2 research outputs found
Efficient Decomposition of Dense Matrices over GF(2)
In this work we describe an efficient implementation of a hierarchy of
algorithms for the decomposition of dense matrices over the field with two
elements (GF(2)). Matrix decomposition is an essential building block for
solving dense systems of linear and non-linear equations and thus much research
has been devoted to improve the asymptotic complexity of such algorithms. In
this work we discuss an implementation of both well-known and improved
algorithms in the M4RI library. The focus of our discussion is on a new variant
of the M4RI algorithm - denoted MMPF in this work -- which allows for
considerable performance gains in practice when compared to the previously
fastest implementation. We provide performance figures on x86_64 CPUs to
demonstrate the viability of our approach
An Improvement over the GVW Algorithm for Inhomogeneous Polynomial Systems
The GVW algorithm is a signature-based algorithm for computing Gr\"obner
bases. If the input system is not homogeneous, some J-pairs with higher
signatures but lower degrees are rejected by GVW's Syzygy Criterion, instead,
GVW have to compute some J-pairs with lower signatures but higher degrees.
Consequently, degrees of polynomials appearing during the computations may
unnecessarily grow up higher and the computation become more expensive. In this
paper, a variant of the GVW algorithm, called M-GVW, is proposed and mutant
pairs are introduced to overcome inconveniences brought by inhomogeneous input
polynomials. Some techniques from linear algebra are used to improve the
efficiency. Both GVW and M-GVW have been implemented in C++ and tested by many
examples from boolean polynomial rings. The timings show M-GVW usually performs
much better than the original GVW algorithm when mutant pairs are found.
Besides, M-GVW is also compared with intrinsic Gr\"obner bases functions on
Maple, Singular and Magma. Due to the efficient routines from the M4RI library,
the experimental results show that M-GVW is very efficient