236 research outputs found
Constructing Cubature Formulas of Degree 5 with Few Points
This paper will devote to construct a family of fifth degree cubature
formulae for -cube with symmetric measure and -dimensional spherically
symmetrical region. The formula for -cube contains at most points
and for -dimensional spherically symmetrical region contains only
points. Moreover, the numbers can be reduced to and if
respectively, the later of which is minimal.Comment: 13 page
Numerical hyperinterpolation over nonstandard planar regions
We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes
Improved cubature formulae of high degrees of exactness for the square
AbstractThe method of constructing minimal cubature rules with high algebraic degrees of exactness is developed by adapting a powerful algorithm for solving the system of nonlinear equations. As a result, new cubature formulae of degrees 15, 17, 19, 21, and 23 are derived for the square. They lead to lower numbers of knots and/or to better quality with respect to those known previously. The formulae obtained should be considered as the most efficient for the calculation of two-dimensional integrals with a high precision
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