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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Some properties of rank-$2$ lattice rules
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by J. N. Lyness and I. H. Sloan PDF
Math. Comp. 53 (1989), 627-637 Request permission

Abstract:

A rank-2 lattice rule is a quadrature rule for the (unit) s-dimensional hypercube, of the form \[ Qf = (1/{n_1}{n_2})\sum \limits _{{j_1} = 1}^{{n_1}} {\sum \limits _{{j_2} = 1}^{{n_2}} {\bar f({j_1}{{\mathbf {z}}_1}/{n_1} + {j_2}{{\mathbf {z}}_2}/{n_2}),} } \] which cannot be re-expressed in an analogous form with a single sum. Here $\bar f$ is a periodic extension of f, and ${{\mathbf {z}}_1}$, ${{\mathbf {z}}_2}$ are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one- and two-dimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 627-637
  • MSC: Primary 65D32
  • DOI: https://doi.org/10.1090/S0025-5718-1989-0982369-6
  • MathSciNet review: 982369