Sakata generalized the Berlekamp -Massey algorithm to n dimensions in 1988. The Berlekamp -Massey -Sakata (BMS) algorithm can be used for finding a Gröbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem using linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequences for 0-dimensional ideals. Under genericity assumptions, we propose a randomized preprocessing of the table that corresponds to performing a linear change of coordinates on the polynomials associated with the linear recurrences. This technique then essentially reduces our problem to using the efficient 1-dimensional Berlekamp -Massey (BM) algorithm. However, the number of probes to the table in this scheme may be elevated. We thus consider the table in the black-box model: we assume probing the table is expensive and we minimize the number of probes to the table in our complexity model. We produce an FGLM-like algorithm for finding the relations in the table, which lets us use linear algebra techniques. Under some additional assumptions, we make this algorithm adaptive and reduce further the number of table probes. This number can be estimated by counting the number of distinct elements in a multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can relate this quantity with the geometry of the final staircase. Hence, in favorable cases such as convex ones, the complexity is essentially linear in the size of the output. Finally, when using the LEX ordering, we can make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp -Massey.

We propose several new schedules for Strassen-Winograd's matrix multiplication algorithm, they reduce the extra memory allocation requirements by three different means: by introducing a few pre-additions, by overwriting the input matrices, or by using a first recursive level of classical multiplication. In particular, we show two fully in-place schedules: one having the same number of operations, if the input matrices can be overwritten; the other one, slightly increasing the constant of the leading term of the complexity, if the input matrices are read-only. Many of these schedules have been found by an implementation of an exhaustive search algorithm based on a pebble game.

We propose different implementations of the sparse matrix-dense vector multiplication (SpMV) for finite fields and rings Z /m Z. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of SpMV in the LinBox library, and henceforth the speed of its black box algorithms. Besides, we use this and a new parallelization of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields.

We consider the problem of solving a full rank consistent linear system A(u)x = b(u) where the m × n matrix A and the m-dimensional vector b has entries that are polynomials in u over a field. We give an algorithm that computes the unique solution x = f (u)/g(u), which is a vector of rational functions, by evaluating the parameter u at distinct points. Those points ξ λ where the matrix A evaluates to a matrix A(ξ λ ), with entries over the scalar field, of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some u = ξ λ results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. Our algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts.We have implemented our algorithms with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use singular values based numeric rank computations. The arising linear systems for the error-corrected parametric solution are demonstrated to be well-conditioned even when the input scalars have noise. In several initial experiments we have shown that our approach is numerically stable even for larger systems m = n = 100, provided the degrees in the solution are small (≤ 2). For smaller systems m = n = 10 with higher degrees (≤ 20) the algorithm works similarly to rational function recovery. Our implementation can correct 13 true errors in both settings.

The so-called Berlekamp-Massey-Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp-Massey algorithm to n-dimensional tables, for n > 1. We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode n-cyclic codes, a generalization in dimension n of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.