Computing minimal interpolation bases

Abstract

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 $m$ vectors of size$\sigma$ it uses $O~( m^{\omega-1} (\sigma + |s|) )$ field operations, where$\omega$ is the exponent of matrix multiplication, and $|s|$ is the sum of theentries of the input shift $s$, with $\min(s) = 0$. 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

Popov Form Computation for Matrices of Ore Polynomials

“…Although in a slightly different context, we remark that a linear algebra point for normal form computation can be found for the commutative case already in [13, Sections 6.7.1 and 6.72]. More recently, in [3] and [12,Section 7] it is shown that the shifted Popov basis of a k[x]-module given by equations exactly corresponds to some specific rows in a reduced row echelon form of the left nullspace of a constant matrix with much larger dimension than the original polynomial equations. The algorithm we present here is based on the linearization technique of Labhalla, Lombardi and Marlin [14].…”

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

“…3] which returns s-Popov bases and is about twice slower than PM-Basis; making this overhead negligible for some usual cases is future work. For completeness, we handle general approximants (with one modulus per column of F) by an iterative approach from [2,47]; faster algorithms are more complex [23][24][25] and use partial linearization techniques.…”

“…. This definition is a specialization of those in [3,24], which consider n sets of points, one for each of the n columns of E 1 , . .…”

“…Then, we describe our implementation of a key building block: multiplication. Section 3 presents the next major part of our work, concerning algorithms for approximant bases [2,15,25,53] and interpolant bases [3,23,24,47]. We focus on a version of interpolants which is less general than in these references but allows for a more efficient algorithm.…”

Fast computation of approximant bases in canonical form

“…5.3 and 6.14). Jeannerod et al (2017) gave an algorithm achieving a cost similar to that in the first item above, in the more general context of Eq. (2) and thus covering the case of arbitrary orders as well; the cost bound above improves upon that given in (ibid., Thm.…”