A new class of equal-weight integration rules on the hypercube
type: Item
instance of: scholarly article
title: A new class of equal-weight integration rules on the hypercube
zbMATH Open document ID: 1067.65005
author: Neil A. Butler
published in: Numerische Mathematik
publication date: 2005-02-08T00:00:00Z
review text: The author considers the approximation problem of the integral on the \(n\)-dimensional hypercube (1) \(I[f] =\int_{[0,1]^n} f(x)dx\). He proposes to approximate the \(n\)-dimensional integral (1) with equal-weight integration rules of the type \(Q[f] = (1/N)\sum_{j=1}^n f(x^{(j)})\), where \(N\) is the number of points, as in quasi-Monte Carlo methods. However, the integration rules generate points cyclically, as with the method of good lattice points. Combining the appealing properties of high (modified) degree with the equally weighted points, one can obtain efficient integration rules in terms of the number of points required to attain a certain degree of the integration rule method. The paper is organized as follows: Section 2 introduces the new equal-weight integration rules. Section 3 defines the method degree and compares it with both algebraic and trigonometric degrees. The main theorem, in Section 4, provides conditions for the existence of the new integration rules for degrees up to 13. In Section 5, actual integration rules are provided for up to 7 million points and up to 500 dimensions.
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 2132463
zbMATH Keywords: integral approximation on hypercube
zbMATH Keywords: equal-weight integration rules
zbMATH Keywords: quasi-Monte Carlo integration
zbMATH Keywords: good lattice points method
MaRDI profile type: Publication
full work available at URL: s00211-004-0562-5
OpenAlex ID: W2084897054
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DOI: 10.1007/S00211-004-0562-5
description: scientific article; zbMATH DE number 2132463