Multi-point evaluation in higher dimensions

Hoeven, Joris van der; Schost, Éric

doi:10.1007/s00200-012-0179-3

Cited by 21 publications

(24 citation statements)

References 15 publications

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“…It is unclear how to achieve this post-multiplication in time quasi-linear in the size of the polynomial support when the evaluation points are arbitrary, as in our case. Existing work achieves quasi-linear complexity for specific points [14,24].…”

Section: Discussionmentioning

confidence: 99%

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Emiris

Kalinka

Konaxis

et al. 2013

Computer-Aided Design

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We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial.

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Section: Discussionmentioning

confidence: 99%

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Emiris

Kalinka

Konaxis

et al. 2013

Computer-Aided Design

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“…They feature quasi-linear complexity in terms of the sizes of the supports of the input and of a given superset for the support of the product. Nevertheless the logarithmic overhead of the latter algorithms is important, and alternative approaches, directly based on the truncated Fourier transform, have been designed in [14,15,19] for supports which are initial segments of N n (i.e. sets of monomials which are complementary to a monomial ideal), with the same order of efficiency as the FFT multiplication for univariate polynomials.…”

Section: Related Workmentioning

confidence: 99%

On the complexity of multivariate blockwise polynomial multiplication

Hoeven

Lecerf

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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In this article, we study the problem of multiplying two multivariate polynomials which are somewhat but not too sparse, typically like polynomials with convex supports. We design and analyze an algorithm which is based on blockwise decomposition of the input polynomials, and which performs the actual multiplication in an FFT model or some other more general so called ''evaluated model". If the input polynomials have total degrees at most d, then, under mild assumptions on the coefficient ring, we show that their product can be computed with O s 1.5337 ring operations, where s denotes the number of all the monomials of total degree at most 2d.

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“…This relationship becomes apparent when combining the algorithms to obtain additive FFTs, since one essentially obtains the algorithms of Gao and Mateer. The techniques developed for additive FFTs have yet to be applied to conversions involving the Newton basis. In the realm of multiplicative FFTs, one has the algorithms of van der Hoeven and Schost [26], which convert between the monomial basis and the Newton basis associated with the radix-2 truncated Fourier transform points [25,24]. Fast conversion between the two basis is a necessary requirement of multivariate evaluation and interpolation algorithms [26,11] and their application to systematic encoding of Reed-Muller and multiplicity codes [11].…”

Section: Introductionmentioning

confidence: 99%

“…In the realm of multiplicative FFTs, one has the algorithms of van der Hoeven and Schost [26], which convert between the monomial basis and the Newton basis associated with the radix-2 truncated Fourier transform points [25,24]. Fast conversion between the two basis is a necessary requirement of multivariate evaluation and interpolation algorithms [26,11] and their application to systematic encoding of Reed-Muller and multiplicity codes [11]. For applications in coding theory, characteristic two finite fields are particularly interesting due to their fast arithmetic.…”

Section: Introductionmentioning

confidence: 99%

Fast Hermite interpolation and evaluation over finite fields of characteristic two

Coxon

2020

Journal of Symbolic Computation

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An additive fast Fourier transform over a finite field of characteristic two efficiently evaluates polynomials at every element of an F 2 -linear subspace of the field. We view these transforms as performing a change of basis from the monomial basis to the associated Lagrange basis, and consider the problem of performing the various conversions between these two bases, the associated Newton basis, and the "novel" basis of Lin, Chung and Han (FOCS 2014). Existing algorithms are divided between two families, those designed for arbitrary subspaces and more efficient algorithms designed for specially constructed subspaces of fields with degree equal to a power of two. We generalise techniques from both families to provide new conversion algorithms that may be applied to arbitrary subspaces, but which benefit equally from the specially constructed subspaces. We then construct subspaces of fields with smooth degree for which our algorithms provide better performance than existing algorithms.

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Multi-point evaluation in higher dimensions

Cited by 21 publications

References 15 publications

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

On the complexity of multivariate blockwise polynomial multiplication

Fast Hermite interpolation and evaluation over finite fields of characteristic two

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