251 research outputs found
On Polynomial Multiplication in Chebyshev Basis
In a recent paper Lima, Panario and Wang have provided a new method to
multiply polynomials in Chebyshev basis which aims at reducing the total number
of multiplication when polynomials have small degree. Their idea is to use
Karatsuba's multiplication scheme to improve upon the naive method but without
being able to get rid of its quadratic complexity. In this paper, we extend
their result by providing a reduction scheme which allows to multiply
polynomial in Chebyshev basis by using algorithms from the monomial basis case
and therefore get the same asymptotic complexity estimate. Our reduction allows
to use any of these algorithms without converting polynomials input to monomial
basis which therefore provide a more direct reduction scheme then the one using
conversions. We also demonstrate that our reduction is efficient in practice,
and even outperform the performance of the best known algorithm for Chebyshev
basis when polynomials have large degree. Finally, we demonstrate a linear time
equivalence between the polynomial multiplication problem under monomial basis
and under Chebyshev basis
Formal proof for delayed finite field arithmetic using floating point operators
Formal proof checkers such as Coq are capable of validating proofs of
correction of algorithms for finite field arithmetics but they require
extensive training from potential users. The delayed solution of a triangular
system over a finite field mixes operations on integers and operations on
floating point numbers. We focus in this report on verifying proof obligations
that state that no round off error occurred on any of the floating point
operations. We use a tool named Gappa that can be learned in a matter of
minutes to generate proofs related to floating point arithmetic and hide
technicalities of formal proof checkers. We found that three facilities are
missing from existing tools. The first one is the ability to use in Gappa new
lemmas that cannot be easily expressed as rewriting rules. We coined the second
one ``variable interchange'' as it would be required to validate loop
interchanges. The third facility handles massive loop unrolling and argument
instantiation by generating traces of execution for a large number of cases. We
hope that these facilities may sometime in the future be integrated into
mainstream code validation.Comment: 8th Conference on Real Numbers and Computers, Saint Jacques de
Compostelle : Espagne (2008
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
Solving Sparse Integer Linear Systems
We propose a new algorithm to solve sparse linear systems of equations over
the integers. This algorithm is based on a -adic lifting technique combined
with the use of block matrices with structured blocks. It achieves a sub-cubic
complexity in terms of machine operations subject to a conjecture on the
effectiveness of certain sparse projections. A LinBox-based implementation of
this algorithm is demonstrated, and emphasizes the practical benefits of this
new method over the previous state of the art
Relaxing order basis computation
International audienceThe computation of an order basis (also called sigma basis) is a fundamental tool for linear algebra with polynomial coefficients. Such a computation is one of the key ingredients to provide algorithms which reduce to polynomial matrices multiplication. This has been the case for column reduction or minimal nullspace basis of polynomial matrix over a field. In this poster, we are interested in the application of order basis to compute minimal matrix generators of a linear matrix sequence. In particular, we focus on the linear matrix sequence used in the Block Wiedemann algorithm
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
Block projections have been used, in [Eberly et al. 2006], to obtain an
efficient algorithm to find solutions for sparse systems of linear equations. A
bound of softO(n^(2.5)) machine operations is obtained assuming that the input
matrix can be multiplied by a vector with constant-sized entries in softO(n)
machine operations. Unfortunately, the correctness of this algorithm depends on
the existence of efficient block projections, and this has been conjectured. In
this paper we establish the correctness of the algorithm from [Eberly et al.
2006] by proving the existence of efficient block projections over sufficiently
large fields. We demonstrate the usefulness of these projections by deriving
improved bounds for the cost of several matrix problems, considering, in
particular, ``sparse'' matrices that can be be multiplied by a vector using
softO(n) field operations. We show how to compute the inverse of a sparse
matrix over a field F using an expected number of softO(n^(2.27)) operations in
F. A basis for the null space of a sparse matrix, and a certification of its
rank, are obtained at the same cost. An application to Kaltofen and Villard's
Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an
integer matrix yields algorithms requiring softO(n^(2.66)) machine operations.
The derived algorithms are all probabilistic of the Las Vegas type
Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation
We consider space-saving versions of several important operations on
univariate polynomials, namely power series inversion and division, division
with remainder, multi-point evaluation, and interpolation. Now-classical
results show that such problems can be solved in (nearly) the same asymptotic
time as fast polynomial multiplication. However, these reductions, even when
applied to an in-place variant of fast polynomial multiplication, yield
algorithms which require at least a linear amount of extra space for
intermediate results. We demonstrate new in-place algorithms for the
aforementioned polynomial computations which require only constant extra space
and achieve the same asymptotic running time as their out-of-place
counterparts. We also provide a precise complexity analysis so that all
constants are made explicit, parameterized by the space usage of the underlying
multiplication algorithms
Essentially Optimal Sparse Polynomial Multiplication
We present a probabilistic algorithm to compute the product of two univariate
sparse polynomials over a field with a number of bit operations that is
quasi-linear in the size of the input and the output. Our algorithm works for
any field of characteristic zero or larger than the degree. We mainly rely on
sparse interpolation and on a new algorithm for verifying a sparse product that
has also a quasi-linear time complexity. Using Kronecker substitution
techniques we extend our result to the multivariate case.Comment: 12 page
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