21,784 research outputs found
Computing minimal interpolation bases
International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between vectors of size it uses field operations, where is the exponent of matrix multiplication, and is the sum of theentries of the input shift , with . This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
Fast algorithm for border bases of Artinian Gorenstein algebras
Given a multi-index sequence , we present a new efficient algorithm
to compute generators of the linear recurrence relations between the terms of
. We transform this problem into an algebraic one, by identifying
multi-index sequences, multivariate formal power series and linear functionals
on the ring of multivariate polynomials. In this setting, the recurrence
relations are the elements of the kerne l\sigma of the Hankel operator
$H$\sigma associated to . We describe the correspondence between
multi-index sequences with a Hankel operator of finite rank and Artinian
Gorenstein Algebras. We show how the algebraic structure of the Artinian
Gorenstein algebra \sigma\sigma yields the
structure of the terms $\sigma\alpha N nAK[x 1 ,. .. , xnIHIA$ and the tables of multiplication by the variables in these
bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with
improved complexity bounds. We present applications of the method to different
problems such as the decomposition of functions into weighted sums of
exponential functions, sparse interpolation, fast decoding of algebraic codes,
computing the vanishing ideal of points, and tensor decomposition. Some
benchmarks illustrate the practical behavior of the algorithm
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