Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation

A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomial

A Node Elimination Algorithm for Cubatures of High-Dimensional Polytopes

Node elimination is a numerical approach for obtaining cubature rules for the approximation of multivariate integrals over domains in Rn. Beginning with a known cubature, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. In this work, a new node elimination criterion is introduced that is based on linearization of the moment equations. In addition, a penalized iterative solver is introduced that ensures positivity of weights and interiority of nodes. We aim to construct a universal algorithm for convex polytopes that produces efficient cubature rules without any user intervention or parameter tuning, which is reflected in the implementation of our package gen-quad. Strategies for constructing the initial rules for various polytopes in several space dimensions are described. Highly efficient rules in four and higher dimensions are presented. The new rules are compared to those that are obtained by combining transformed tensor products of one dimensional quadrature rules, as well as with known analytically and numerically constructed cubature rules

Weighted quadrature formulas for semi-infinite range integrals

Weighted quadrature formulas on the half line $(a,+\infty)$, $a>0$, for non-exponentially decreasing integrands are developed. Such $n$-point quadrature rules are exact for all functions of the form $x\mapsto x^{-2}P(x^{-1})$, where $P$ is an arbitrary algebraic polynomial of degree at most $2n-1$. In particular, quadrature formulas with respect to the weight function $x\mapsto w(x)=x^\beta\log^m x$ ($0\le \beta<1$, $m\in \mathbb{N}_0$) are considered and several numerical examples are included