Diversification improves interpolation

Abstract

We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations.

Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values

“…One may assume that some polynomials residues f (ω i ) mod (x p−1 + · · · + 1) are faulty. The Chinese remaindering of the term exponents with several p can be done by diversification [9]. The sparsest shift algorithms in [6] can be modified similarly.…”

“…Hybrid symbolic-numeric algorithms have traditionally assumed that the input scalars have with some relative error been deformed. For instance, if one allows substantial oversampling, a sparse polynomial can be recovered from numerical noisy values, where each value has a relative error of 0.5 [9], perhaps even 0.99, with still more oversampling. Here we consider errors in the Hamming distance sense, i.e., several incorrect values without assumption on any accuracy.…”

A new deterministic algorithm for sparse multivariate polynomial interpolation

“…Finding coefficients after finding the exponents is quite straightforward. Giesbrecht and Roche [13] studied polynomial interpolation over remainder black box which is somewhat similar to the modular black box introduced in [12]. A Las Vegas randomized algorithm that uses fewer black box evaluations was presented in [13] for this remainder black box.…”

“…Giesbrecht and Roche [13] studied polynomial interpolation over remainder black box which is somewhat similar to the modular black box introduced in [12]. A Las Vegas randomized algorithm that uses fewer black box evaluations was presented in [13] for this remainder black box. Arnold, Giesbrecht and Roche [2] devised a Monte Carlo algorithm for interpolating polynomials represented by straight-line programs.…”

Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

“…Our investigations are motivated by numeric sparse polynomial interpolation algorithms [11,12,20,13]. In the Ben-Or/Tiwari version of this algorithm, one first computes for a sparse polynomial f (x) ∈ C[x] the linear generator for h l = f (ω l+1 ) for l = 0, 1, 2, .…”