# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Lattice rules: projection regularity and unique representationsHTML articles powered by AMS MathViewer

by I. H. Sloan and J. N. Lyness
Math. Comp. 54 (1990), 649-660 Request permission

## Abstract:

We introduce a unique characterization for lattice rules which are projection regular. Any such rule, having invariants ${n_1},{n_2}, \ldots ,{n_s}$, may be expressed, uniquely, in the form $Qf = \frac {1}{{{n_1}{n_2} \cdots {n_s}}}\sum {\sum \cdots \sum {\bar f} \left ( {\frac {{{j_1}{{\mathbf {z}}_1}}}{{{n_1}}} + \frac {{{j_2}{{\mathbf {z}}_2}}}{{{n_2}}} + \cdots + \frac {{{j_s}{{\mathbf {z}}_s}}}{{{n_s}}}} \right )} ,$ where the matrix $Z = {({{\mathbf {z}}_1},{{\mathbf {z}}_2}, \ldots ,{{\mathbf {z}}_s})^T}$ is upper unit triangular and individual elements satisfy $0 \leq z_r^{(c)} < ({n_r}/{n_c}),r < c$.
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