Sign determination in residue number systems

Brönnimann, Hervé; Emiris, Ioannis Z.; Pan, Victor Y.; Pion, Sylvain

doi:10.1016/s0304-3975(98)00101-7

Cited by 36 publications

(41 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Previous work on integer determinant evaluation includes [ABM99,EGV00,-Kal02, KV01,Pan02b]; other works such as [BEPP99,PY01] focus on sign determination but do not improve upon the current record complexity bounds (although the output-sensitive approach in [BEPP99, Section 4] was instrumental in improving these bounds). The best complexities for a general scalar matrix are due to the algorithms in [EGV00,KV01]; cf.…”

Section: The Known and New Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Improved algorithms for computing determinants and resultants

Emiris

2004

Journal of Complexity

View full text Add to dashboard Cite

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.

show abstract

Section: The Known and New Resultsmentioning

confidence: 99%

“…Resultant values and signs also capture important tests in computational geometry, including the case of infinitesimal perturbations of the input. These diverse applications are discussed in [BEPP99,Can88,CLO98,DE01,Man93].…”

Section: Our Subjects and Techniquesmentioning

confidence: 99%

Improved algorithms for computing determinants and resultants

Emiris

2004

Journal of Complexity

View full text Add to dashboard Cite

show abstract

“…One can bound the precision of the error-free computations by performing them modulo sufficiently many reasonably bounded coprime moduli m i and then recovering the value det A mod m, m = i m i , by applying the Chinese Remainder algorithm. Some relevant techniques are elaborated upon in (34). In particular, to minimize the computational cost, one can select random primes or prime powers m i recursively until the output value modulo their product stabilizes.…”

Section: Computing the Signs And The Values Of Determinantsmentioning

confidence: 99%

Algebraic and Numerical Algorithms

Emiris¹,

Pan²,

Tsigaridas³

2009

Chapman &Amp; Hall/CRC Applied Algorithms and Data Structures Series

View full text Add to dashboard Cite

“…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.

View full text Add to dashboard Cite

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.

show abstract

Sign determination in residue number systems

Cited by 36 publications

References 13 publications

Improved algorithms for computing determinants and resultants

Improved algorithms for computing determinants and resultants

Algebraic and Numerical Algorithms

On the complexity of computing determinants

Contact Info

Product

Resources

About