A BLAS based C library for exact linear algebra on integer matrices

Chen, Zhuliang; Storjohann, Arne

doi:10.1145/1073884.1073899

Cited by 41 publications

(61 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the purposes of the Steinmann heptagon bootstrap, we have further cut down the weight-2 basis yielded by this procedure to only those 28 symbols that satisfy the Steinmann relations before proceeding to weight 3. We have carried out the large linear algebra problems necessary for the heptagon bootstrap with the help of the SageMath system [112], which employs the IML integer matrix library [113]. As a double check, we also fed the weight-7 integrability constraint matrix into A. von Manteuffel's FinRed program, which independently generated a basis for the 9570-dimensional weight-7 Steinmann heptagon space reported in table 1.…”

Section: B a Matrix Approach For Computing Integrable Symbolsmentioning

confidence: 99%

Heptagons from the Steinmann cluster bootstrap

Dixon

Drummond

Harrington

et al. 2017

J. High Energ. Phys.

165

312

View full text Add to dashboard Cite

Abstract:We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar N = 4 supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal (Q) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.

show abstract

Section: B a Matrix Approach For Computing Integrable Symbolsmentioning

confidence: 99%

Heptagons from the Steinmann cluster bootstrap

Dixon

Drummond

Harrington

et al. 2017

J. High Energ. Phys.

165

312

View full text Add to dashboard Cite

show abstract

“…Our implementation relies primarily on the level 3 BLAS routines provided by the Automatically Tuned Linear Algebra Software (ATLAS) library [18]: a widely used, portable, highly optimized implementation of the BLAS. In addition, we make limited use of the Integer Matrix Library (IML) [3] and the GNU MultiPrecision Arithmetic (GMP) library [7] for matrix inversion over a finite field and arbitrary precision integer arithmetic respectively.…”

Section: Resultsmentioning

confidence: 99%

“…We first recall the standard linear and quadratic lifting algorithms in §2.1 and §2.2, respectively, and in §2. 3 we give a variation of quadratic lifting that yields a straight line formula…”

Section: Liftingmentioning

confidence: 99%

Deterministic unimodularity certification

Pauderis

Storjohann

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Self Cite

View full text Add to dashboard Cite

The asymptotically fastest algorithms for many linear algebra problems on integer matrices, including solving a system of linear equations and computing the determinant, use high-order lifting. Currently, high-order lifting requires the use of a randomized shifted number system to detect and avoid error-producing carries. By interleaving quadratic and linear lifting, we devise a new algorithm for high-order lifting that allows us to work in the usual symmetric range modulo p, thus avoiding randomization. As an application, we give a deterministic algorithm to assay if an n × n integer matrix A is unimodular. The cost of the algorithm is O((log n)n ω M(log n + log ||A||)) bit operations, where ||A|| denotes the largest entry in absolute value, and M(t) is the cost of multiplying two integers bounded in bit length by t.

show abstract

“…Results in that direction are proved in [8]. This strategy can be seen as a dual of the probabilistic technique recently introduced by Chen and Storjohann [13] to decrease the number of input vectors when they are linearly dependent: their technique decreases the number of input vectors while the one above decreases the number of coordinates of the input vectors.…”

Section: Open Problemsmentioning

confidence: 93%

Floating-Point LLL: Theoretical and Practical Aspects

Stehlé

2009

Information Security and Cryptography

View full text Add to dashboard Cite

The text-book LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the Gram-Schmidt orthogonalisation by floating-point approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floating-point approximations seems to be natural for LLL even from the theoretical point of view: it is the key to reach a bit-complexity which is quadratic with respect to the bit-length of the input vectors entries, without fast integer multiplication. The latter bit-complexity strengthens the connection between LLL and Euclid's gcd algorithm. On the practical side, the LLL implementer may weaken the provable variants in order to further improve their efficiency: we emphasise on these techniques. We also consider the practical behaviour of the floating-point LLL algorithms, in particular their output distribution, their running-time and their numerical behaviour. After 25 years of implementation, many questions motivated by the practical side of LLL remain open.

show abstract

A BLAS based C library for exact linear algebra on integer matrices

Abstract: Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.

Cited by 41 publications

References 21 publications

Heptagons from the Steinmann cluster bootstrap

Heptagons from the Steinmann cluster bootstrap

Deterministic unimodularity certification

Floating-Point LLL: Theoretical and Practical Aspects

Contact Info

Product

Resources

About