13,849 research outputs found
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Exact Sparse Matrix-Vector Multiplication on GPU's and Multicore Architectures
We propose different implementations of the sparse matrix--dense vector
multiplication (\spmv{}) for finite fields and rings \Zb/m\Zb. 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
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Asymptotically fast polynomial matrix algorithms for multivariable systems
We present the asymptotically fastest known algorithms for some basic
problems on univariate polynomial matrices: rank, nullspace, determinant,
generic inverse, reduced form. We show that they essentially can be reduced to
two computer algebra techniques, minimal basis computations and matrix fraction
expansion/reconstruction, and to polynomial matrix multiplication. Such
reductions eventually imply that all these problems can be solved in about the
same amount of time as polynomial matrix multiplication
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