Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

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

Computing the Kalman form

We present two algorithms for the computation of the Kalman form of a linear control system. The first one is based on the technique developed by Keller-Gehrig for the computation of the characteristic polynomial. The cost is a logarithmic number of matrix multiplications. To our knowledge, this improves the best previously known algebraic complexity by an order of magnitude. Then we also present a cubic algorithm proven to more efficient in practice.Comment: 10 page

Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices

Rank-profile revealing Gaussian elimination and the CUP matrix decomposition

Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the well known variations of such decompositions and transformations that have been proposed in the literature. We present an algorithm to compute the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra, Moran and Hui (1982), and show reductions from the other most common Gaussian elimination based matrix transformations and decompositions to the CUP decomposition. We discuss the advantages of the CUP algorithm over other existing algorithms by studying time and space complexities: the asymptotic time complexity is rank sensitive, and comparing the constants of the leading terms, the algorithms for computing matrix invariants based on the CUP decomposition are always at least as good except in one case. We also show that the CUP algorithm, as well as the computation of other invariants such as transformation to reduced column echelon form using the CUP algorithm, all work in place, allowing for example to compute the inverse of a matrix on the same storage as the input matrix.Comment: 35 page

Efficient Dense Gaussian Elimination over the Finite Field with Two Elements

In this work we describe an efficient implementation of a hierarchy of algorithms for Gaussian elimination upon dense matrices over the field with two elements. We discuss both well-known and new algorithms as well as our implementations in the M4RI library, which has been adopted into Sage. The focus of our discussion is a block iterative algorithm for PLE decomposition which is inspired by the M4RI algorithm. The implementation presented in this work provides considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the alacrity of our approach

Elimination-based certificates for triangular equivalence and rank profiles

International audienceIn this paper, we give novel certificates for triangular equivalence and rank profiles. These certificates enable somebody to verify the row or column rank profiles or the whole rank profile matrix faster than recomputing them, with a negligible overall overhead. We first provide quadratic time and space non-interactive certificates saving the logarithmic factors of previously known ones. Then we propose interactive certificates for the same problems whose Monte Carlo verification complexity requires a small constant number of matrix-vector multiplications, a linear space, and a linear number of extra field operations, with a linear number of interactions. As an application we also give an interactive protocol, certifying the determinant or the signature of dense matrices, faster for the Prover than the best previously known one. Finally we give linear space and constant round certificates for the row or column rank profiles

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

International audienceWe present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures.Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization.Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebraover a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies