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On computing the determinant and Smith form of an integer matrix
 In Proceedings of the 41st Annual Symposium on Foundations of Computer Science
, 2000
"... A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A ¡£ ¢ n ¤ n the algorithm requires O ¥ n 3 ¦ 5 ¥ logn § 4 ¦ 5 § bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using as ..."
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Cited by 43 (9 self)
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A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A ¡£ ¢ n ¤ n the algorithm requires O ¥ n 3 ¦ 5 ¥ logn § 4 ¦ 5 § bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using
Integer Matrix factorization and its Application
, 2005
"... Matrix factorization has been of fundamental importance in modern sciences and technology. This work investigates the notion of factorization with entries restricted to integers or binaries, where the “integer” could be either the regular ordinal integers or just some nominal labels. Being discrete ..."
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Matrix factorization has been of fundamental importance in modern sciences and technology. This work investigates the notion of factorization with entries restricted to integers or binaries, where the “integer” could be either the regular ordinal integers or just some nominal labels. Being discrete
Fast Computation of the Smith Normal Form of an Integer Matrix
 In Proc. Int'l. Symp. on Symbolic and Algebraic Computation: ISSAC '95
, 1995
"... We present two new probabilistic algorithms for computing the Smith normal form of an A 2 Z m\Thetan . The first requires an expected number of O(m 2 n \Delta M(m log kAk)) bit operations (ignoring logarithmic factors) and is of the Las Vegas type; that is, it never produces an incorrect answer. ..."
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Cited by 18 (0 self)
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. Here kAk = max ij jA ij j and M(l) bit operations are sufficient to multiply two lbit integers (M(l) = l 2 using standard arithmetic) . This improves on the previously best known (deterministic) algorithm of Hafner and McCurley, which requires about O(m 3 n log kAk \Delta M(m log kAk)) bit
An engineered algorithm for the Smith form of an integer matrix
"... A variety of algorithms for computing Smith normal forms of integer matrices are known. Their worst case asymptotic complexities have steadily improved over the past four decades. However, in practice, an asymptotically inferior algorithm often outperforms an asymptotically better one. We oer an \en ..."
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A variety of algorithms for computing Smith normal forms of integer matrices are known. Their worst case asymptotic complexities have steadily improved over the past four decades. However, in practice, an asymptotically inferior algorithm often outperforms an asymptotically better one. We oer
Consider a 2n × 2n integer matrix
"... be a 2n × 2n integer matrix with the principal block A square and nonsingular. An algorithm is presented to determine if the Schur complement D−CA−1B is equal to the zero matrix in O (̃nω log M ) bit operations. Here, ω is the exponent of matrix multiplication and M  denotes the largest en ..."
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be a 2n × 2n integer matrix with the principal block A square and nonsingular. An algorithm is presented to determine if the Schur complement D−CA−1B is equal to the zero matrix in O (̃nω log M ) bit operations. Here, ω is the exponent of matrix multiplication and M  denotes the largest
Probabilistic Computation of the Smith Normal Form of a Sparse Integer Matrix
, 1995
"... We present a new probabilistic algorithms to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"; A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requir ..."
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Cited by 19 (4 self)
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We present a new probabilistic algorithms to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"; A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm
MAIK “Nauka /Interperiodica ” (Russia). On the Number of Irreducible Coverings of an Integer Matrix
, 2004
"... —The metric (quantitative) properties of the set of coverings of an integer matrix are examined. An asymptotic estimate for the logarithm of the typical number of irredundant σcoverings is obtained in the case when the number of rows in the matrix is not smaller than the number of its columns. As ..."
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—The metric (quantitative) properties of the set of coverings of an integer matrix are examined. An asymptotic estimate for the logarithm of the typical number of irredundant σcoverings is obtained in the case when the number of rows in the matrix is not smaller than the number of its columns
Fast Computation Of The Smith Form Of A Sparse Integer Matrix
 Computational Complexity
, 1996
"... . We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"  A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm requ ..."
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Cited by 15 (4 self)
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. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A 2 Z m\Thetan . The algorithm treats A as a "blackbox"  A is only used to compute matrixvector products and we don't access individual entries in A directly. The algorithm
On efficient sparse integer matrix Smith normal form computations
, 2001
"... We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth. W ..."
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Cited by 42 (20 self)
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We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of wordsize primes. Consequently, the algorithm does not suffer from coefficient growth
computational complexity FAST COMPUTATION OF THE SMITH FORM OF A SPARSE INTEGER MATRIX
"... Abstract. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ Z m×n. The algorithm treats A as a “black box”—A is only used to compute matrixvector products and we do not access individual entries in A directly. The algorithm requires about O(m2 ..."
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Abstract. We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix A ∈ Z m×n. The algorithm treats A as a “black box”—A is only used to compute matrixvector products and we do not access individual entries in A directly. The algorithm requires about O(m2
Results 1  10
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