Multi-point evaluation in higher dimensions

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 finite-dimensional -algebras of the form ,...

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Bibliographic Details
Published in:Applicable algebra in engineering, communication and computing Vol. 24; no. 1; pp. 37 - 52
Main Authors: van der Hoeven, Joris, Schost, Éric
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.01.2013
Subjects:
Artificial Intelligence
Computer Hardware
Computer Science
Symbolic and Algebraic Manipulation
Theory of Computation
68W30
68W40
Multi-point interpolation
65F99
Multi-point evaluation
12Y05
13P10
Algorithm
Power series multiplication
Complexity
ISSN:0938-1279, 1432-0622
Online Access:Get full text
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Summary: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 finite-dimensional -algebras of the form , where is generated by monomials of the form ; one particularly important example is the algebra of truncated power series . Similarly 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 Schost (ISSAC’05, ACM, New York, NY, pp 293–300, 2005 ), we obtain fast multiplication algorithms for algebras of the above form.
ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-012-0179-3