W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In IEEE, editor, Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 675-685. IEEE, IEEE Computer Society Press, 12-14 November 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1. INTRODUCTION In the first half of 2000, two new algorithms were discovered for the e#cient computation of the determinant of a (dense) matrix with integer entries. Suppose that the dimension of the matrix is n n and that the maximum bit length of all entries is b. The algorithm by [10] requires 2.5 fixed precision, that is, bit operations. Here and in the following we use the exponent o(1) to capture missing polylogarithmic factors O( log n) log b) where C1 , C2 are constants ( soft O ) As it has turned out an algorithm in [15] which in turn is based on one by ....

.... algorithm in [15] which in turn is based on one by [31] and which uses the baby steps giant steps algorithm design technique, can be adapted to the dense integer matrix determinant problem and then has bit complexity (n [20, Section 2] Both algorithms use randomization and the algorithm in [10] is Monte Carlo always fast and probably correct and the one in [20] is Las Vegas always correct and probably fast. Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithms a la Strassen [8, 7, 14] By blocking [6, 16, 29, 30] the baby steps giant steps ....

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Eberly, W., Giesbrecht, M., and Villard, G. On computing the determinant and Smith form of an integer matrix. In Proc. 41stAnnual Symp. Foundations of Comp. Sci. (Los Alamitos, California, 2000), IEEE Computer Society Press.

....multiple of the identity matrix, the running time is again that of Chinese remaindering. The techniques developed in [32] for computing the characteristic polynomial of a sparse matrix lead to a speedup for the general, dense determinant problem. For integer matrices, the bit complexity was shown [16] to be n , where log measures the length of the entries in A and the exponent adjustment by o(1) captures missing log factors ( soft O ) The algorithms of [32, 16] are randomized of the Monte Carlo kind always fast, probably correct and can be further speeded by a ....

....sparse matrix lead to a speedup for the general, dense determinant problem. For integer matrices, the bit complexity was shown [16] to be n , where log measures the length of the entries in A and the exponent adjustment by o(1) captures missing log factors ( soft O ) The algorithms of [32, 16] are randomized of the Monte Carlo kind always fast, probably correct and can be further speeded by a Strassen Coppersmith Winograd sub cubic time matrix multiplication algo # This material is based on work supported in part by the National Science Foundation under Grants Nos. CCR 9988177, ....

[Article contains additional citation context not shown here]

W. Eberly, M. Giesbrecht, and Gilles Villard. On computing the determinant and Smith form of an integer matrix. In Proc. 41stAnnual Symp. Foundations of Comp. Sci., Los Alamitos, California, 2000. IEEE Computer Society Press.

....1. INTRODUCTION In the first half of 2000, two new algorithms were discovered for the e#cient computation of the determinant of a (dense) matrix with integer entries. Suppose that the dimension of the matrix is n n and that the maximum bit length of all entries is b. The algorithm by [10] requires 2.5 fixed precision, that is, bit operations. Here and in the following we use the exponent o(1) to capture missing polylogarithmic factors O( log n) C 1 (log b) C 2 where C1 , C2 are constants ( soft O ) As it has turned out an algorithm in [15] which in turn is based on ....

.... algorithm in [15] which in turn is based on one by [31] and which uses the baby steps giant steps algorithm design technique, can be adapted to the dense integer matrix determinant problem and then has bit complexity (n [20, Section 2] Both algorithms use randomization and the algorithm in [10] is Monte Carlo always fast and probably correct and the one in [20] is Las Vegas always correct and probably fast. Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithms a la Strassen [8, 7, 14] By blocking [6, 16, 29, 30] the baby steps giant steps ....

[Article contains additional citation context not shown here]

Eberly, W., Giesbrecht, M., and Villard, G. On computing the determinant and Smith form of an integer matrix. In Proc. 41stAnnual Symp. Foundations of Comp. Sci. (Los Alamitos, California, 2000), IEEE Computer Society Press.

....multiple of the identity matrix, the running time is again that of Chinese remaindering. The techniques developed in [32] for computing the characteristic polynomial of a sparse matrix lead to a speedup for the general, dense determinant problem. For integer matrices, the bit complexity was shown [16] to be n 3.5 o(1) log #A#) 2.5 o(1) where log #A# measures the length of the entries in A and the exponent adjustment by o(1) captures missing log factors ( soft O ) The algorithms of [32, 16] are randomized of the Monte Carlo kind always fast, probably correct and can be further ....

....for the general, dense determinant problem. For integer matrices, the bit complexity was shown [16] to be n 3.5 o(1) log #A#) 2. 5 o(1) where log #A# measures the length of the entries in A and the exponent adjustment by o(1) captures missing log factors ( soft O ) The algorithms of [32, 16] are randomized of the Monte Carlo kind always fast, probably correct and can be further speeded by a # This material is based on work supported in part by the National Science Foundation under Grants Nos. CCR 9988177, DMS 9977392 and INT 9726763 (Kaltofen) Appears in the Computer ....

