An output-sensitive variant of the baby steps/giant steps determinant algorithm

Kaltofen, Erich

doi:10.1145/780506.780524

Cited by 16 publications

(20 citation statements)

References 29 publications

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“…The recent works [Kal02] (which yields a bound of the order of m 3 q) and [Sto03] do not apply efficiently to the quasi-Toeplitz structure and thus are not sufficient to support our improvements. In particular, [Sto03] considers univariate matrices and proposes a Las Vegas algorithm for their determinant computation with arithmetic complexity in O Ã ðmd 1 Þ; where OðmÞ bounds the arithmetic complexity of matrix multiplication and d 1 bounds the degree of the entries.…”

Section: The Known and New Resultsmentioning

confidence: 88%

Improved algorithms for computing determinants and resultants

Emiris

2004

Journal of Complexity

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Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvestertype resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems.

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Section: The Known and New Resultsmentioning

confidence: 88%

Improved algorithms for computing determinants and resultants

Emiris

2004

Journal of Complexity

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“…For the classical matrix multiplication exponent ω = 3, the bit complexity of integer matrix determinants is thus proportional to n η+o(1) as follows: η = 3 + 1 2 (Eberly et al 2000;Kaltofen 1992Kaltofen , 2002, η = 3 + 1 3 (Theorem 4.2 on page 111), η = 3 + 1 5 (line 4 in Table 6.1 on page 119), η = 3 (Storjohann 2004). Together with the algorithms discussed in Section 1 on page 94 that perform well on propitious inputs, such a multitude of results poses a problem for the practitioner: which of the methods can yield faster procedures in computer algebra systems?…”

Section: Discussionmentioning

confidence: 99%

“…A detailed description of this algorithm, with an early termination strategy in case the determinant is small (cf. Brönnimann et al 1999;Emiris 1998), is presented by Kaltofen (2002).…”

Section: Introductionmentioning

confidence: 99%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

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Abstract. We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in (n 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

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“…In the previous section, the number m of modular computations was fixed, and only the correction capacity was made output-sensitive In order to limit the number of modular computations, early termination Chinese remaindering is commonly used [10,3]: an increasing number of modular residues are computed until the reconstructed value in R stabilizes. In the context of parallel computing, a chunk of modular computations will be done in parallel at each step and if the stabilization condition is not met, then another chunk will be computed.…”

Section: Early Terminationmentioning

confidence: 99%

“…The certification consists in testing if X (j) = ri mod Pi and return the residue that succeeds every tests. By choosing an appropriate number of ri's, any probability of success can be guaranteed (refer to [10] for a detailed analysis).…”

Section: Certifiermentioning

confidence: 99%

Output-sensitive decoding for redundant residue systems

Khonji¹,

Pernet²,

Roch³

et al. 2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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We study algorithm based fault tolerance techniques for supporting malicious errors in distributed computations based on Chinese remainder theorem. The description holds for both computations with integers or with polynomials over a field. It unifies the approaches of redundant residue number systems and redundant polynomial systems through the Reed Solomon decoding algorithm proposed by Gao. We propose several variations on the application of the extended Euclid algorithm, where the error correction rate is adaptive. Several improvements are studied, including the use of various criterions for the termination of the Euclidean Algorithm, and an acceleration using the Half-GCD techniques. When there is some redundancy in the input, a gap in the quotient sequence is stated at the step matching the error correction, which enables early termination parallel computations. Experiments are shown to compare these approaches.

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An output-sensitive variant of the baby steps/giant steps determinant algorithm

Cited by 16 publications

References 29 publications

Improved algorithms for computing determinants and resultants

Improved algorithms for computing determinants and resultants

On the complexity of computing determinants

Output-sensitive decoding for redundant residue systems

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