An application of diophantine approximation to the construction of rank-1 lattice quadrature rules

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dc.contributor.author Langtry, TN
dc.date.accessioned 2012-10-12T03:32:49Z
dc.date.issued 1996-10
dc.date.issued 1996-10
dc.identifier.citation Mathematics of Computation, 1996, 65 (216), pp. 1635 - 1662
dc.identifier.citation Mathematics of Computation, 1996, 65 (216), pp. 1635 - 1662
dc.identifier.issn 0025-5718
dc.identifier.other C1UNSUBMIT en_US
dc.identifier.uri http://hdl.handle.net/10453/17950
dc.description.abstract Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in ℝs and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962) -themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variaole, these bounds are conveniently characterized by the figure of merit ρ, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of ρ. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.
dc.description.abstract Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in ℝs and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962) -themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variaole, these bounds are conveniently characterized by the figure of merit ρ, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of ρ. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.
dc.language eng
dc.language eng
dc.relation.isbasedon 10.1090/S0025-5718-96-00758-2
dc.title An application of diophantine approximation to the construction of rank-1 lattice quadrature rules
dc.type Journal Article
dc.description.version Published
dc.parent Mathematics of Computation
dc.parent Mathematics of Computation
dc.journal.volume 216
dc.journal.volume 65
dc.journal.number 216 en_US
dc.publocation Boston en_US
dc.identifier.startpage 1635 en_US
dc.identifier.endpage 1662 en_US
dc.cauo.name SCI.Mathematical Sciences en_US
dc.conference Verified OK en_US
dc.for 0802 Computation Theory and Mathematics
dc.for 0103 Numerical and Computational Mathematics
dc.for 0102 Applied Mathematics
dc.personcode 830105
dc.percentage 34 en_US
dc.classification.name Applied Mathematics en_US
dc.classification.type FOR-08 en_US
dc.edition en_US
dc.custom en_US
dc.date.activity en_US
dc.location.activity en_US
dc.description.keywords Continued fractions
dc.description.keywords Continued fractions
dc.description.keywords Diophantine approximation
dc.description.keywords Diophantine approximation
dc.description.keywords Lattice rules
dc.description.keywords Lattice rules
dc.description.keywords Multiple integration
dc.description.keywords Multiple integration
dc.description.keywords Numerical cubature
dc.description.keywords Numerical cubature
dc.description.keywords Numerical quadrature
dc.description.keywords Numerical quadrature
pubs.embargo.period Not known
pubs.organisational-group /University of Technology Sydney
pubs.organisational-group /University of Technology Sydney/Faculty of Science
utslib.copyright.status Open Access
utslib.copyright.date 2015-04-15 12:23:47.074767+10
pubs.consider-herdc false
utslib.collection.history School of Mathematical Sciences (ID: 340)
utslib.collection.history General (ID: 2)


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