The determination of canonical forms for lattice quadrature rules

Triangular canonical forms for lattice rules of prime-power order

In this paper we develop a theory of t-cycle D−Z representations for s-dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a D-matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their Z-matrix is a column permuted version of ...

Quadrature Rules for Fuzzy Integrals

Learning Canonical Forms of Entailment Rules

We propose a modular approach to paraphrase and entailment-rule learning that addresses the morphosyntactic variability of lexical-syntactic templates. Using an entailment module that captures generic morpho-syntactic regularities, we transform every identified template into a canonical form. This way, statistics from different template variations are accumulated for a single template form. Add...

Explicit forms of weighted quadrature rules with geometric nodes

a f (x)w(x) dx = n − k=0 wkf (xk) + Rn+1[f ], where w(x) is a weight function, {xk}k=0 are integration nodes, {wk} n k=0 are the corresponding weight coefficients, and Rn+1[f ] denotes the error term. During the past decades, various kinds of formulae of the above type have been developed. In this paper, we introduce a type of interpolatory quadrature, whose nodes are geometrically distributed ...

Quadrature rules for rational functions

It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turr an, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received...