Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

Abstract

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Padé approximation and constrained multivariate interpolation, and has applications in coding theory and security.This problem asks to find univariate polynomial relations between m vectors of size σ; 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 O˜(m ω−1 σ) field operations, where ω is the exponent of matrix multiplication and the notation O˜(·) indicates that logarithmic terms are omitted.Earlier works, in the case of Hermite-Padé approximation [34] and in the general interpolation case [18], compute non-normalized bases. Since for arbitrary shifts such bases may have size Θ(m 2 σ), the cost bound O˜(m ω−1 σ) 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 O(mσ). 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.

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

“…Approximant basis algorithms often require P to be in s-reduced form [47]; although the s-ordered weak Popov form is stronger, obtaining it involves minor changes in these algorithms, without impact on performance according to our experiments. Recent literature shows that this stronger form reveals valuable information for further computations with P [23,25], in particular for finding s-Popov bases [4].…”

“…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.…”

“…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.…”

Computing Popov and Hermite Forms of Rectangular Polynomial Matrices

“…[3] relies on a fractionfree algorithm for the latter computation, and hence lends itself well to cases where K is not finite. In our context, following this approach with the fastest known approximant basis algorithm [12] yields the cost bounds O˜((m + n) ω−1 nmd) for the Popov form and O˜((m + n) ω−1 n 2 md) for the Hermite form. For the latter this is the fastest existing algorithm, to the best of our knowledge.…”

“…In the nonsingular case, exploiting information on the pivots has led to algorithmic improvements for normal form algorithms [10,12,14,23]. Following this, we put our effort into two computational tasks: finding the location of the pivots in the normal form (the pivot support), and using this knowledge to compute this form.…”