr., Hermite Normal form Computation Using Modulo Determinant Arithmetic, Math. of Operating Research, Vol. 12, No. 1, February 1987, pp. 50-59.

....computability of Hermite normal forms of integer matrices was first proved by Kannan and Bachem [75] using delicate and complicated analysis of the problems of intermediate swell. Subsequently, a much easier argument based on modulo arithmetic was given by Domich, Kannan and Trotter [36]. As consequences, we have that: ffl Linear Diophantine Systems can be solved in polynomial time. Assuming A has been preprocessed to have full row rank, to solve Ax = b; x 2 Z n we first obtain HNF(A) AK = Lj0] The input system has a solution if and only if L Gamma1 b is integral and if ....

P.D.Domich, R. Kannan and L.E. Trotter, Hermite normal form computation using modulo determinant arithmetic, Mathematics of Operations Research, 12, (1987), 50-59.

.... an algebraic number eld (see [1] There are also applications to group theory, since abelian groups are Z modules (see for example [9] The problem of computing the image of (that is, the HNF basis of the image of ) has been studied extensively, for example in [4] 10] 14] 15] 11] [7], 6] 12] and [8] The rst ve of these algorithms su er, to one degree or another, from an explosion in the size of integers used in intermediate stages, a phenomenon known as entry explosion which a ects many algorithms over Z. The last four of these algorithms use modular arithmetic, with ....

Domich, P. D., Kannan, R., and Trotter, Jr., L. E. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research 12 (1987), 50-59.

....be found in the area of normal forms of matrices. Normal forms of matrices, like the Hermite and Smith normal form, can be used to classify matrices and solve Diophantine systems (cf. 13] 8] These normal forms can be computed more eOEciently modulo the determinant of a particular matrix (cf. [7], 9] So faster determinant algorithms contribute to faster algorithms for solving several kinds of problems. Perhaps the two most commonly used algorithms for determinant computations are fractionfree Gaussian elimination and the Chinese remaindering method. Using standard arithmetic, the ....

Domich, P., Kannan, R., and Trotter, L. Hermite normal form computation using modulo determinant arithmetic. Math. Oper. Res. 12 (1987), 5059.

....There are several choices for computing a Hermite normal form and it is not clear which algorithm is best in practice. We do not emphasize on that point. We used in our final implementation a non modular implementation of Havas [9] and a modular version due to Domich, Kannan and Trotter [7], taking account of the additional information we have about the maximal value of the determinant. 4.2 Computing relations We now recall the original idea of Seysen [19] used in the method of Hafner and McCurley for getting a relation. The following proposition is an immediate generalization of ....

P. D. Domich, R. Kannan, and L. E. Trotter jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operation Research 87 (1987).

....a slightly modified version of the structured Gaussian elimination algorithm [8] to decrease the size of the matrix. Here we use the sparsity of the relation matrix. 3. 2) By Gaussian elimination compute a multiple Delta of the lattice determinant and then perform modular Hermite reduction [6] modulo Delta. For details see [3] In very few cases it turns out in step (3) that the lattice found in step (2) is not of dimension k. Step (4) The additional relations are found by the same technique as in step (2) The computation of the new determinant is done by updating the HNF using ....

....[3] In very few cases it turns out in step (3) that the lattice found in step (2) is not of dimension k. Step (4) The additional relations are found by the same technique as in step (2) The computation of the new determinant is done by updating the HNF using standard techniques as described in [6]. Step (5) For the computation of the SNF of the basis of L we use standard techniques (see [7] Since the size of the matrix to be considered is very small, no special treatment is necessary. 3 Improvements of the sequential algorithm 3.1 Finding one relation First of all we describe the ....

P.D. Domich, R. Kannan, L.E. Trotter, Hermite normal form computation using modulo determinant arithmetic, Math. Op. Res. 12 (1987), 50-59.

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P.D. Domich, R. Kannan, L.E. Trotter, Hermite normal form computation using modulo determinant arithmetic, Math. Op. Res. 12 (1987), 50-59.

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Domich, P. D., Kannan, R., and Trotter, Jr., L. E. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research 12 (1987), 50-59.

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P. D. Domich, R. Kannan, and L. E. Trotter jr. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operation Research 87 (1987).

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P.D. Domich, R. Kannan, L.E. Trotter, Hermite normal form computation using modulo determinant arithmetic, Math. Op. Res. 12 (1987), 50-59.

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P.D. Domich, R. Kannan and L.E. Trotter, Hermite Normal Form Computation using Modulo Determinant Arithmetic, Math. Op. Res. 12 (1987), S. 50--59.

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P.D. Domich, R. Kannan, R. and L.E. Trotter Jr., Hermite normal form computation using modulo determinant arithmetic, Mathematics of Operations Research, 12:1, (1987) 50-59.

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P. Domich, R. Kannan, and L. E. Trotter. Hermite normal form computation using modulo determinant arithmetic. Mathematics of Operations Research, 12(1):50-- 59, 1987.

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P.D. Domich, R. Kannan, R. and L.E. Trotter Jr., Hermite normal form computation using modulo determinant arithmetic, Mathematics of Operations Research, 12:1, (1987) 50-59.

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P.D. Domich, R. Kannan, R. and L.E. Trotter Jr., Hermite normal form computation using modulo determinant arithmetic, Mathematics of Operations Research, 12:1, (1987) 50-59.

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