In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε−1 and d. Strong tractability means that ...

We discuss the numerical integration of polynomials times exponential weighting functions arising from multiscale finite element computations. The new rules are more accurate than standard quadratures and are better suited to existing codes than formulas computed by symbolic integration. We test our approach in a multiscale finite element method for the 2D reaction-diffusion equation. Standard ...

Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the s-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions s > 2, one usually has to resort to computer search algorithms. The fast component-by-component approach is a useful algorithm for finding suitable quadrature rules. We present a modifica...

We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in Lp norms and the weighted coefficients introduced via `q norms, where p, q ∈ [1,∞]. The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule us...

Abs t r ac t -An account is given of the role played by moments and modified moments in the construction of quadrature rules, specifically weighted Newton-Cotes and Gaussian rules. Fast and slow Lagrange interpolation algorithms, combined with Gaussian quadrature, as well as linear algebra methods based on moment equations, axe described for generating Newton-Cotes formulae. The weaknesses and ...