scholarly journals The number of lattice rules having given invariants

1992 ◽  
Vol 46 (3) ◽  
pp. 479-495 ◽  
Author(s):  
Stephen Joe ◽  
David C. Hunt
Keyword(s):  
Normal Form ◽  
Quadrature Rule ◽  
Unit Cube ◽  
Numerical Results ◽  
Smith Normal Form ◽  
Lattice Rules ◽  
Dimensional Unit ◽  
Lattice Rule ◽  
Positive Integers ◽  
Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

scholarly journals Constructions of general polynomial lattice rules based on the weighted star discrepancy

2007 ◽  
Vol 13 (4) ◽  
pp. 1045-1070 ◽  
Author(s):  
Josef Dick ◽  
Peter Kritzer ◽  
Gunther Leobacher ◽  
Friedrich Pillichshammer
Keyword(s):  
Lattice Rules ◽  
General Polynomial ◽  
Star Discrepancy

scholarly journals Extensions of Fibonacci Lattice Rules

2009 ◽  
pp. 259-270 ◽  
Author(s):  
Ronald Cools ◽  
Dirk Nuyens
Keyword(s):  
Lattice Rules ◽  
Fibonacci Lattice

scholarly journals Construction algorithms for polynomial lattice rules for multivariate integration

2005 ◽  
Vol 74 (252) ◽  
pp. 1895-1922 ◽  
Author(s):  
J. Dick ◽  
F. Y. Kuo ◽  
F. Pillichshammer ◽  
I. H. Sloan
Keyword(s):  
Lattice Rules ◽  
Multivariate Integration ◽  
Construction Algorithms

scholarly journals Richardson Extrapolation of Polynomial Lattice Rules

2019 ◽  
Vol 57 (1) ◽  
pp. 44-69 ◽  
Author(s):  
Josef Dick ◽  
Takashi Goda ◽  
Takehito Yoshiki
Keyword(s):  
Richardson Extrapolation ◽  
Lattice Rules
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