Lagrange interpolation on subgrids of tensor product grids

Sauer, Thomas

doi:10.1090/s0025-5718-03-01557-6

Cited by 29 publications

(28 citation statements)

References 22 publications

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“…, Xn]: in particular, for 1 ≤ r ≤ R, the leading term of Gr is gr. Then the following results are proved in [25] (see also [29]). …”

Section: Proof Of Theoremmentioning

confidence: 87%

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Multivariate power series multiplication

Schost

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M . In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.

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“…, Xn]: in particular, for 1 ≤ r ≤ R, the leading term of Gr is gr. Then the following results are proved in [25] (see also [29]). …”

Section: Proof Of Theoremmentioning

confidence: 87%

“…Our approach requires to solve specific but well-known [29] evaluation / interpolation problems. Obtaining sharp complexity estimates for these tasks is of independent interest.…”

Section: Conclusion Open Problemsmentioning

confidence: 99%

Multivariate power series multiplication

Schost

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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“…We found no previous references dedicated to the evaluation problem (a naive solution obviously takes quadratic time). As to our form of interpolation, an early reference is [Wer80], with a focus on the bivariate case; the question has been the subject of several subsequent works, and one finds a comprehensive treatment in [Sau04]. However, the algorithms mentioned previously do not have quasi-linear complexity.…”

Section: Problem Statementmentioning

confidence: 99%

“…In this paper, following the terminology of [Sau04], we consider evaluation points that are subgrids of tensor product grids. We prove that for some suitable monomial bases, evaluation and interpolation can both be done in time O(n |I | log 2 |I | log log |I |), where n is the number of variables and |I | is the size of the evaluation set (and of the monomial basis we consider).…”

Section: Introductionmentioning

confidence: 99%

Multi-point evaluation in higher dimensions

Hoeven

Schost

2012

AAECC

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In this paper, we propose efficient new algorithms for multi-dimensional multi-point evaluation and interpolation on certain subsets of so called tensor product grids. These point-sets naturally occur in the design of efficient multiplication algorithms for finitedimensional C-algebras of the form A = C[x 1 , , x n ]/I, where I is generated by monomials of the form x 1 i1 x n in ; one particularly important example is the algebra of truncated power seriesSimilarly to what is known for multi-point evaluation and interpolation in the univariate case, our algorithms have quasi-linear time complexity. As a known consequence [Sch05], we obtain fast multiplication algorithms for algebras A of the above form.

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“…The next corollary answers this question. If an ideal projector P A is a Lagrange projectors then the family of Lagrange projectors P A with bidiagonal A coincides with the family of Lagrange projectors that interpolate on a triangular subgrid of a tensor-product grid, studied in [14].…”

Section: Proposition 36 Let P Be An Ideal Projector Onto Fmentioning

confidence: 99%