[Article contains additional citation context not shown here]

W. Eberly, M. Giesbrecht, and Gilles Villard. On computing the determinant and Smith form of an integer matrix. In Proc. 41stAnnual Symp. Foundations of Comp. Sci., Los Alamitos, California, 2000. IEEE Computer Society Press.

....which has the cost shown. Computation of determinants is done using the algorithm of [1] This takes O(r 4 (log(r) k) r 3 k 2 ) time to compute the determinant of an r by r integer matrix each of whose entries is at most k bits long. Although somewhat faster algorithms are known [10], they are randomized, and for simplicity, we use a deterministic algorithm. We use this algorithm in Step 9. In the worst case, e (and hence q) is k bits long. So the entries of C are at most k bits long, and the cost of computing det(C) is O(s 4 (log(s) k) s 3 k 2 ) which is O(s ....

W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In Proceedings of the 41st Symposium on Foundations of Computer Science. IEEE, 2000.

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W. Eberly, M. Giesbrecht and G. Villard, Computing the determinant and Smith form of an integer matrix, In Proc. 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA USA, IEEE Computer Society Press, November 2000, pp. 675--685.

....expression swell. Several authors have successfully addressed the question during the last three years. Our aim is to survey these studies, especially around the determinant, the matrix inverse and matrix canonical forms. New progresses for the determinant complexity have been obtained in [3, 6, 7]. It is now known that the determinant and the Smith normal form of a polynomial matrix can be computed by a certified randomized algorithm in O(n in K [7] We prove in [4] that the same operation count is valid for column reduction. # CNRS Laboratoire LIP, Ecole Normale Superieure de Lyon, ....

W. Eberly, M. Giesbrecht and G. Villard, Computing the determinant and Smith form of an integer matrix, in Proc. The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA USA, nov. 2000, IEEE Computer Society Press, pp. 675--685.

....lower running time than the worst case inputs. The lowest known exponent of n in the bit complexity of the sign or of the determinant is decreasing. In particular, for the determinant this bit complexity is known to be below the algebraic complexity times the maximum bit size of the output (see [33, 24, 36] and section 6) This has motivated this survey to focus on the sequential time complexity rather on other aspects such as memory resources, parallel time or practical considerations. We will discuss deterministic and randomized algorithms. The usage of random bits leads to Monte Carlo algorithms ....

.... The use of system solution can be generalized to computing the whole determinant by applying the same technique iteratively to perturbations of A [56] This approach initially proposed for computing the characteristic polynomial of a sparse matrix is developed in the integer case by Eberly et al. [24]. The resulting randomized Monte Carlo algorithm is sensitive to the size of the determinant and to a parameter (A) the number of distinct invariant factors, which characterizes the Smith form. The number of distinct invariant factors satis es (A) O( j det Aj) O n log kAk) ....

[Article contains additional citation context not shown here]

W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA, pages 675685. IEEE Computer Society Press, Nov. 2000.

....lower running time than the worst case inputs. The lowest known exponent of n in the bit complexity of the sign or of the determinant is progressing. In particular, for the determinant this bit complexity is known to be below the algebraic complexity times the maximum bit size of the output (see [33,24,36] and section 6) This has motivated this survey to focus on the sequential time complexity rather on other aspects such as memory resources, parallel time or practical considerations. We will discuss deterministic and randomized algorithms. The usage of random bits 2 leads to Monte Carlo ....

.... use of system solution can be generalized to computing the whole determinant by applying the same technique iteratively to perturbations of A [56] This approach initially proposed for computing the characteristic polynomial of a sparse matrix is developed in the integer case by Eberly et al. [24]. The resulting randomized Monte Carlo algorithm is sensitive to the size of the determinant and to a parameter #(A) the number of distinct invariant factors, which characterizes the Smith form. The number of distinct invariant factors satisfies #(A) O( # det A ) # O # ( # n log #A#) ....

[Article contains additional citation context not shown here]

W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA, pages 675--685. IEEE Computer Society Press, Nov. 2000.

....calls to InvFact. Lemma 5 then imply the cost and storage assertion. In the same way, the failure probability is in O(n 2 p n log n) ut Our strategy, based on rank k perturbations combined to binary searches, may be applied in other situations. A paper of Eberly, Giesbrecht and the author [5] demonstrates the technique over the integers. Acknowledgements. Grateful thanks to Erich Kaltofen for his questions. ....

W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In Proc. 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA, 2000.

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W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In IEEE, editor, Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pages 675-685. IEEE, IEEE Computer Society Press, 12-14 November 2000.

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